Hard Arithmetic

Lang’s theorem — Pt. IV: a mixed-characteristic variant

alexyoucis — Sat, 22 Nov 2025 20:29:23 +0000

This is a continuation of this post, this post, and this post. In it we formulate the correct mixed characteristic version of Lang’s theorem and give a proof using ideas from -adic Hodge theory. In doing so, we give an alternative understanding of Lang’s theorem giving rise to a reformulation that works far beyond the finite field case.

The two candidates to replace and

Of course, one would like to imagine that the most naive version of a mixed-characteristic Lang’s theorem says that there are mixed-characteristic analogues

of ,
of with Frobenius ,

such that for sufficiently nice, the ‘Lang map’

is surjective.

The interesting thing is that there are two ‘natural’ choices for and . For one choice all analogues of Lang’s theorem holds, and for the other the naive analogue of Lang’s theorem always fails (although other reinterpretations hold).

The ring of unramified -adic integers

We begin with the more naive guess for how to ‘deform’ and to mixed characteristic.

Namely, we begin by making the following simple observation (where we write ):

Then taking as the sort canonical/primitive mixed-characteristic deformation of , we set

We call this the unramified extension of of degree and the maximal unramified extension of , respectively.

We observe that has a natural -Frobenius map which acts as the identity on and sends each to . This lifts the usual -Frobenius map on .

Let us state the obvious properties of these rings.

Proposition: The following statements hold true:

Both and are Henselian regular local rings of dimension ,

both and are -adically separated, but only the former is -adically complete,

is the unique unramified extension of degree and is the maximal unramified extension of .

All of these are fairly basic, and are left to the reader (to either prove them or consult any book on algebraic number theory). But, purely for fun, let us try to explicitly exhibit a non-convergent Cauchy sequence in .

As is well-known, as is a non-archimedean ring if it were complete then an infinite series converges if and only if . In particular, as is a unit, it’s easy to see that . That said, the sequence does not actually converge to an element of . Namely, assume it did. Then, it would need to lie in some . But, then we see that if we set

then the image of in would lie in . But, , and thus we arrive at a contradiction.

Remark: Of course, there is the following well-known vast generalization of the above non-completeness: an algebraic extension is complete if and only if it is actually a finite extension. This follows from a Baire category theorem argument (e.g., see this math.stackexchange.com answer).

The Witt vectors

We now recall the much more robust way of trying to deform and to mixed-characteristic objects: the Witt vectors. I won’t go too much into the ring of Witt vectors here, but one can see this post for a discussion of them. Instead, I just summarize the main properties of the Witt vector construction below.

Remark: The above post is just scratching the surface of the incredibly deep arithmetic properties that the Witt vectors hold. If you really want to understand these properties, I cannot recommend strongly enough the lecture notes [Borger] and the videos [BorgerVideos].

Before we do so, let us recall that a ring is called a strict -ring if it

is -adically complete,
flat over (i.e., -torsion free), and
the -algebra is perfect (i.e., the Frobenius map on is an automorphism).

With this, we have the following conceptual description of the Witt vectors.

Proposition/Definition (prop:Witt-vector-properties): The following statements are true:

for a perfect -algebra there exists a unique (up to isomorphism) strict -ring with called the (-typical) Witt vectors of ,

there exists a map of multiplicative monoids called the Teichmuller lift and any element of can be written uniquely as for elements in ,

for any -adically complete ring and map of -algebras the map

is a -adically continuous ring map,

the Witt vector construction forms an equivalence of categories

Proof: See this post and the references therein.

Remark: Let me try to give some highfalutin meaning to . Associated to any map of rings is a complex of -modules, called the cotangent complex, which is something like a ‘higher-order’ or ‘simplicial’ refinement of the usual Kahler differentials . Its major important is that it ‘completely controls deformation theory’ (e.g., see Tag 08U6). Just like the Kahler differentials, the Frobenius map on induces the multiplication by -map on which, as it is a complex of -modules, is zero. But, as is an isomorphism, we deduce that the zero map is an isomorphism of . Thus, induces both the zero map and an isomorphism on , thus this complex must be (quasi-isomorphic to) .

What this means is that admits unique deformation along (pro-)thickenings of . Thus, is the free/universal (pro-)deformation of along the pro-thickening .

As is hopefully clear by the above proposition and remark, the Witt vectors give rise to a much more systematic, canonical way of deforming the objects and to mixed-characteristic.For notational/historical reasons, we shall denote by (read ‘Zee-pee breve‘).

Applying the universal property of the Witt vectors to -Frobenius morphism produces a natural -Frobenius morphism

This lifts the -Frobenius on .

Of course, it is still useful and interesting to relate the Witt vectors to the more naive deformation given by the unramified extensions. The exact relationship is then given by the following.

Proposition (prop:Z_pur-completion-witt): There are natural compatible isomorphisms

,

and the second is -equivariant.

Proof: Given part 3. of Proposition/Definition (prop:Witt-vector-properties), to define these maps it suffices to observe that and . Thus, by part 4. of Proposition/Definition (prop:Witt-vector-properties) we will be done if we can show that and are strict rings. But, this is clear.

Lang’s theorem for the maximal unramified extension

We now examine to what extent some of the equivalent versions of Lang’s theorem over finite fields can have analogies using the rings and .

Failure of naive Lang’s theorem

As alluded to above, we observe that the naive version of Lang’s theorem does not hold if one chooses to use the maximal unramified extension, and that this lack of holding is quite dire. This is made precise by the following.

Proposition (prop:no-Lang-for-max-unram): For every quasi-compact smooth reductive group -scheme , such that contains an element of infinite order (which is a vacuous condition, see the remark below), then the Lang map

is not surjective.

Proof: Let us observe that as is finite presentation over that

(e.g., see Tag 00QO). But, if and then

where this last equality holds as .

In particular, suppose that . Then, if for some , the above shows that there exists some such that . But, as , it is clear that . Thus, has finite order. Thus, we see that consists only of finite-order elements, from where the claim follows.

Remark: It is interesting to compare this with the proof of Proposition (prop:cohomological-reinterpretation) in Pt. III where we use the opposite observation, that every element of is finite order, to prove the result!.

Remark: The condition on the existence of a non-torsion element in Proposition (prop:no-Lang-for-max-unram) is essentially vacuous. Namely, suppose that is a non-trivial smooth connected group scheme with connected fibers, then contains a non-torsion element. Indeed, one can argue this directly using ‘standard algebraic group theory’ with some work, but a quick proof follows from the theory of -adic Lie groups.

Namely, it’s clear that is an analytic group over of positive dimension, in the sense of [Serre, Chapter IV]. But then, as is of characteristic , we see by [Serre, Chapter IV, §9, Theorem 5] that has an open torsion-free subgroup . As is of positive dimension, we have that cannot be trivial (as it is open) and thus contains (infinitely many) non-torsion elements.

A positive cohomological result

While the naive version of Lang’s theorem can’t hold using and , it turns out that the analogue of Proposition (prop:cohomological-reinterpretation) actually from Pt. III does hold in this characteristic setting.

Proposition (prop:cohomology-of-Z_q): Suppose that is a smooth group -scheme with connected fibers, Then, the cohomology set is trivial.

Proof: Let be a -torsor over . We must show that . By Proposition (prop:all-torsors-PHS) of Pt. II in the case that it applies (resp. Remark (rem:alg-spaces) from that post in general) we have that is represented by a smooth -scheme (resp. is a smooth algebraic space over . In particular, it is easy to see that is a principal homogenous space for . As is a connected group variety over , we know by Lang’s theorem that is non-empty. But, by Hensel’s lemma we have that is surjective, and thus is non-empty, and thus is trivial, as desired.

Remark: So, what the above tells us implicitly (and explicitly examining the proofs) is that the cohomological version of Lang’s theorem holds over , but that its connection to the naive version of Lang’s theorem was special to the finite field case.

A crystalline reinterpretation of Lang’s theorem

As a warm-up to our correct mixed-characteristic version of Lang’s we give yet another interpretation of the classical Lang’s theorem in terms of so-called mod etale -crystals.

This is particularly significant for us as we shall see that, in contrast to the classical case, we shall actually use a crystalline reasoning to deduce the naive analogue of a mixed-characteristic version of Lang’s theorem using and .

We begin with the following definition.

Definition (mod-p-etale-crystals):A mod etale -crystal over is a pair where

is a finite-dimensional -vector space, and

is a -linear isomorphism (called the Frobenius isomorphism),

where is the -power Frobenius.

The category of such is denoted (leaving the choice of implicit). To be explicit, the Hom sets are given by

We abuse notation and write the set of such as , suppressing the Frobenii .

Let us observe that has the natural structure of an exact:-linear -category:

the subset
,

is actually an -linear subspace,
one defines
,

where the latter makes sense given the natural identification

,
we say a sequence of mod etale -crystals over is exact if the underlying sequence of -vector spaces is exact.

In particular, it makes sense to consider the category of -objects in for a finite type affine group -variety , as discussed in Pt. II.

Recall, by definition, that such a -object means an exact -linear -functor

and that a morphism between them is a natural transformation of such functors. In particular, it’s clear that we may identify with pairs where is an object of and is an isomorphism in where, here, is computed pointwise.

We denote the set of isomorphism classes of such -objects in by .

Construction: For an element of consider the functor

associating to a representation the -space with Frobenius

with the first isomorphism the natural one. Then, is an object of .

The above construction, in particular, gives us a map

However, this map is not injective. To make this precise, let us define the notion of -conjugation on where, recall, we are denoting by

the -Frobenius map. Namely, we say that two elements are -conjugate, denoted , if there exists some such that .

Proposition (prop:mod-p-crystals-and-B(G)): For a finite type affine group -scheme , the map induces a bijection

.

Proof: Let us first show that this map is injective. This entails describing the set of isomorphisms . But, observe that forgetting the Frobenius structure, and , are precisely just the trivial -object on -vector spaces. It follows from Theorem (thm:Broshi) from Pt. II that the set of such isomorphisms (again ignoring Frobenius structure) is given by , where explicitly, the isomorphism is such that for any representation of the induced map

is given by .

Now, for this isomorphism to be Frobenius-equivariant we require that for all such one has that

Now, one can check that and so we see that this is equivalent to

As this must hold for all representations , it follows (e.g., by Theorem (thm:Broshi) from Pt. II) that , and so , from where the claim follows.

To show surjectivity, we must show that every in is isomorphic to one of the form for some . But, observe as in the first part of the argument, forgetting the Frobenius shows that may be thought of as an element of . But, by Theorem (thm:Broshi) from Pt. II such a thing is classified by , which is trivial by Example (eg:Galois-cohomology) from Pt. II. Thus, as an object of , we may identify with the trivial object

Using the natural identification , we see from Theorem (thm:Broshi) from Pt. II that

must be given by some element of . It is then not hard to see that , as desired.

An immediate corollary of this proposition is the following reinterpretation of Lang’s theorem.

Corollary (cor:mod-p-crystal-Lang’s-theorem-equiv): For a finite type affine group -scheme , Lang’s theorem for is equivalent to the claim that .

Proof: The fact that is a singleton is equivalent to the claim that for some (equiv. any) in we have that for all other in . But, taking we see that this is equivalent to the claim that for every in there exists some in such that or, in other words, that . The claim follows.

Lang’s theorem for the Witt vectors

In this final section, we state and prove the naive version of Lang’s theorem using and and give the natural generalization to other bases.

The naive version of Lang’s theorem and etale -crystals

Let us begin by stating the precise naive analogue of Lang’s theorem that we wish to prove.

Theorem (thm:naive-Lang-for-Witt): Let be a smooth affine group -scheme with connected fibers. Then, the Lang map

,

is surjective.

Remark: One sees that in the above we have restricted our attention to affine group schemes, wheras the original Lang’s theorem made no such restriction. This is not because the result is false in that generality (though I have frankly not checked it), but more because I am secretly (or not so secretly) interested more in the -crystal perspective we will develop below.

Similarly, there are other ways to try and prove Theorem (thm:naive-Lang-for-Witt). Notably, one can try to iteratively building a solution to in in for all . It seems likely one can do this actually by applying the classical Lang’s theorem to the Greenberg schemes (e.g., see [Greenberg] or [CLNS, Chapter 4]).

But, I feel as though this -crystal is the more interesting/generalizable statement.

Let us observe that any naive approach to analogize the proof of Lang’s theorem from Pt. I will meet obvious resistance. First and foremost, unlike the case of finite fields, the Lang map is not algebraic. So any naive method trying to apply algebraic geometry to this situation will fail (though cf. the notion of Greenberg schemes mentioned in the above remark). Thus, we will need a different way to approach this problem.

The approach we choose to take is that generalizing the notion of mod etale -crystals from the previous subsection, which will yield an equivalent version of the naive version of Lang’s theorem as in Corollary (cor:mod-p-crystal-Lang’s-theorem-equiv). We will then prove this equivalent crystal statement by using -adic Hodge theory to transform the crystal equestion into a much easier one.

To this effect, we begin with the following definition.

Definition (etale-crystals): An etale -crystal over is a pair where

is a finite-free -module, and

is a -linear isomorphism (called the Frobenius isomorphism),

where is the -power Frobenius.

The category of such is denoted (leaving the choice of implicit). Again to be explicit, the Hom sets are given by

We abuse notation and write the set of such as , suppressing the Frobenii .

Similar to, , one sees that has the natural structure of an exact -linear -category:

the subset
,

is actually an -linear subspace,
one defines
,

where the latter makes sense given the natural identification

,
we say a sequence of etale -crystals over is exact if the underlying sequence of -modules is exact.

In particular, it again makes sense to consider the category of -objects in for a flat and finite type affine group -variety , as discussed in Pt. II.

Similar to the mod p case, we denote the set of isomorphism classes of such -objects in by .

We then also have the following similar construction.

Construction: For an element of consider the functor

associating to a representation the -module with Frobenius

with the first isomorphism the natural one. Then, is an object of .

The above construction, in particular, gives us a map

However, this map is not injective, and the non-injectivity is related to the notion of -conjugation on , defined in the exact analogous way to the finite-field situation.

One can then show the following precisely as one showed Proposition (prop:mod-p-crystals-and-B(G)), with the only change being that instead of needing to be trivial, you now need that is trivial. But, you can show this easily using the same proof as used above for Proposition (prop:cohomology-of-Z_q).

Proposition: For a flat and finite type affine group -scheme , the map induces a bijection

.

We then deduce the following corollary exactly the same as we did for Corollary (mod-p-crystal-Lang’s-theorem-equiv).

Corollary (cor:mixed-char-Lang-equiv-to-trivial-crystals): For a flat and finite type affine group -scheme , the naive version of Lang’s theorem holding for is equivalent to the claim that .

Katz’s version of the Riemann–Hilbert correspondence

Our original proof of Lang’s theorem from Pt. I ultimate took advantage of an interesting phenomenon: the lines between topology and arithmetic/geometry over were blurred. Unsurprisingly, our approach to this mixed-characteristic version of Lang’s theorem will take advantage of a similar blurring. But the classical case used the fact that the topological Frobenius map (a generator of a Galois group, i.e., an etale fundamental group) could be manifested as a map of schemes, and as we mentioned above, this is just not possible here. We need a different blurring.

The blurring we choose to use is a characteristic version of the Riemann–Hilbert correspondence (see also [§2.7, Szamuely] for a discussion of Riemann–Hilbert in a way focused on the aspects close to how we want to use it here) due to Katz .

To state this in the correct generality, we shall use the -truncated Witt functor , and do the standard abuse of notation and write to mean , and to mean .

Finally, we will use the notion of mod etale -crystals over , the category of which is denoted . These are defined as the obvious intermediaries between mod etale -crystals over (i.e., the and case) and etale -crystals (i.e., the and case).

Theorem (cf. [Katz, Proposition 4.1.1)]: Let be a perfect algebra and . Then, the functor

where means -fixed points and is a (variable) object of , is an exact -linear -equivalence.

How to exactly understand Katz’s theorem as an incarnation of a ‘-adic Riemann–Hilbert correspondence is a serious matter, and thematically occupies a non-trivial amount of modern number theory in the form of the subject known as -adic Hodge theory. To get a better understanding of this I think [LurieVideos] is a great and user-friendly start.

That said, we can sketch the idea in very broad terms. Namely, there has long been a feeling that the Frobenius (or rather ) serves as the -adic analogue of a connection (i.e., differential operator) in the archimedean setting. For instance, this has been a unifying work of Buium, and one can consult [Buium] for a discussion of this. But, the idea is that a connection tells you how to ‘parallel transport’ elements of the fiber of a vector bundle in an ‘infinitesimal neighborhood’. For us our vector bundle is something like our -vector space, and the infinitesimal neighborhoods are given by the .

In any case, if you buy that, then you immediately see the analogy between the Riemann–Hilbert correspondence and Katz’s theorem:

in the archimedean setting one takes a vector bundle with connection (thought of as being defined on an infinitesimal neighborhood) and gets a local system via the horizontal sections ,
in the non-archimedean (or mod ) setting one takes a module with Frobenius and gets a local system via the ‘horizontal sections’ (or rather, its associated ‘sheaf on this infinitesimal neighborhood’).

Etale -crystals over simply connected bases

Using Katz’s version of the Riemann–Hilbert correspondence, we are now ready to state our mixed-characteristic generalization of Lang’s theorem to bases much more general than originally considered.

To this end, we make the following definition.

Definition: A perfect -algebra is simply connected if it is connected and its etale fundamental group is trivial.

NB: Here we are defining the etale fundamental group of to be the etale fundamental group as in Tag 0BNC for any geometric point of . The choice of here is irrelevant by Tag 0BND (cf. also this math.stackexchange answer).

Example: Suppose that is an integral domain and let be the integral closure of in an algebraic closure of . Then, is evidently perfect: it’s reduced so clearly the Frobenius map is injective, and if then has a solution in as it is algebraically closed, but as this polynomial is monic the solution actually lies in .

But, in fact, is also simply connected. Indeed, it’s normal (by definition) and so one can apply Tag 0BQM.

Remark: For simplicity we have chosen to work with affine schemes. This has one very marked disadvantage–in characteristic affine schemes are very rarely simply connected. For example, you can take a finitely generated -algebra and take its perfection . This will be a perfect -algebra, and has Frobenius is a universal homeomorphism will satisfy (e.g., see Tag 0BTT). But, is essentially never trivial–for example, see this answer of Qing Liu on mathoverflow).

This can be easily remedied by working with non-affine schemes, in which case one can obtain similar results to what we discuss below, but for which there are many more examples. For what this would look like, one can actually consult Katz’s original article, or [Emerton–Kisin].

We can now state our vast generalization of Lang’s theorem. To do so, for a perfect -algebra and , let us denote by the isomorphism classes in . So, our previous notation was .

Theorem (thm:crystals-over-sc-bases): Let be a simply connected and perfect -algebra and let be a smooth affine group -scheme. Then, for all

.

Proof: Let us observe that by Katz’s version of the Riemann–Hilbert correspondence we have an equivalence of categories

That said, as is simply connected, we know that there is a natural exact -linear -equivalence

where the first equivalence follows from the discussion in Tag 03VD, and the second because is trivial. In particular, we deduce an equivalence of categories

But, applying Broshi’s theorem from Pt. II the isomorphism classes in this category are classified by . But, arguing as in Proposition (prop:cohomology-of-Z_q) shows that this is actually a singleton by Lang’s theorem.

In particular, combining Corollary (cor:mixed-char-Lang-equiv-to-trivial-crystals) with Theorem (thm:crystals-over-sc-bases) gives us a proof of Theorem (thm:naive-Lang-for-Witt) as desired!

References

[Borger] https://maths-people.anu.edu.au/~borger/classes/copenhagen-2016/LectureNotes.pdf

[BorgerVideos]

Video 1: https://youtu.be/_y2Tcu-iJV4?si=4-L68es72FwZ-LEm
Video 2: https://youtu.be/BP5I-dB9TYk?si=2q4l4X-t-k2_sbSW
Video 3: https://youtu.be/liiBRBN-5aM?si=6JhdtZUh5Os67LSb

[BLR] Bosch, S., Lütkebohmert, W. and Raynaud, M., 2012. Néron models (Vol. 21). Springer Science & Business Media.

[Buium] Buium, A., 2015. Differential calculus with integers. Arithmetic and Geometry, London Mathematical Society Lecture Note Series, 420, pp.139-187.

[CLNS] Chambert-Loir, A., Nicaise, J. and Sebag, J., 2018. Motivic integration. In Motivic Integration (pp. 305-361). New York, NY: Springer

[Emerton–Kisin] Emerton, M. and Kisin, M., 2004. The Riemann-Hilbert correspondence for unit F-crystals. Société mathématique de France.

[Greenberg] Greenberg, M.J., 1963. Schemata over local rings: II. Annals of Mathematics, 78(2), pp.256-266.

[Hindry–Silverman] Hindry, M. and Silverman, J.H., 2013. Diophantine geometry: an introduction (Vol. 201). Springer Science & Business Media.

[Katz] Katz, N.M., 1973. P-adic properties of modular schemes and modular forms. In Modular Functions of One Variable III: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972 (pp. 69-190). Berlin, Heidelberg: Springer Berlin Heidelberg.

[Kedlaya–Liu] Kedlaya, K.S. and Liu, R., 2015. Relative -adic Hodge theory: foundations. Asterisque, 2015(371), pp.9-+.

[LurieVideos]

Video 1: https://youtu.be/xPkzcU85SLg?si=R5XxRo12G4PrmAuK
Video 2: https://youtu.be/dqJGp-OOsxc?si=Zzjcbg1WlRxX5ESg
Video 3: https://youtu.be/EV7nGvPveww?si=x_jcmPV2qD3oqmZA
Video 4: https://youtu.be/kzX-KYDS0fQ?si=UHvTnWhIfjAGiEIA
Video 5: https://youtu.be/ZWaAAvUR7rk?si=5cflFHpPU-6f6IaR

[Pstragowski] https://www.kurims.kyoto-u.ac.jp/~piotr/291y_notes.pdf

[Serre] Serre, J.P., 2009. Lie algebras and Lie groups: 1964 lectures given at Harvard University. Springer.

[Silverman] Silverman, J.H., 2009. The arithmetic of elliptic curves (Vol. 106, pp. xx+-513). New York: Springer.

[Szamuely] Szamuely, T., 2009. Galois groups and fundamental groups (Vol. 117). Cambridge university press.

[VD] Veĭsfeĭler, B.J. and Dolgačev, I.V., 1974. Unipotent group schemes over integral rings. Mathematics of the USSR-Izvestiya, 8(4), p.761.

Lang’s theorem — Pt. III a torsor reinterpretation and manifold applications

alexyoucis — Sat, 15 Nov 2025 22:41:55 +0000

This is a continuation of this post and this post. Namely, use our discussion of torsors from the previous post to give a reinterpretation of Lang’s theorem. This will allow us to more concretely explain a multitude (but still just a small slice) of interesting implications of Lang’s theorem, and thereby understand how significant this result is for the study of finite fields.

A cohomological interpretation of Lang’s theorem

With all of this build-up, we can now finally explain the cohomological interpretation of Lang’s theorem.

Proposition (prop:cohomological-reinterpretation): Let be a smooth quasi-compact and connected group over . Then, Lang’s theorem is equivalent to the claim that is trivial, i.e., that every -torsor on is trivial.

Proof: Let us observe from Example (eg:Galois-cohomology) from Pt. II that the set of torsors is nothing but the Galois cohomology set . We first claim that is naturally identified with .

Indeed, an element of is a continuous -cocycle . As is dense in we see that is determined by as ranges over . But, by the -cocycle condition, is determined by . So, the map

is injective. To see it’s surjective, let in . This must lie in for some , where is the unique degree extension. We may try to define a cocycle

via the following inductive scheme:

But, this is well-defined if and only if

is trivial.

This triviality may not be true, but observe that for any we have that

as . But, as is an element of the finite group it has a finite order, say . Then, we see that is trivial. Thus, we can define the cocycle as in not on , but on but, in particular, this implies that is in the image of the map

as desired.

Moreover, it’s easy to see that if and only if and are -conjugate: there exists in such that . Thus, all in all, we see that we have an identification

In particular, we see that is trivial if and only if every is -conjugate to the identity or, said differently, if for every there exists some such that . The claim follows.

Division algebras and Brauer–Severi varieties: the case of the projective linear group

From Proposition (prop:cohomological-reinterpretation) we know that if is a finite field, then is trivial. This has many interesting reinterprations, basically utilizing Proposition (twists=torsors), which we give two of here.

Division algebras

Let be a field. Recall that a division algebra over is a (possibly non-commutative, but associative and unital) ring in which

every non-zero element has an inverse,
,
and .

More generally, one says that a (possibly non-commutative, but associative and unital) ring is a central simple algebra over if

,
has no non-trivial proper two-sided ideals ,
and .

Such things arise naturally and frequently in arithmetic (e.g., see [Gille–Szamuely]).

Example (eg:division-algebras-over-R): It is famous theorem of Frobenius that the only central division algebras over are and the Hamiltonian quaternions .

One might ask what the possible division algebras are over a finite field . This is answered by the following famous theorem of Wedderburn.

Theorem (Wedderburn’s little theorem): If is a finite field and is a central simple algebra over , then for some . In particular, if is a division algebra over , then .

We now explain that Wedderburn’s little theorem is a consequence of Lang’s theorem.

The only real facts about central simple algebras we need are the following.

Lemma 1 (Skolem–Noether, see [Gille–Szamuely, ):For any field , there is a natural identification

.

Lemma 2: For a central simple algebra over a field , the following two statements hold :

if is separably closed then must be isomorphic to for some (see [Gille–Szamuely, Theorem 2.2.1]),

if is an extension, then is a central simple algebra over (see [Gille–Szamuely, Lemma 2.2.2]).

Now, it is not hard to check that for any field , the association

with transition functors for given by is a stack on , well defined by the second statement in Lemma 2.

So, one may then reinterpret the Skolem–Noether Lemma as the statement that the automorphism sheaf for the object of is . Moreover, every central simple algebra is a twist (in the stack ) of by the first statement of Lemma 2.

Thus, by Proposition (twists=torsors) of Pt. II we see that

But, Proposition (prop:cohomological-reinterpretation) implies that this last set is a singleton. Wedderburn’s little theorem follows.

Remark: It turns out that equivalence classes of central simple algebras over a field form a group, called the Brauer group of and denoted (e.g., see [Gille–Szamuely]). Thus, we can reinterpret the above discussion as saying that if is finite then is trivial.

Brauer–Severi varieties

We present another interesting manifestation of the triviality of . Namely, let us recall that a variety over a field is called an -dimensional Brauer–Severi variety if

i.e., is ‘geometrically’ projective space, but need not be so over . We say a Brauer–Severi variety is trivial if it is isomorphic to some projective space.

Again, such things naturally occur in nature. For example, the following proposition tells you exactly which curves are Brauer–Severi varieties, which gives many examples.

Proposition (prop:BS-curves): Let be a field and a smooth projective geometrically connected curve over . Then, the following are equivalent

is a Brauer–Severi variety,

(i.e., has genus ),

is a conic section, i.e., isomorphic to , for some elements .

Proof: That 1. implies 2. is clear as

To see that 2. implies 3. first observe that the cotangent sheaf has degree

by the Riemann–Roch theorem. Thus, is a line bundle on of degree . But, as

we see by [Gortz–Wedhorn II, Proposition 26.59] that is very ample and so the choice of a basis of defines a closed embedding .

But, again by Riemann–Roch, we have that , and thus . But, a hypersurface in must be of the form for some homogenous polynomial, say of degree . But, by the genus-degree formula we know that

and so or . In the latter case we’re done, and if then evidently

and so we’re also done in this case.

To see that 3. implies 1., we may assume that is separably closed, and show that such a conic section is isomorphic to . Let be a closed point on and let . Then, , and the exact same argument as in the last paragraph gives us a closed embedding , which must evidently be an isomorphism.

The claim then is that every Brauer–Severi variety over a finite field is trivial.

Proposition (prop:BS-over-finite-field): Let be a finite field. Then, every Brauer–Severi variety over is trivial.

The proof of this proposition is analgous to the above proof of Wedderburn’s little theorem, where the Skolem–Noether lemma is replaced by the following.

Lemma (lem:automorphisms-of-P^n): Let be a field. Then, there is a natural isomorphism

.

Proof: Observe that naturally acts on in the obvious way:

This action factorizes through and gives rise to a homomorphism

which is evidently injective. To see it’s surjective, let be an automorphism. Then, from the well-known description of the functor of points for (e.g., see [Gortz–Wedhorn I, §(8.6)]) that is uniquely determined by the line bundle and the -sections for . Now, as is an automorphism it’s easy to see that is a generator of and

0" class="latex" />.

But, the only such line bundle on with these two properties is . Thus, we see that . Then, for we can write

as these are what all global sections of look like. Moreover, as is a basis of the global sections we see that is invertible. This matrix is well-defined up to scalar multiples, and so defines a unique element of , and one has , as desired.

Equivalence classes of quadratic forms: the case of special orthogonal groups

We now would like to give a reinterpretation of Proposition (prop:cohomological-reinterpretation) in the case of the special orthogonal group .

Namely, let us recall that for a -algebra a (non-degenerate -ary) quadratic form over is homogenous polynomial in of degree . In other words, it’s an element of of the form

where the matrix is invertible and satisfies if ). Here we are using to denote the ‘vector’ of variables . Recall that an equivalence between two quadratic forms and is an element of such that .

The special orthogonal group is the algebraic subgroup of defined as the group of self-equivalences of , i.e.,

for any -algebra , where we are abusing notation and using also for the induced element of . This is a smooth (in fact reductive) algebraic subgroup of with connected fibers (over ).

Now, it is very far from true that any two -ary quadratic forms over a field are equivalent.

Example (eg:qudratic-forms-over-R): Over the quadratic forms given by

represent a set of non-redundant representatives for the equivalence classes of -ary quadratic forms with no repeats.

That said, it’s a simple exercise to show that all quadratic forms over are equivalent if is separably closed: all quadratic forms are equivalent to the trivial one . Over an arbitrary field we abuse notation and say a quadratic form is trivial if it is equivalent to . We also write .

Moreover, one can easily check that

where we think of as a groupoid with morphisms given by equivalences is a stack on when equipped with the natural transition maps.

In particular, since every object of becomes trivial in , we see that consists of just the twists of in . In particular, by Proposition (twists=torsors) we have a natural bijection

But, as is connected, we then immediately deduce the following from Proposition (prop:cohomological-reinterpretation).

Proposition (prop:qf-trivial-finite-field): Suppose that is a finite field. Then, every -ary non-degenerate quadratic form is trivial.

Homogenous varieties

Let us fix a field and a connected group variety over . We say that a finite type scheme over is a homogenous space for if there exists an action which is transitive, i.e., the action is transitive.

Note that it is not always true that is non-empty.

Example (eg:quadratic-form-homogenous-space): Consider the quadratic form

over Then, is a homogenous variety for , but .

One remarkable (but simple) consequence of Lang’s theorem, is that homogenous spaces always admit a rational point over a finite field.

Proposition (prop:homogenous-space-Lang): Suppose that is a finite field, is a connected group variety over , and is a homogenous space for . Then, .

Proof: As before, let denote the Frobenius map . Then, the natural map is a bijection, and so it suffices to show that is non-empty. Fix a point in . Then, may differ from , but by the transitivity of the action of there exists some in such that . But, by Lang’s theorem, there exist some in such that . So then, it’s easy to see (using the fact that the action of on is defined over , and so Frobenius acts compatibly) that , and thus is an element of .

That said, we can actually further apply Lang’s theorem to better understand the rational points of a homogenous space over a finite field. Namely, let us temporarily go back to the general situation of a field , a connected group variety , and a homogenous space for .

Observe that if is a homogenous variety for , and if is in , we obtain a stabilizer algebraic subgroup , and we obtain a natural isomorphism

(e.g., see [Milne, Proposition 7.11]). This connects are notion of homogenous varieties to the more classical notion of ‘homogenous varieties’: homogenous varieties admitting rational points are the same as quotients of a(n algebraic) group by a subgroup.

Example: We can now clarify Example (eg:quadratic-form-homogenous-space). Namely, one can show that , where is a certain parabolic subgroup of . But, is not defined over , which is is what stops from being a quotient of over (or, what amounts to the same thing, having an -rational point).

So now, if is a point of , then we have an isomorphism . This isomorphism gives rise to an injection . But, in general, this map is not surjective, i.e., need not act transitively on even though acts transitively on .

To help clarify this, fix a point in and for a finite separable extension of , set

It is easy to see that this defines a sheaf on which comes equipped with an action of . Moreover, it’s easy to see that this is actually a torsor.

In particular, if is a connected group variety, then Lang’s theorem tells us that is trivial, at least when is finite, and as was arbitrary that

is a bijection.

We summarize the above discussion as follows.

Proposition (prop:Lang-homogenous-full-rational-points): Suppose that is a finite field, is a connected group variety over , and is a homogenous space for . Let be any point of . Then, if is a connected group variety, the natural map

,

is a bijection. In particular, acts transitively on .

Conjugacy classes of semisimple elements

While Propositions (prop:homogenous-space-Lang) and (prop:Lang-homogenous-full-rational-points) have many interesting applications, perhaps one of the most interesting comes from thinking about different notions of conjugacy inside of .

Rational vs. geometric conjugacy

Let us again temporarily go back to the general situation of a (perfect) field and a connected group variety over . Fix two elements and of . Let us say that and are

rationally conjugate if they are conjugate in ,
geometrically conjugate if they are conjugate in .

For , it is well-known that these two notions coincide, but this is false in general.

Example (eg:rationally-and-geometrically-conjugate-are-different): Consider the two elements and in . These are easily seen to be conjugate if and only if is a sum of two squares in . So, taking shows they are geometrically conjugate but not rationally conjugate.

To help us understand this phenomenon better, let us fix an element of and let denote its centralizer. We can then define its scheme-theoretic conjugacy class as follows

Observe then that we have a natural map identifying with the orbit subscheme of defined by via the conjugation action (see [Milne, §17.(c)-(e)]). In particular, we may think of as a subset of , but which subset?

Using the basic equality

and using the fact that , it’s easy to see that is precisely those elements of which are geometrically conjugate to , i.e., conjugate to in .

In other words, the scheme-theoretic conjugacy class cannot detect rational conjugacy (i.e., conjugacy in , only geometric conjugacy. This makes sense: belonging to some subvariety is about vanishing/non-vanishing loci of polynomials, which cannot tell the difference between and , and so the algebro-geometric version of conjugacy class can only be this ‘geometric’ version.

With an eye toward applying Proposition (prop:Lang-homogenous-full-rational-points), let us observe that need not be connected (e.g., take in ). We can then find an intermediary notion between rational and geometric conjugacy. Namely we have natural maps (none of which are necessarily surjective)

Elements in lying in the image of under this full composition are those rationally conjugate to , those lying in the image of are those geometrically conjugate to , and we define those lying in the image of to be those which are stably conjugate to .

Remark (rem:stably-conjugate): The reason for the name ‘stable’ is due to the most common application of this idea, to the study of so-called stable distributions in the harmonic analysis of reductive groups. For more information see [Labesse].

Applying Proposition (prop:Lang-homogenous-full-rational-points) gives us the following result over finite fields.

Proposition (prop:stable-conj): Suppose that is a finite and a connected group variety. Suppose that is smooth over . Then, if and in are stably conjugate they are rationally conjugate. In particular, if is additionally connected, then if and are geometrically conjugate they are rationally conjugate.

Example (eg:rationally-and-geometrically-conjugate-are-different-continued): Note that in Example (eg:rationally-and-geometrically-conjugate-are-different) the element has centralizer given by the diagonal torus, as long as (but in that case ). In particular, Proposition (prop:stable-conj) actually implies that over a finite field of odd characteristic, we have that and are conjugate.

By the contents of Example (eg:rationally-and-geometrically-conjugate-are-different) this implies that Lang’s theorem ‘knows’ that is a sum of two squares in . This result is not hard to show by hand using a counting argument. Namely, if had no solutions in (let alone ) then the sets and are disjoint. But, observe that as is odd the squaring map has kernel of size . So, the image has size . Thus, there are squares in , and thus if and were disjoint, then there would be at least elements in , which is absurd. So, while this sum-of-squares fact is not so surprising, it is interesting that Lang’s theorem ‘knows it’.

Conjugacy classes of semisimple elements in certain reductive groups over finite fields

We would like to apply the ideas from the last subsection to give a concrete and computable way to understand conjugacy classes of certain elements in certain algebraic groups over finite fields.

The Steinberg map

To begin, let us assume that is any perfect field and is a (connected) reductive group over . I won’t go into precisely what reductive means here (see this post for a quick introduction), but morally (but not precisely in characteristic !) it means that is affine, smooth, and the representation category of is semi-simple. Examples of such groups are

, ,

(see loc. cit. for the definitions of these groups), and the reader should feel free to imagine is one of these groups.

Now for any algebraic group over , it is tempting to want to study conjugacy classes in algebro-geometrically by taking the ‘quotient ‘ where is acting on itself by conjugation. This quotient is very ill-behaved if interpreted in any sort of naive sense, essentially because the stabilizers of this action (i.e., the centralizer subgroups of ) are so large. In particular, for this to have any sort of well-behaved geometry one needs to take a quotient that remembers these stabilizers, i.e., one wants to take a stack quotient , see Tag 04UI for the precise definition.

The geometry of this stack quotient can be quite complicated stack-theoretically, and so it is perhaps desirable to cut down in the complexity by asking:

Question: Is there a scheme which best approximates ?

The notion of ‘best approximates’ is of course quite wheedly, and frankly quite subtle, but at least one thing you should require is that there is a map which is initial amongst maps to schemes (or even to reasonable algebraic spaces)–this would uniquely classify it if it exists. What else you might hope the map to satisfy is then a matter of application-minded taste.

The reason that being reductive is valuable is that there there is such a best approximation. Namely, let us set

where is the map on global functions induced by the conjugation action

In words, is the ring of conjugation invariant functions on . We then define the GIT quotient of by to be

which is, a priori, just a -scheme equipped with a natural map

which we call the Steinberg map.

When is reductive, this GIT quotient and the steinberg map, are very well-behaved.

Theorem (see [Popov] and [Mumford, Chapter 1, §2]): Suppose that is reductive. Then,

is an irreducible, normal, rational, Cohen–Macaualy variety,

where is a maximal torus of and ,

the map is smooth if and only if is a product of simply connected semi-simple groups, and in this case is isomorphic to where is the rank of and is a product of copies of ,

the map is surjective realizes as a universal categorical and GIT quotient of acting by , and factorizes through a map which is initial amongst maps to schemes.

While this is all very nice, it’s clear that this really helps us understand conjugacy classes in any more concretely. But, this is very much not the case, as the following famous theorem of Chevalley tells us.

Theorem (Chevalley): Let be a maximal torus with associated Weyl group scheme . Then, the natural map corresponding to the map

,

where is the natural action, is an isomorphism.

Remark: It is very convenient to note that a large portion of this story holds over an arbitrary base scheme . For this I recommend looking at [Alper2, §9] and [Česnavičius–Bouthier, §4.1].

In fact, when is a split, simply connected, semisimple group we can soup this up even more. Namely, let us now fix a split maximal torus of , and recall that inside of are the subset of fundamental weights (see [Milne, §22.8]) of size , where is the rank of . Using heighest weight theory (see [Milne, Chapter 22, §a]) we can associate to these fundamental weights a set of fundamental (irreducible) representations and then by taking the trace obtain a set of fundamental (irreducible) characters

We can assemble these into a morphism given by sending

One can then show (e.g., see [Popov, Theorem 3.12]) that one can identify this with the Steinberg map .

Example (eg:SL2-Steinberg): Consider the case when . This is a split simply connected semisimple group over of rank . In this case, one may show that if denotes the characteristic polynomial of , then (up to reordering)

where are the fundamental characters of . In other words, as these coefficients can also be described as we see that, up to reordering,

for .

So, for example, taking shows that there is an isomorphism

corresponding (of course) to the trace function.

Example: For the group , another split simply connected semisimple group over , one can similarly describe the fundamental characters . For some variety, let me slightly change the presentation of the answer though. Via the tautological embedding it makes sense to speak of the eigenvalues of an element of . These eigenvalues come in reciprocal pairs . Let us then define morphisms

for given by

which are actually -polynomials in the entries of . Then, these are the fundamental characters of and thus we can concretely describe the Steinberg map.

For example, taking , we see that we have a unique isomorphism

such that the composition

is given by

as one can explicitly check.

Parameterizing conjugacy classes

So all of the above is very nice, but one will notice a glaring omission: how is explicitly related to conjugacy classes in , and how does any of this relate back to the actual topic of these posts, i.e., how does it relate to Lang’s theorem?

The first question has a nice answer, but one that may not be what one expects. To help understand it, we make two observations:

Suppose that is an element of , and . Observe then that as is a condition stable under scalar extension, we see that not only does take constant values on the rational conjugacy class of , but that it takes constant values on the geometric conjugacy class of ! In other words, the restriction of to the scheme-theoretic conjugacy class is constant, i.e., .
But, this scheme-theoretic conjugacy class may not be closed and, as is closed we actually see that .

Because of this, the best one could possibly hope for is that the Steinberg map can help us study not rational conjugacy, not even geometric conjugacy, but the relation if .

To start to slightly edulcerate this, let us first note that it’s easy to show that for any the closure of the scheme-theoretic conjugacy class will always contain a unique closed conjugacy class with one very important caveat: this is, a priori, only defined over (e.g., see [Milne, Proposition 7.6] and the suceeding discussion or [Humprheys, Chapter 1, §1.5]).

So, the first question we need to ask is: which group elements give rise to close conjugacy classes? This, it turns out, has a really beautiful answer. Namely, recall that an element is called semisimple if any of the equivalent conditions hold:

for every representation the matrix in is diagonalizable,
for a single faithful representation the matrix in is diagonalizable,
there exists a (possibly non-split) maximal torus such that ,
the centralizer has reductive connected component.

(e.g., see [Humphreys, Chapter 2, §2.2]).

Theorem (e.g., see [Humphreys, Chapter 1, §1.7]): An element has the property that is closed if and only if it’s semisimple.

Given all of this, and some standard facts about GIT quotients (e.g., see [Mumford, Chapter 0 and §1.1-1.2]), the following is not hard to show.

Theorem: Suppose that is a reductive group. Then, the natural map

,

given by sending to the unique semisimple conjugacy class in contained in is a bijection.

Note that if is a conjugacy class, then for any element in then (applied elementwise) is still a conjugacy class: it’s . We say that is Galois invariant if for all .

In particular, we see as a corollary of this that the Steinberg map induces an injection

with image those Galois invariant semisimple conjugacy classes in that contain an element of .

This highlights an important point here, the Steinberg map is surjective as a map of schemes, but the induced map is not necessarily surjective (if is not algebraically closed). In particular, it’s not necessarily true that every Galois-invariant conjugacy class contains a Galois-invariant element.

We can actually use the above theory to give quite a nice example of such a conjugacy class.

Example: Let and consider , i.e., the quaternion division algebra over ramified exactly at . This defines a reductive group , i.e., the group of (reduced) norm elements in .

Consider then the field and . Inside consider the element Its characteristic polynomial is so the -conjugacy class of is Galois-invariant by the above discussion as the trace is Galois-fixed.

That said, the quadratic field does not embed into . Indeed, is ramified at , so where is the Hamiltonian quaternions, whose only finite etale subalgebra of degree is . On the other hand,

Thus fails to embed at and so cannot embed rationally.

Now, if there were an element lying in the conjugacy class , then would be semisimple with characteristic polynomial , so , giving an embedding , a contradiction. Therefore, . This gives a Galois-invariant conjugacy class with no -rational representative.

That said, we have the following amazing theorem of Kottwitz.

Theorem ([Kottwitz]): Suppose that is quasi-split and that is simply connected. Then, every Galois-invariant conjugacy class in contains an element of , i.e., the Steinberg map induces a surjection .

Here being quasi-split means that admits a Borel subgroup over (see [Milne, §17.66]). That said, for soon-to-be-explained applications of Lang’s theorem, let me point out that Lang’s theorem actually implies that every reductive group is quasi-split over a finite field! Indeed, by [Milne, Corollary 23.54] every group is an inner form of a quasi-split group, and non-trivial inner forms give rise (with the choice of more information) to non-trivial classes in (e.g., see [Milne, §3.52]), but this is impossible by Lang’s theorem!

Finally, before we reap the fruits of all of this labor, we state one last definition and one last theorem which will allow us to fully take advantage of Proposition (prop:stable-conj). Namely, let us say that a semisimple element is regular semisimple if its centralizer is connected.

Theorem (e.g., see [Humphreys, §2.11]): Suppose that is a reductive group such that is simply connected. Then, every semisimple element of is regular semisimple.

Using Lang’s theorem, one can assemble the above discussion to say many interesting things about semisimple conjugacy classes of reductive groups over finite fields. But, for aesthetics, let me state the cleanest theorem possible under some (mild) hypothesis.

Namely, from the above discussion we deduce the following.

Theorem: Let be a split simply connected semisimple group over a finite field of rank . Then, if are the fundamental characters of , then the map

,

is a bijection.

Let us end with a very simple example that came from this math.stackexchange.com question.

Example: Suppose that is a finite field and and are elements of such that and that this common trace is not equal to . Then, and are conjugate.

Indeed, first observe that because (and similarly for ) has trace , it must be semisimple. Indeed, it suffices to show that its characteristic polynomial does not have repeated roots. But, its characteristic polynomial is the form . If this factored as then we would see that and so and thus . But, and thus this cannot happen by assumption.

Thus, we see that and are semisimple, and so by the above theorem (as is split semisimple and simply connected) we have that they are conjugate if and only if they have the same image under the Steinberg map. But, by Example (eg:SL2-Steinberg) this is equivalent to the claim that , as desired.

Remark: Note that this example really does use that we are over a finite field as the Example (eg:rationally-and-geometrically-conjugate-are-different) produces examples of elements and in such that but they are not (rationally) conjugate (only geometrically conjugate)!

References

[Alper2] Alper, J., 2014. Adequate moduli spaces and geometrically reductive group schemes.

[Česnavičius–Bouthier] Bouthier, A. and Česnavičius, K., 2022. Torsors on loop groups and the Hitchin fibration. Ann. Sci. École Norm. Sup, 55, pp.791-864.

[Gille–Szamuely] Gille, P. and Szamuely, T., 2017. Central simple algebras and Galois cohomology (Vol. 165). Cambridge University Press.

[Gortz–Wedhorn I] Görtz, U. and Wedhorn, T., 2020. Algebraic geometry I: schemes. Springer Fachmedien Wiesbaden.

[Gortz–Wedhorn II] Görtz, U. and Wedhorn, T., 2023. Algebraic Geometry II: cohomology of schemes. Wiesbaden: Springer Spektrum.

[Humphreys] Humphreys, J.E., 1995. Conjugacy classes in semisimple algebraic groups (No. 43). American Mathematical Soc..

[Kottwitz] Kottwitz, R.E., 1982. Rational conjugacy classes in reductive groups. Duke Mathematical Journal, 49(4), p.785.

[Labesse] https://www.math.utah.edu/~ptrapa/src2006/labesse.pdf

[Milne] Milne, J.S., 2017. Algebraic groups: the theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.

[Mumford] Mumford, D., Fogarty, J. and Kirwan, F., 1994. Geometric invariant theory (Vol. 34). Springer Science & Business Media.

[Popov] Popov, V.L., 2011. Cross-sections, quotients, and representation rings of semisimple algebraic groups. Transformation Groups, 16(3), pp.827-856.

Lang’s theorem — Pt II: an interlude on torsors

alexyoucis — Sat, 15 Nov 2025 22:10:20 +0000

This is a continuation of this post. Here we will discuss the various ways one can view torsors in anticipation of applying this to deduce some interesting corollaries of Lang’s theorem.

Torsors

Let us temporarily work more generally and fix:

to be a base scheme,
set to be the the small étale site of (e.g., see [Poonen, §6.2] or Tag 03XB),
and let be a sheaf of groups on .

NB: The vast majority of what we say below works if is replaced by any site/topos. For instance, one can consult [Giraud] which is the bible for this topic.

Definition (def:torsor): A -torsor on is a sheaf of sets on equipped with an action such that

the map is an isomorphism of sheaves,

there exists a cover in such that is non-empty for all .

With the obvious notion of morphisms we let denote the category of -torsors on , and denote the set of isomorphism classes in .

NB: The first condition in being a torsor can be rephrased in perhaps more familiar terms: if is non-empty, then the action of on is simply transitive.

I won’t spend any real time discussing the motivation for torsors (you can see more here). Instead, we will now several equivalent ways of understanding/parameterizing torsors.

Twists

The most instructive way to think about torsors is that they are classifying twists of some given object. Unfortunately, to make this precise requires a little bit of setup, which I don’t want to go into here. So, let us instead make an informal definition which, in all cases of interest, can be actually upgraded to something more rigorous.

Definition (def:prestack): A prestack on is a contravariant functor

where is the category of groupoids.

In more concrete terms we obtain:

for every object of we get a groupoid of objects ,
and for every map a morphism of groupoids .

Remark: The reason this is informal is that, in practice, nothing is ever a prestack in the way I have defined it (e.g., all the examples of prestacks below are really not actually prestacks). The issue is that in general for a map the pullback morphism usually requires a choice. For example, if is quasi-coherent sheaves on , then this pullback morphism would be base extension/tensor product, which is something defined by universal property: there is no such thing as the tensor product, only something satisfying the tensor product property which only specifies it up to unique isomorphism.

One might make choices of pullback maps for every morphism, but then one gets into difficulty: are those choices compatible with composition? This becomes such an overwhelming headache that, in practice, if one is trying to be perfectly rigorous the correct framework to talk about what I mean by prestack here is a category fibered in groupoids, see Tag 003S. That said, such an object is always equivalent to a prestack in the way we’ve defined it here, see Tag 0048, so the inaccuracy is mostly harmless. One should just note that such a prestack might give rise to a category fibered in groupoids equivalent to the one that you started with but that looks ostensibly very different!

Here are some simple examples.

Example (eg:schemes-prestack): We can set to be the set -schemes which is a groupoid using isomorphisms -schemes over . This then becomes a prestack when we declare that the functors is the pullback functor .

NB: You can spice this example up by replacing with any subgroupoid of morphisms satisfying property , where is any property stable under pullback.

Example (eg:VBs-prestack) One can set to be the set of vector bundles on , which is a groupoid using isomorphisms of vector bundles. This becomes a prestack when we declare that the functor is just the normal pullback of vector bundles.

Example (eg:torsors-prestack): In fact, one can consider to be the groupoid . This again becomes a prestack with the obvious pullback functors.

Now, prestacks are like a -categorical version of a presheaf, and what we really need is the notion of a stack — the -categorical version of a sheaf. I won’t define the notion here, see Tag 02ZH, but intuitively the idea should be clear: if is a cover and I have objects and compatible identifications then I can glue these objects to an object of , and similarly for morphisms.

NB: I suggest consulting the references [Alper] and [Bertin] to learn more about stacks if you aren’t familiar.

The operative definition we’ve been alluding to is the following.

Definition (twists): Let be a prestack on . Then, for an object of , a twist of in is another element of such that and are locally isomorphic (i.e., there exists a cover in such that for all ). We denote the -isomorphism classes of twists of by , or just .

As suggested by the terminology ‘twist’, one should think of as the set of things that locally look like or, said differently, different ways of cutting up (e.g., looking at ) and reassembling them in a different (i.e., ‘twisted’) way.

The upshot of all of this is that if is a stack you can exactly classify the set in terms of torsors.

Proposition (twists=torsors)[Giraud, Chapitre III, Corollaire 2.5.2]: Suppose that is a stack. Then, the map

is a bijection.

Here we are considering the

isomorphism sheaf given by
the automorphism group sheaf with its natural group structure.

The map in Proposition (twists=torsors) then is sensical as has a natural action by .

Let us give some examples of this phenomenon.

NB: I will elide the, often non-trivial, claim that the I write down below are stacks.

Example (eg:vb-twists): Let be the stack of vector bundles over , and consider , the free rank -vector bundle, an element of . Then, essentially by definition, twists of are rank -vector bundles on . But, is . So, rank -vector bundles are naturally classified by the set of -torsors on .

Here the isomorphism sheaf is often called the frame bundle associated to .

Example (eg:twists-ell-curves): Let be a perfect field of characteristic not or . Let be the stack of elliptic curves over , and fix in . Then, twists of are just elliptic curves over such that .

We can parameterize these by the set . In most cases (e.g., see [Silverman, Theorem 10.1]), is something very simple — it’s the group (sheaf) . Then, Kummer theory says this cohomology set is precisely , i.e., with the elements of up to squares.

Writing as (the projectivization of) , the twist of associated to an element is the quadratic twist with equation .

Example (eg: twists-torsors): Let be the stack of -torsors. Then, we always have in the trivial -torsor, i.e., acting on itself by left multiplication.

It’s a good exercise to check that if is a -torsor, then it is trivial (i.e., isomorphic to ) if and only if is non-empty. So, we see that every -torsor is a twist of . So, in fact, . On the other hand, one explicitly checks that . Thus, we arrive at the tautological looking

Remark: Example (eg: twists-torsors) shows that in fact one can always understand -torsors as twists by taking the silly choice of stack (the stack of -torsors). So, as preivously hinted at, twists and torsors are just two different names for the same thing. Of course, the point is that twists have much more flexibility, as there can be many stacks and points such that –maybe one can think of this as a ‘model for -torsors’. Not being married to one particular model (i.e., the model given in Example (eg: twists-torsors)) is very powerful.

Čech cohomology

We next give the most down-to-earth description of torsors, in terms of Čech cocycles.

Definition (def:cech-cocycles): Let be a cover in , and write and . Then, the Čech cocycle set is given by

We define an equivalence relation on by declaring that if there exists a collection such that for all . We define the Čech cohomology set to be

The Čech cohomology set is functorial in , where we think of morphisms of covers as refinements, and in fact we can form its colimit

as ranges over all covers of .

The relationship between Čech cohomology and torsors is summed up neatly as follows.

NB: For notational convenience, for a cover in , let us write for the set of -torsors such that is non-empty for all , i.e., such that becomes trivial over (each constituent of) .

Proposition (prop:torsors-cech) [Giraud, Chapitre III, Proposition 3.6.3]: There is a bijection

functorial in . In particular, there is a bijection

Proof idea: This bijection is easily interpreted using Proposition (twists=torsors) and Example (eg: twists-torsors). Indeed, every -torsor trivial over is obtained by gluing the trivial torsors together along the overlaps in a compatible way. As the group automorphisms of is these identifications on overlaps give rise to an element . One then checks that the compatibility condition on the overlap isomorphisms is precisely the condition for this tuple to belong to , and that two tuples of gluing data give isomorphic -torsors if and only if they are -equivalent.

We end by giving a concrete interpretation of the Čech cohomology set in the case when is a finite Galois cover, i.e., is faithfully flat (or, equivalently, finite etale) and there is a finite group acting on by -morphisms such that

is an isomorphism. Here is the constant group scheme/sheaf over associated to .

Let us observe that in this case for a -torsor on , the group acts on by functoriality. Recall that we can then always define the non-abelian group cohomology set

Here consists of all -valued -cocyles on , i.e., maps satisfying the relationship

for all in . We then write that if there exists some element such that for all in .

We can soup this definition up slightly. Namely, suppose that is a projective system of Galois covers, with Galois groups . Let us set (which exists as the transition maps are affine, see Tag 01YX) and, for simplicity, let’s assume that is locally of finite presentation so that . Then, if we set , a profinite group, we can define

where, by definition, the left-hand side is the continuous group cohomology, and this equality is a claim opposed to a definition. This continuous group cohomology defined exactly the same as in group cohomology, but we require that be continuous, where the source is given the topology inherited from its profinite group structure, and is given the discrete topology.

So then, one can then use the explicit isomorphism of with to show the following.

Proposition (prop:group-cohomology): Let be a finite Galois cover with Galois group . Then, there is a functorial bijection

.

If is locally of finite presentation and is a projective system of finite Galois covers with Galois group , then setting and we have

Thus, if every -torsor trivializes on some (i.e., for some ), then we have a natural identification

Example (eg:Galois-cohomology): Let us consider now the case when , where is any field and fix a separable closure . Then, let us observe that we have , where ranges over all finite Galois subextensions of , is a projective system of Galois covers with Galois groups . In this case the limit is .

We claim that any -torsor trivializes on some . Indeed, we know that there exists some étale cover over which each is non-empty. But, by Tag 02GL we can write for finite separable extensions . By the sheaf property we know that

and as this is non-empty each is non-empty. But, let be a Galois extension of dominating . Then, as we have a map we have that the target is non-empty, from where the claim follows.

Thus, applying Proposition (prop:group-cohomology) we see that if is locally of finite presentation,

where the final cohomology set is, by definition, the Galois cohomology of .

Principal homogenous spaces

In the natural case when the sheaf is itself representable and locally of finite presentation, it is natural to ask the question: in which situations is a -torsor itself then automatically representable as well?

To this end, let us make the following definition.

Definition (def:PHS): Let be a locally of finite presentation and flat group -scheme. Then, a principal homogenous space for is a faithfully flat -scheme together with an action map such that

the morphism

is an isomorphism,

the map is faithfully flat.

We denote the category of principal homogenous spaces for on by .

The relationship to torsors on is then the following, which we leave as an exercise (see the nota bene below).

Proposition (prop:torsors-PHS): Let be a smooth group -scheme. Then, the functor

is an equivalence onto the full subcategory of representable sheaves in the target.

NB: the smoothness here is a red herring, if one uses the fppf site of this is always true. In the context of the above is an easy exercise. The smoothness of just ensures that -torsors on and are the same.

So, we see that we can really ask when the functor from Proposition (prop:torsors-PHS) is an equivalence of categories. A partial answer is given as follows.

Proposition (prop:all-torsors-PHS)(see [Milne2, Theorem 4.3]): Suppose that is a smooth group -scheme satisfying one of the following conditions:

is affine,

is separated and has dimension at most ,

is smooth proper with geometrically connected fibers and is regular.

Then, the functor in Proposition (prop:torsors-PHS) is an equivalence.

Using Proposition (prop:all-torsors-PHS): we can actually give another example where Proposition (prop:group-cohomology) applies.

Example (eg:henselian-group-cohomology): Suppose that is a Henselian local ring with residue field , and set . Suppose further that is a smooth group -scheme satisfying one of the equivalent conditions of Proposition (prop:all-torsors-PHS).

Consider the strict Henselization of (see Tag 04GN), which is a Henselian local ring with residue field . By Tag 0BSL where are finite Galois covers with Galois groups , and , the étale fundamental group of .

We claim that any -torsor on satisfies for some , and thus by Proposition (prop:group-cohomology) that

Indeed, as we are assuming that Proposition (prop:all-torsors-PHS) applies, we know that is representable. In fact, as is étale locally on isomorphic to , and is -smooth, we know that is -smooth (e.g., see Tag 02VL). Thus, in fact, is surjective by Hensel’s lemma (see [Gortz–Wedhorn II, Theorem 20.12]). But, as we already discussed in Example (eg:Galois-cohomoloy) this latter set is non-empty. But, as is locally of finite presentation (see Tag 02KY) we have that and so one of the is non-empty as desired.

Remark (rem:alg-spaces): In some sense, I think the technicalities of the above discussion are illusory, bordering on misleading. Namely, while it is true that one requires some non-trivial assumptions on to ensure that any -torsor is representable, this is the wrong thing to focus on. Namely, while a torsor may not always be representable, it is always an algebraic space (e.g., see Tag 04SK). Algebraic spaces are, for essentially all intents and purposes, just as good as schemes. For example, they still satisfy Hensel’s lemma, and so the conclusion of Example (eg:henselian-group-cohomology) actually holds without any assumptions on .

Tannakian interpretation

In this final subsection we reinterpret -torsors on in so-called Tannakian terms, at least when satisfies some stronger base assumptions. We will, at points, be somewhat vague here, and we suggest the reader consult [Imai–Kato–Y., Appendix A] for more information, in an even greater generality.

We begin by stating our necessary stronger condition, which essentially always holds in practice (below, the phrase ‘Dedekind domain’, should be allowed to include fields).

Condition: : Suppose that is a Dedekind domain, there is a morphism of schemes , and is the pullback along of a flatt finite type affine group -scheme.

Remark: These conditions may seem strange, but let me quickly say what the technical point that breaks generalizations is. We will be dealing with representations of over . For general this category can be quite subtle. Namely, in general not every finitely generated -stable submodule of a representation is a representation. Indeed, we we require representations to be on finite projective -modules, and a submodule of a finitely projective -module need not be projective. But, this is true if is a Dedekind domain (e.g., see Tag 0AUW). This fact is an important point in the proofs of the results below.

Let us denote by the category of all homomorphisms , where is a finite projective -module. As per usual, we will think of such an object as an together with an (implicitly defined) -linear action of and usually just write it as .

This category has the natural structure of an -linear -category. Namely, the bi-functor

has a natural factorization through (i.e., there is the functorial structure of an -module on the -sets) and there is an -bilinear functor

which satisfies reasonable coherence properties. This category also has an (Quillen) exact structure. In other words, we can say when an exact sequence of representations of is exact which, as one would expect, is when the underlying sequence of -modules is exact.

Remark: This category is not abelian of course, in particular the exact structure here cannot be understand internally to the category. For instance, taking to be trivial, one sees that multiplication by is an epimorphism in the category of finite projective (i.e., free) -modules, but is not so in the category of all -modules). In general the notion of an ‘exact category’ is extra structure in the non-(abelian category) setting. In particular, equivalences of categories need not preserve this exactness structure, and even if the equivalence preserves the exact structure, its quasi-inverse may not!

While we won’t define it here, one can imagine how one would define, in general, an exact -linear category . With this, we can make the operative definition.

Definition (def:G-objects): Let be an exact -linear -category. Then, a -object in is a functor

which is

exact: if

is an exact sequence of -representations over then

is an exact sequence in ,

-linear: for all and in the map

is -linear,

a -functor: the functor is symmetric monoidal (see Tag 0FFL and Tag 0FFY).

We denote the category of -objects in by , where here a morphism of -objects is an invertible natural transformation.

We now consider our chief example of an exact -linear -category.

Example (eg:vect-cat): Let be a scheme over . Then, the category of vector bundles on is an exact -linear -category. Here the -linear structure is the natural one coming from the fact that is an -scheme, and the tensor product is the usual one over . Finally, we say that a sequence

of vector bundles on is exact if the underlying sequence of -modules is exact.

So, we obtain from the category of -objects in . As is standard, we often refer to such an object as a -bundle on .

What is the relationship between -bundles and -torsors? The answer starts to reveal itself when we make the following observation.

Proposition (prop:GL_n-obj-vb): Consider a rank finite projective -module . Then, there is a natural equivalence between and , the full subcategory of consisting of rank vector bundles.

Proof idea: Assume that is actually free (the general case comes from ‘twisting’), so isomorphic to . Then, the functor

is given by , as expected, with the only non-trivial claim that this is actually a rank vector bundle. Conversely, given a rank vector bundle , we can build a -bundle by the following rule:

This right-hand side is the sheaf given

i.e., the quotient of the product sheaf given by the equivalence relation that allows you to ‘shift’ the action from one factor to the other. One can check that this quotient sheaf actually has the structure of a vector bundle on . These functors are quasi-inverse to each other.

From Proposition (prop:GL_n-obj-vb) we can explain how to build a functor

Namely, let us fix a -torsor . Then, using the natural equivalence from Proposition (prop:GL_n-obj-vb) we can associate the following object of :

Here

is the natural functor (e.g., see [Giraud, Chapitre III, Proposition 1.4.6]) which, concretely, can be understood in terms of Čech cohomology: it associates to a -cocycle the -cocyle .

The non-trivial fact is then the following.

Theorem (thm:Broshi) ([Broshi, Theorem 1.2]): The functor

is an equivalence of categories.

Remark: there are several remarks to make here.

In [Broshi, Theorem 1.2] it is stated that one needs (in our notation) to be faithfully flat over but, this is not necessary (see [Scholze, Theorem 19.5.1] and the footnote).
There is an ideologically substantial, but easy to fix, mistake in the proof of [Broshi, Theorem 1.2] — see [Imai–Kato–Y., Remark A.19] for details.
In my personal opinion it’s helpful to understand that a version of Proposition of Theorem (thm:Broshi) holds in a much vaster generality, even without many parts of Condition , as long as you are willing to replace the target of the equivalence by a full subcategory. The theorem of Broshi then says that, in the setting of the theorem, this full subcategory is, in fact, the entire category — see [Imai–Kato–Y., §A.5] for details.
Again, in far greater generality one can understand the objects here in terms of twists, as above, for a very natural stack — the stack of ‘vector bundles with tensors’. This is a very helpful, intuitive way to understand torsors, but as it won’t be useful in this article, we ignore this here — again see [Imai–Kato–Y., §A.5] for details.)

What Theorem (thm:Broshi) allows us to do (especially in conjunction with point 4. of the above remark) is view -torsors on as -bundles or, more evocatively, vector bundles with -structure.

References

[Alper] https://sites.math.washington.edu//~jarod/moduli.pdf

[Bertin] Bertin, J., 2012. Algebraic stacks with a view toward moduli stacks of covers. In Arithmetic and Geometry Around Galois Theory (pp. 1-148). Basel: Springer Basel.

[Broshi] Broshi, M., 2013. G-torsors over a Dedekind scheme. Journal of Pure and Applied Algebra, 217(1), pp.11-19.

[Giraud] Giraud, J., 2020. Cohomologie non abélienne (Vol. 179). Springer Nature.

[Gortz–Wedhorn] Görtz, U. and Wedhorn, T., 2023. Algebraic Geometry II: cohomology of schemes. Wiesbaden: Springer Spektrum.

[Imai–Kato–Y.] https://arxiv.org/abs/2406.08259

[Milne2] Milne, J.S., 1980. Etale cohomology (PMS-33) (No. 33). Princeton university press.

[Scholze] Scholze, P. and Weinstein, J., 2020. Berkeley Lectures on p-adic Geometry:(AMS-207).

[Silverman] Silverman, J.H., 2009. The arithmetic of elliptic curves (Vol. 106, pp. xx+-513). New York: Springer.

Lang’s theorem — Pt. I: the classical result

alexyoucis — Sat, 15 Nov 2025 21:50:25 +0000

In this post we discuss Lang’s theorem (i.e., that the Lang isogeny on is surjective, for a connected group variety) and an integral analogue of this result. This will allow us to discuss a lot of interesting arithmetic geometry, including the usefulness for the multitude of ways to interpret torsors.

Lang’s theorem in the classical setting

We begin by recalling (and proving) Lang’s theorem in the classical setting over a finite field.

Rigorous statement

Our main players will be:

a finite field of order , for a prime,
a connected group -variety (i.e., a connected reduced finite-type group -scheme).

Note that here need not be a linear algebraic group (e.g., a reductive group or a unipotent group like the additive group ), but is free to be an abelian variety. In fact, one should feel mostly free to pretend that it is one of those two types, as the famous theorem of Barsotti–Chevalley (e.g., see [Milne, Theorem 8.27]) says that every such is an (essentially unique) extension of an abelian variety by a linear algebraic group.

Write for the -Frobenius element of : the map of fields given by

As is well-known, the -Frobenius is the important element of the Galois group , as one has that , i.e., that is a (free) topological generator of .

Now, for any -scheme , one can naturally associate the function

which modifies an element of by . More rigorously, it associates to a morphism over its precomposition with .

In the case when is our group, we have more room to play. Namely, in this case has a natural group structure, and so it makes sense to multiply the inverse of this map with the identity to arrive at the so-called Lang map:

In some sense we may think of as measuring the relative position of and and so, in some sense, measuring its rationality (e.g., ).

It is this map for which Lang’s theorem pertains to.

Theorem (Lang, see [Mumford–Oda, Theorem 6.2.1]): The Lang map is surjective, i.e., every element of can be written in the form for some in .

The two remarks that should immediately be made are:

this seems ostensibly surprising (why should this be true?),
it may not yet be clear how powerful of a statement this is.

The latter we will return to below, but we now immediately try to address the former.

Some intuition

Forgetting the full version of Lang’s theorem for a moment, we pause to ask the following even simpler question:

Question: Why should the image of even be ‘large’?

It’s not immediately obvious that doesn’t lie in some positive codimension subset of – or, for that matter, that it’s even infinite!

Remark: In fact, the entirety of Lang’s theorem follows from the ‘largeness’ of its image by exploiting the geometry of groups (i.e., ‘spreading out by group multiplication’ type arguments).

But, we can intuitively see why at least the image can’t be too small.

Plausibility argument for Lang’s theorem: Suppose that were induced by an algebraic map over . Then, it should be the case that is étale. Namely, at least intuitively, the derivative is as it differs from the identity by , and the derivative of should be as it is something like (and as and we’re in characteristic ). So, if is étale that means, in particular, that has open image and so the image of can’t be too small (e.g., can’t be contained a positive codimension subset–all connected groups are irreducible, see [Milne, Corollary 1.35]).

Let me point out the subtlety in this ‘argument’: we assumed that the map was algebraic. But this is rarely case for other fields. For instance, for a variety over we get a ‘complex conjugation map’

which is not holomorphic, and therefore cannot possibly be induced by a morphism of -schemes .

And, in fact, this plausibility argument can’t work in the real case, because Lang’s theorem is false in this context!

Example: Consider . Then, the map

is not surjective. Indeed, by Hilbert’s Theorem 90, , where

is the norm map.

Thus, we see the pivotal point for Lang’s theorem: over (the finite field) , the boundary between arithmetic and geometry becomes blurred due to the existence of a Frobenius morphism of -schemes.

Definition: For a -scheme the -Frobenius morphism is the morphism of -schemes

which is the identity on the underlying topological space, and on sheaves is the map -power map.

Observe that induces a morphism and, as expected, this agrees with . Indeed, it suffices to prove this for affine, in which case corresponds to the map

and corresponds to the map

where is the map . But, of course, these are equal: .

Remark (rem:only-for-finite): Let me point out another, but related, reason why the plausibility argument has to fail outside the case of finite fields. If the map

for some (perfect) field and some element is meant to be etale then, in particular, (where is the identity element), should be discrete. But, as is Noetherian, discrete implies finite. On the other hand, is visually the same as : the points of valued in the fixed field . For this to be finite, the only real option is that that is finite.

Proof of Lang’s theorem

Consider the action of on itself given as follows:

where is the -Frobenius. Then, for any in we have the orbit map given by . We claim that the map is étale.

Indeed, it is a standard calculation (see the proof in [Mumford–Oda]) that the derivative of at the identity

is equal to, identifying by the translation by map , . But, is zero (e.g., see [Mumford–Oda, Lemma 6.2.2]). So, is an isomorphism. But, observe that if is translation by , then which allows you to traslate this calculation that is an isomorphism to deduce htat is an isomorphism for any in . This implies that is étale by [Mumford–Oda, Criterion 5.4.6].

But, as is étale, this implies that the -orbit of (i.e., the image of ) is open. But in fact, this implies that it’s all of . Indeed, suppose not, then we can choose a point of not contained in . But, by the same argument we have that is open. But, as is connected, it’s irreducible (e.g., see [Milne, Corollary 1.35]), and so and intersect, but this is a contradiction as and are disjoint.

In particular, we see that and so

But, as acts on as , Lang’s theorem follows.

Remark: As we remarked in Remark (rem:only-for-finite) the étaleness of the Lang’s map can only be true for finite fields. But, upon first glance, it seems all we used in the above proof was that is characteristic . It is a good exercise to pinpoint exactly where the finiteness of was used.

A souping up of Lang’s theorem

It’s clear from the proof of Lang’s theorem that what we’re really analyzing is the Lang morphism:

We may then provide the following geometric version of Lang’s theorem.

Proposition (Lang’s theorem, geometric redux): Let be a connected group variety over . Then, the Lang morphism is a finite Galois cover with Galois group .

Proof: We have already seen in the proof of Lang’s theorem that is étale and surjective. Let us observe that acts on by left translation, and this action is trivialized by : for in we have

as . Moreover, a simple argument shows for any field and any morphism the fibers of over are acted on simply transitively by . Thus, we will be done (by the definition of a finite Galois cover of schemes) if we can show that is finite.

To show it’s finite we argue as follows. Since is étale, it is generically finite (see Tag 0BAI), say it is finite over the open subset . It suffices to show that for any point of there exists a neighborhood of over which is finite. But, take a point of . Then, is a neigborhood of and over and differ by translation morphisms (which are isomorphisms), so must be finite over as desired.

Remark: Associated to any reductive group over an algebraically closed field is a so-called algebraic fundamental group:

Here is the universal cover of the derived subgroup (see [Milne, Definition 18.7]), is a maximal torus of , and is a maximal torus of containing the image of under the composition .

Alternatively, one has

where is a maximal torus of , and is the coroot lattice–i.e., the lattice inside of the cocharacters generated by the coroots.

Example (eg:alg-fun-group): Using this last description it is easy to check, for example, for any , but that .

A natural question then becomes:

Question: Is the same as the etale fundamental group of when is semisimple?

The assumption is semisimple is because, else, the algebraic fundamental group is an infinite discrete group (e.g., see Example (eg:alg-fun-group)), and so can’t be the etale fundamental group–one should then really consider its profinite completion for such a comparison.

The answer to this question is

yes if the characteristic of is zero (e.g., see this wonderful discussion by Brian Conrad),
no in positive characteristic by the souped up version of Lang’s theorem.

Example: it follows from Lang’s theorem that the etale fundamental group of admits as a quotient for every !

Appendix: Deligne–Rapoport’s lemma

Just for fun, let us give an alternative means of proving finiteness of , once we know that is étale with kernel . This will be via the following beautiful observation of Deligne–Rapoport which should be more well-known, and for which we provide a (probably very overcomplicated) proof as there doesn’t seem to be one in the literature dealing with the non-Noetherian case.

NB: In the following we use the following terminology for an étale morphism : we say that has constant fiber size if there is an integer such that for all fields and all morphisms one has or, equivalently (cf. Tag 02GL), if for all such with separably closed one has . In fact, it suffices to check this for maps with a point of .

Lemma (lem:DR) [cf. Deligne–Rapoport, §II, Lemme 1.19]: Suppose that is a quasi-finite, separated, and étale morphism. Then, the following are equivalent:

is finite,

for each connected component , the induced function has constant fiber size.

Proof: If is finite then it is finite locally free (see Tag 02KB). Thus, so is . But, then is a vector bundle on , and so evidently of constant rank. This implies that has constant fiber dimension (e.g., use Tag 0CKW as overkill) over .

To show the converse it suffices by Tag 02LS to show that is proper. By Tag 0BX5 it suffices to show the valuative criterion for properness. Let be a valuation ring and a morphism. As this must facotrize through a connected component, we see that replacing by may assume that and that has constant fiber size. In fact, replacing by its Henselization (cf. Tag 0ASK), we may assume that is Henselian.

By Zariki’s Main Theorem (see Tag 05K0) there exists a finite morphism and a quasi-compact open embedding over . Note is reduced (see Tag 03PC), so we are free to replace by the Zariski closure of in with the reduced subscheme structure. Then, is injective (see Tag 056D), and so has no -torsion as doesn’t (see Tag 0539). Thus, by a deep theorem of Raynaud–Gruson (see [Raynaud–Gruson, Premiere partie, Corollaire (3.4.7)]) one actually has that is finite presentation. Thus, it is finite locally free (again by Tag 02KB).

Write . As dominance is preserved by flat base change (e.g., see Tag 03HV and Tag 03HF) we have that is a dense open subset of but, as is a discrete set of points, this implies that . Now, as is Henselian, we can write , where are local (see Tag 04GG) and, by set up, reduced, so domains. As is finite locally free we see that is a finite injection of local domains. As we are only interested in questions concerning we may assume that for all . Thus, is a local subring of dominating , and so equal to it. Thus, is an open subscheme of . If it were a proper open subset, it’s clear that the fiber dimension would not be constant. Thus, and the conclusion follows.

References

[Deligne–Rapoport] Deligne, P. and Rapoport, M., 1973. Les schémas de modules de courbes elliptiques. In Modular Functions of One Variable II: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972 (pp. 143-316). Berlin, Heidelberg: Springer Berlin Heidelberg.

[Milne] Milne, J.S., 2017. Algebraic groups: the theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.

[Mumford–Oda] https://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf

[Raynaud–Gruson] Raynaud, M. and Gruson, L., 1971. Critères de platitude et de projectivité: Techniques de≪ platification≫ d’un module. Inventiones mathematicae, 13(1), pp.1-89.

Shimura varieties and their integral canonical models (Pt. 2)

alexyoucis — Sat, 15 Nov 2025 21:41:40 +0000

This is a continuation of this post.

§3 How and why?

Before we continue, I think it’s worth pausing to ask the above two questions about Shimura varieties.

§3.1 How?

While the moduli description in Expectation (exp:Shim-var) might provide a workable theory (at least in a theoretical sense), it’s entirely non-evident how people work with Shimura varieties given the harsh, arithmetically opaque, nature of their construction. So, how do people work with them?

Of course, there are a multitude of answers to this question. But, let me list a few very general ideas for how people work with Shimura varieties in practice.

Archimedean-type methods

Shimura varieties, by their very design are objects of an Archimedean (Hodge theory) provenance. It’s perhaps not shocking that some of the most powerful methods to actually proving this about Shimura varieties exploit fundamentally Archimedean theories. For example, the following ideas are routinely used to understand general Shimura varieties:

Archimedean Hodge theory,
differential-geometric and geometric-group-theoretic ideas (e.g., Margulis superridigity),
methods of logic applied to the analytic world (i.e., -minimality).

Special points

As indicated by Step 4 of the construction of Shimura varieties, the only truly tractable approach to them in an (obviously) arithmetically useful way is to use special points, where we do completely understand the arithmetic. This applies beyond the construction, and reducing to special points is one of the most common ways to prove something about general Shimura varieties.

“Type theory”

As will be discussed below, Shimura varieties can be ‘filtered by complexity’ into certain ‘types’. For low-complexity Shimura varieties (although they are not so simple!) one can really define an honest-to-god arithmetically-useful moduli problem in the vision of Expectation (exp:Shim-var). It is a common technique people apply to try and prove things about higher-complexity types by reducing them down the complexity ladder to these more approachable Shimura varieties.

Remark: I have explicitly left out a tantalizing direction here: -adic (Hodge-theoretic) methods. That is because this will essentially be the perspective taken in this series of lectures, and so I will delay their discussion until later.

§3.2 Why?

Given all of this to-do about Shimura varieties, especially given the relative distance we are from making Expectation (exp:Shim-var) real, you have to wonder: is the juice is worth the squeeze?

Of course, at least for me, the answer is yes. Let me give three (of many possible more) reasons why I personally feel this is the case.

§3.2.1 Langlands

The canonical, and ‘correct’, answer is the Langlands program. In remarkably vague terms, Langlands posits a rough triangular correspondence between three of the most important pillars in modern mathematics:

I won’t attempt to try and define either -valued Galois representations or automorphic representations of –for more information on them (and what I am about to say below), see this post.

Remark: In fact, the more familiar reader will know that it’s not even -valued Galois representations that appear here (even at a rough approximation). Namely, it should be -parameters for –in particular it’s closer to -valued Galois representations. That said, for the sake of space I will ignore this important point.

But, Langlands gave something even finer than the above picture. He posited what sort of more primitive object actually is connecting the above three pillars: Shimura varieties. Namely, one should have something as indicated by the following diagram:

The vertical arrow should be obvious given Expectation (exp:Shim-var), but let me give a small idea of how Shimura varieties are related to Galois representations and automorphic representations and, in fact, connects them. (For more details see the above-linked post.)

Conjecture (Langlands–Kottwitz): There exists a decomposition

,

such that is .

Let us tease out the meaning of the terms here:

Via the Hecke action, and the normal Galois action on étale cohomology, the left-hand carries an action of .
The right-hand side is the decomposition of the left-hand side into irreducibles for the action of , so is a representations of and is a representation of .
The index set travels over automorphic representations of . These are, by definition, representations of and, as it turns out, always decompose into a box product for the decomposition .
Finally, is a certain representation of , and is the global Langlands correspondence applied to .

In other words, the cohomology of Shimura variety realizes the global Langlands conjecture.

Other applications to algebraic geometry/number theory/harmonic analysis/differential geometry

While Langlands is maybe the marquee application of Shimura varieties, they have found nearly endless applications to other areas of algebraic geometry, number theory, and harmonic analysis.

Let me just give an example of an application for each of these fields which I find interesting:

Algebraic geometry: Madapusi used Shimura varieties to prove the Tate conjecture (a conjecture of fundamental importance in algebraic geometry) for K3 surfaces over finite fields (see [Madapusi]).
Number theory: The proof of the averaged Colmez conjecture by Andreatta–Goren–Howard–Madapusi (also proved independently by Yuan–Zhang), a conjecture relating heights on CM abelian varieties to special values of -functions, made use of orthogonal Shimura varieties (see [AGHM]).
Harmonic analysis: It was long a question of whether or not automorphic forms admit -adic interpolations. The answer (in many cases) is yes, and the definition involves Shimura varieties and their cohomology (e.g., see [Hida]).

A measuring stick for understanding of motives

Lastly, and most vaguely, but also close to my own heart, is the following. Over time we are getting closer and closer to trying to make Expectation (exp:Shim-var), and its correct ‘arithmetic generalizations’, a reality. For instance, the theme of this series of posts is that one can get an understanding of certain ‘canonical integral models’ of Shimura varieties using something tanatlizingly in the directions of a -adically integral analogue of Expectation (exp:Shim-var).

I find this ability of Shimura varieties as a metric for how far along our inexorable journey towards understanding ‘motives’ we are, to be one of the most beautiful aspects of the subject.

§4 “Type theory”

We mentioned above that there is a sort of “type theory” for Shimura varieties that naturally filters them by complexity. We would now like to explore this (to shallow depth), especially as the type of penultimate complexity, those of ‘abelian type’, factor heavily into future posts.

Modular curves

We start with the most basic (non-trivial), and classic class of Shimura varieties: modular curves.

Namely, let us fix:

,
which is the -conjugacy class of the map

(assume for simplicity).

Then, one may show that is a Shimura datum. And in fact, the associated Shimura varieties are the classical modular curves. Namely,

Siegel type

Modular curves are part of a general family of Shimura varieties, called Siegel type. In essence, they are the natural higher-dimensional generalizations of modular curves.

Namely, let us fix

–the symplectic similitude group, i.e., the group of linear automorphisms of which preserves the standard symplectic form up to a scalar (the ‘similitude’),
is the so-called Siegel upper and lower half-spaces which can be shown to be the -conjugacy class of a cocharacter,
().

Then is a Shimura datum, and the associated Shimura varieties are classifying the natural higher-dimensional analogues of what modular curves classify. Namely,

In other words, Shimura varieties of Siegel type are the same as Mumford’s moduli space of principally polarized abelian varieties together with level structure.

Remark: Let me roughly say what it means for to be a symplectic similitude. Namely, comes equipped with the standard symplectic pairing, and obtains the symplectic pairing

Because of this last isomorphism, this pairing is really only well-defined up to a scalar, but this doesn’t effect whether is a symplectic similitude: i.e., whether it takes the standard pairing to this modified Weil pairing up to scalar.

Remark: One may naturally expect the higher-dimensional analogues of modular curves should have group . But, there is no Shimura variety of the form when 2" class="latex" />. This is easy enough to prove given the (unstated) conditions on , but perhaps less obvious is why is this the case from the algebro-geometric perspective.

Let me give two answers:

For , we would have which has (real) dimension . In particular, this is an odd-dimensional real manifold for unless and so cannot be given a complex structure, and thus neither can the associated double quotient.
Secondly, the double quotient associated to would be classifying (cf. the Siegel case) complex tori . But, as observed in Remark (rem:Deligne-explained), such complex tori are generally not algebraizable, and thus neither can their moduli space. To fix this we need to add polarization as in loc. cit., but this then takes us to as in the Siegel case.

(Take note of the following coincidence at play here: .)

In fact, it is perhaps reasonable to expect that with enough imagination one can construct ‘moduli spaces of global -shtukas’ for any , in analogy to the local picture, but which only exist as some sort of analytic stack in the flavor of Clausen–Scholze. Shimura varieties then should constitute, essentially, those such moduli spaces of shtukas which are algebraizable.

From this perspective, it’s shocking that Shimura varieties were discovered in the 60s and 70s: that long ago they identified an object whose ‘true context’ would not be definable for at least(!) 60 years.

PEL, Hodge, Abelian, and General type

Beyond the Siegel case, Shimura varieties fall into ever increasingly more complicated ‘types’. It doesn’t quite make sense for me to spell out exactly what they mean rigorously here, and so instead let me just give a rough idea what they mean. Below, each class contains the previous one, and we give an example indicating a class not contained in any previous type.

PEL type

These classify principally Polarized abelian varieties with Level structure as in the Siegel case, but keep track of extra Endomorphism data in the form a ring map , for certain rings . In particular, these do have moduli problems which work over any .

Example: The Shimura varieties of type considered by Kottwitz, and used by Harris–Taylor to prove the local Langlands conjecture for , are of this type.

Hodge type

These are those Shimura data which admit a closed embedding (called a Hodge embedding) into a Siegel datum . From this embedding one is able to (non-canonically) define (via pullback) an abelian variety , and thus a motive, over .

In the context of Expectation (exp:Shim-var), if is the ‘universal motive with -structure’ over , then is roughy : the motive one gets by applying to the (chosen) Hodge embedding. But, we don’t know how to describe the motive for other representations of . In other words, we can non-conjecturally describe a motive on , but we aren’t able to non-conjecturally put -structure on it.

Example: Certain Shimura varieties , where is the general spin group, are considered in [Madapusi] in the course of the proof for the Tate conjecture for K3 surfaces.

Abelian type

I won’t try to give the rigorous definition of Shimura varieties of abelian type, but let me tell you the point. In the context of Expectation (exp:Shim-var), they are exactly those Shimura varieties whose associated motives with -structure take values in the full subcategory of abelian motives from §0.1.1.

Remark: As mentioned at the end of §0.1.2, abelian motives are objects that we can (at least conjecturally) define over smooth -varieties , using the category . In fact, using those, one can actually formulate and prove a version of Expectation (exp:Shim-var) as in [Milne7] and [Yang]. This is quite remarkable, and really lends some serious credence to Expectation (exp:Shim-var).

Unfortunately, for reasons mentioned at the end of §1, this is actually less useful than it first appears from an arithmetic perspective. One needs to beef up Expectation (exp:Shim-var) to handle, for example, -adic situations. That is why, in practice, even though Expectation (exp:Shim-var) does hold (at least up to blurring the lines between and ) it is rarely used.

Example: This contains Shimura varieties of toral type, the famous Shimura curves (compact analogues of modular curves), and many more. In some sense, Shimura varieties of abelian type are perhaps the most ‘practically useful’ class of Shimura varieties (besides those of ‘preabelian type’), in the sense that they are large enough to contain most every example, but tractable enough to work with.

General type

This is, in some sense, the wild west. Essentially nothing is known about such Shimura varieties from the perspective of Expectation (exp:Shim-var). In fact, in this case it’s not even clear (unlike the abelian case) how to single out what should be the smallest -linear -subcategory of , for a quasi-comopact smooth -variety, containing the motives that should show up in this context.

Example: Most famously, this contains the Shimura varieties with groups and .

§5 Integral canonical models

In this final section we discuss one of the most interesting arithmetic aspects of Shimura varieties: the existence of ‘canonical integral models’ over various rings of integers.

Setup

To state things correctly, we first need to fix some extra notation:

let be a place lying over
set (i.e., the completion of at ), a finite extension of ,
let be the ring of integers of , a finite extension of ,
and write .

We then want to define a certain ‘integral canonical model’ of over . In fact, at least for now, I want this model to actually also be quasi-projective and smooth.

The smoothness condition for this model is quite serious and, in particular, shouldn’t happen for most levels . Thus, we want to single out which sort of levels have a chance for giving us smooth models. This is encapsulated by the following definition.

Definition (defn:hyperspecial): A compact open subgroup is called hyperspecial if it is of the form for a reductive model of over .

Remark: As it turns out, all reductive models of over are abstractly isomorphic (see [Conrad, Theorem 7.2.16]). That said, this abstract isomorphism actually changes the associated hyperspecial subgroup . Thus, while there is only one reductive group scheme model of , there are generally several non-conjugate classes of hyperspecial subgroups of (e.g., see [Conrad, Example 7.2.15]).

As per usual, write . Then, we call a level of the form ,where and is hyperspecial, hyperspecial at .

“Conjecture”(Langlands, Milne,…): If is hyperspecial at , then there exists a “canonical” smooth quasi-projective -model of .

The reason that the word conjecture is in scare quotes is because of this wheedly word “canonical”. What does it mean? The most naive answer is “it should have the same moduli problem” (irrespective of Expectation (exp:Shim-var), I just mean this in a more naive sense). But, obviously this isn’t even approaching rigorous.

Let me give three possible better answers, none of which are maybe truly satisfying.

Answer 1: the Langlands–Rapoport conjecture

Recall from the Langlands–Kottwitz conjecture above, that the cohomology of Shimura varieties is meant to realize the global Langlands correspondence in some sense.

Of course, a natural question is: how does one even begin to approach proving such a statement? In fact, it’s helpful to instead step back and ask another even more fundamental question: how does one even understand any continuous Galois representations ? Namely, the source is a famously un-understandable group.

But, up to questions about semi-simplicitly (crucial/difficult ones we gloss over here!), there is a natural sort of numerical approach. Namely, by the Chebotarev density theorem the set

consisting of choices of Frobenius lifts at all finite places of is dense. Thus, it is enough to understand the set of numbers by the Brauer–Nesbitt theorem.

OK, this seems nice. But, how do we understand these numbers. Well, let’s now assume that actually comes the étale cohomology groups for some (for simplicity) smooth proper -scheme . Then, assuming has a smooth proper model at , it follows from the famous smooth proper base change theorem that

where is the usual (geometric) -Frobenius map, where the residue field has size .

We make one last simplifying change: instead of an indvidual cohomology group , we take the alternating sum . Thus, formally, we get

But, this last term is something that has some concrete meaning! Namely, it’s the famous result of the Grothendieck–Lefschetz trace formula that this is just the cardinality which, at least ostensibly, is something one may be able to actually realistically compute!

So, how does this connect back the notion of ‘canonicalness’ of the model ? Well, in short, Langlands (and later made precise by Langands–Rapoport, see [Milne8], which is why it is now called the Langlands–Rapoport conjecture) expect that such a model being ‘canonical’ should mean that there is a motivic description for each of the sets for all . And, more to the point, that this motivic description could lend itself to explicitly computing the size of this set. To what end? Well, that this point count is explicitly enough that, as part of the general discussion above, one could understand cohomology of Shimura varieties finely enough to prove the Langlands–Kottwitz conjecture.

Remark: There is a subtlety being elided here, but for which there is a brilliant trick to fix! What is the subtlety? It suffices to prove the Langlands–Kottwitz conjecture level-by-level, i.e., to show that for each level there is an isomorphism

where, as per usual, means fixed points. As the above indicates, one could try to prove this by performing point counting arguments at hyperspecial places of . But, of coruse, is not going to be hyperspecial for all places (i.e., the Shimura variety is not going to have good reduction at all places). So it’s not immediately evident that the above idea is enough.

But, the finer point of Chebotarev density saves the day. Not only are the powers of Frobenii at all places dense, but actually one can omit finitely many places and still maintain density. As is hyperspecial at for almost all (i.e., the Shimura variety is good reduction at almost all places), this is actually enough!

Answer 2: motivic expectation

Given Expectation (exp:Shim-var), it is tantalizing to imagine that one could qualify ‘canonicalness’ by giving a motivic description of the -points of for a -scheme . But, how do we make this preicse?

First, observe that as we want to be flat and separated over it suffices, by Yoneda, to restrict our attention to shemes flat and separated over . Then is (schematically) dense in . Thus, the map

is injective. The upshot of this is that the data of a map is actually a property about the associated map . Let us call this property good reduction. Thus, we see that giving a motivic understanding of boils down to giving a motivic criterion for the map to be of good reduction.

But, how do we give such a good reduction criterion? An answer may not be immediately forthcoming, but a good hint is given by our setup: we should consider the reductive model of . Namely, let us say that -motive with -structure over has good reduction if it has an integral model, i.e., there exists a -motive with -structure on such that .

A tantalizing conjecture may then be the following. To state it, let be the hypothetical universal -motive with -structure on , and let be the associated -motive with -structure.

Meta-conjecture:A morphism has good reduction if and only if the -motive with -structure has good reduction.

Of course, as of now, we are very far from making this precise: we don’t know how to define a -motive for a smooth -scheme, so the prospect of defininig a -motive on a -scheme seems entirely out of reach…or is it? We will discuss this more in subsequent posts.

Answer 3: an extension property

While historically there has not been a precise definition of what it means for a single model to be canonical, there has been a precise definition for what an entire system (i.e., as the level away from varies) of models .

Here by a ‘model’ of the system, I mean not only that we have models for each indvidual level, but we have models for the Hecke action. We say that this action is by finite étale morphisms if each and are finite étale morphisms (even integrally).

Definition (Langlands, Milne,…): A system of smooth -models is integral canonical if the Hecke action is by finite étale morphisms, and the following extension property holds: for every formally smooth -algebra , the natural map

,

is a bijection.

It’s a little bit hard to understand what this extension property, something like a Néron model property at infinite level, has anything to do with being a ‘canonical integral model’. In essence, one should think of it as encoding a sort of Néron–Ogg–Shafarevich property, as indicated by the following Siegel-type example.

Example (eg:ICM-Siegel): Let us fix and a prime . Then, the level

is hyperspecial at if and only if . In this case, the integral canonical model of the system is…the same thing. What I mean is that the exact same moduli problems for each such represents a smooth quasi-projective system over , and that together they form an integral canonical models. The only non-obvious point here is the following: why does the extension property hold? The anser: the Néron–Ogg–Shafarevich theorem (see [FaltingsChai, Theorem 6.7]).

Namely, suppose that we have a sequence of points in . Then, we have compatible isomorphisms for all . Passing to the limit we see, for example, that for any prime we have an isomorphism –an isomorphism of étale sheaves. In particular, being trivial means certainly that is unramified (in the appropriate suitable sense), and what the Néron–Ogg–Shafarevich theorem tells us is that this implies that has good reduction, i.e., it admits an abelian variety model over . It’s simple to check that both and (for all ) naturally lift to , and so the sought after point of is .

This example helps clarify the relationship betwen the extension property and the Meta-conjecture. Namely, through the lens of the Meta-conjecture the extension property says something simple: at infinite level the pullback of the motive always has good reduction. Indeed, in the Siegel-type case this is exactly the relevant consequence of the Néron–Ogg–Shafarevich criterion.

So, to underscore the point, one can think of the extension property as a distillation of this fact: it says something weaker than the Meta-conjecture–it doesn’t tell us which maps have the property that has good reduction at finite level, but declares that this always happens at infinite level.

Remark (rem:role-of-hyperspecial): A natural question is what role did the hyperspecialness play in the above discussion? Saying it rigorously is mildly difficult without a large diversion, but you can get the idea just by looking at the last step of Example (eg:ICM-Siegel). Namely, we said that the level structures naturally lift to level structures . The fact that this was possible was due to the fact that . In fact, if , then this moduli problem makes no sense: its special fiber would be empty as is never isomorphic to .

In general, -adic étale theory does not work well when is not invertible, and so any hope of having the ‘same moduli problem’ as one might hope for integral canonical models (e.g., as happened in Example (eg:ICM-Siegel)) will require that nothing can be happening at with the leve lstructure. And, as hinted at in Remark (rem:lvl-str), one should think that hyperspecial-at- level structure indeed corresponds to the ‘partial trivialization’ imposing no conditions at .

Known cases

Now, while the idea of integral canonical models is highly interesting, and potentially highly profitable, it seems almost impossible to imagine how to rigorously approach their construction. Indeed, as we have emphasized in this post, Shimura varieties, and particularly their arithmetic, are quite opaque. As we have now mentioned several times, even Expectation (exp:Shim-var) is not enough to save us as it cannot really deal with varieties over , let alone over .

Given this, it’s perhaps surprising to know that integral canonical models (in the system sense) have been constructed in a huge swath of cases.

Theorem: An integral canonical model of exists when is of:

Siegel type ([FaltingsChai], 1990),

abelian type ([Kisin], 2009),

general type when ([BST], 2024).

Remark: There is also the relevant work of Vasiu (e.g., as in [Vasiu]). See [Moonen, §5] for a discussion of this work and interesting ideas it contains.

So, in essence, the only remaining cases of Langlands integral canonical models conjecture (at least in the system of models sense) are the caes of general type Shimura varieties for small .

OK, so given the daunting nature of constructing arithmetic models of arithmetically-opaque, Shimura varieties, how were the above results proven? The answer, in incredibly broad terms, is as follows (with corresponding numbering):

classical (the models themselves are quite old even if the Néron–Ogg–Shafarevich theorem in the generality needed here is more recent),
type theory (this is done by a sophisticated devissage to the Siegel type case) and some non-trivial integral -adic Hodge theory (used to show smoothness of the construction),
very powerful algebraic geometry/-adic Hodge theory techniques which applies in a much more general setup, and for which applications to integral canonical models are (in some sense) an afterthough.

So, two basic questions which remain in the conjecture of integral canonical models are:

Question 1: How can one characterize integral canonical models at individual level (opposed to the fully system), and in a more motivic fashion, i.e., rigorously realize the above Meta-conjecture?

Question 2: How can one construct integral canonical models for general type Shimura varieties at small primes (preferably with some addtional motivic structure)?

The goal of this series of lectures is to, essentially try to give one answer to Question 1.

§6 A toy example

In this last section I just want to give an idea for what an answer to the above-stated Question 1 might look like in the simplest of all cases: the case of modular curves.

To this end, I want to state a very contrived-sounding question which serves as a warmup to Question 1. The explicit relationship to Question 1 will be follwed up on in subsequent posts.

Question 1′: Can one characterize the integral canonical model of modular curves using only ‘at data’?

Statement of the theorem

Let me tease out the meaning of Question 1′ a bit more. For simplicitly, let us write instead of . Then, as indicated in Example (eg:ICM-Siegel), the integral canonical model of the modular curve (where and ) over is…the same modular curve.

Namely, it is

In other words, has the identical moduli problem to except the elliptic curve in the former is only allowed to live over a scheme, whereas the elliptic curve in the latter is allowed to live over any -scheme (thus the change in notation).

Given this, the answer to Question 1′, at least at first blush, is obivously no. Very clearly the moduli problem for needs the full elliptic curve and this doesn’t just contain ‘at -information’, e.g., it contains the information of the -adic Tate module for any prime . But, a close inspection shows that the Meta-conjecture does posit something exactly like this: our criterion for good reduction didn’t involve the -motive with -structure, but only its base change to . So, let’s forge ahead undeterred.

To give a better sense of what ‘at information’ means here, let me list two objects associated to .

The -adic Tate module:

Let be the universal family of elliptic curves over . Set

a rank two -local system.

It is useful to note that from the perspective of Expectation (exp:Shim-var), is nothing more than the -adic realization of the universal motive -motive with -structure (i.e., )

The -divisible group

Let be the universal family of elliptic curves over and let be its associated -divisible group. From the perspective the Meta-conjecture, one should view this as the -motive which ‘models’ .

Remark: For reaspons of space I won’t try to justify this last claim here, as it will be the focus of future posts. But, in essence, one should think of -divisible groups as being something like -motives with -coefficients, i.e., that they’re like the ‘at version of abelian varieties’. Thus, the object really is motivic in nature.

We are now ready to state our answer to Question 1′, using only the above two pieces of at data. To state the result it will be convenient to give ourselves a little more notation:

denote -adic completion of , thought of as -adic formal scheme,
as the moduli space of -divisible groups
the map determined by (explicitly, it associates to a point of the -divisible group ).

Theorem (toy example):The following two properties of hold, and uniquely characterize it as a separated model of :

Serre–Tate: the morphism is formally étale,

Crystalline locus: a (classical) point of the rigid analytic space lies in the open subset if and if is crystalline.

There’s a lot to unpack here, and it will require us to to finish the next two posts to fully flesh things out. But, for a bare-bones understanding, let me just say the following two things:

The ‘formal étaleness’ in the Serre–Tate statement is the usual one algebraic geometers are familiar with. But, roughly, just means that the map establishes an identification between the deformation theory of and .
The crystalline locus statement involved the phrase “ is crystalline”. Here one can think of the ‘classical point’ as just an -point of for a finite extension . Then, the -local system pulled back to can be thought of a as the -adic Galois representation of given by .

It is then a classical notion in -adic Hodge theory for such a representation to be crystalline which, in broad terms, one should think of as saying that the local system has ‘good reduction’. We will discuss this idea more in depth in future posts.

What is a scheme over anyways?

Perhaps the most in-your-face part of Theorem (toy example) which is not intially expected is the amount of formal and rigid geometry. We don’t have time to really discuss either of these topics seriously. For that I wholeheartedly recommend the survey [FujiwaraKato] (or this less useful previous post).

Instead, I just want to take a moment to explain why formal and rigid geometry ought to show up here. After all, we were after a model of the scheme over –who decided to introduce -adic analytic things into the mix? My contention is, in fact, that it is impossible to avoid it.

To help understand this claim, let me ask the reader to picture a scheme over . Perhaps the most common picture for such a thing is the following:

You have a base space with two points, and , and your fibers over those two points.

I claim that a much more accurate picture of the situation is, instead, the following (taken from [AchingerYoucis]):

The specifics of the symbols here don’t matter, and neither does the base really. Instead focus on the melting flan on top. The

base of this melting flan is a scheme over ,
the body of the flan constitutes a formal scheme over ,
the scheme in its totality is then the result of gluing to along the open region in labeled or, maybe more commonly written, –the Raynaud generic fiber of .

But, just to nail the point home, this open region we are gluing along is rigid analytic, i.e., it only exists as an open subset of the analytic space associated to .

The upshot of this is the following: a -scheme may be thought of as the result of gluing the formal scheme to the -scheme along the common rigid analytic (i.e., adic space) open subset , which we may call the anchor set of the gluing.

One may equationally express this idea as follows:

This is not pure fantasy either, as can be made precise by the main theorem of [AchingerYoucis] which says that one can, in fact, (up to enlarging to algebraic spaces) perform the reverse: any formal -scheme glued to a -scheme along rigid analytic open embedding is an algebraic object!

The triple is what we call a gluing triple. So, in words, the main theorem of [AchingerYoucis] says that (up to subtleties related to algebraic spaces) schemes over are the same as gluing triples.

Example (eg:gluing-triple-A1): One can physically see the rigid geometry at work even in the simplest example: . Indeed, what is the connection between and its generic fiber ? Well, perhaps the most obvious one is . But, I hope the reader won’t mind me being cute and pointing out

A rigid analytic open subset has appeared, and it’s not a coincidence or trick.

From the gluing triple perspective, the rigid and formal geometry appearing in Theorem (toy example), comes into focus. Characterizing as a model of only requires two pieces of information as one part of the gluing triple is fixed as part of the setup. Those two pieces of data are

the rigid analytic open anchor set in we want to glue along (which ultimately is ),
a way to characterize the formal scheme (which is ultimately ) we will glue to along this anchor set.

From this perspective the crystalline locus part of Theorem (toy example) is exactly addressing the first of these peices of information (i.e., pinning down the anchor set), and the Serre–Tate theorem is (perhaps non-obvious) addressing the second (by characterizing the formal scheme we will glue to the anchor set).

In fact, we also see the inevitability of rigid geometry even in the Meta-conjecture. Namely, the natural generalization of Example (eg:gluing-triple-A1) fact always holds: if is a scheme over , then the subset

is always rigid analytic and, moreover, an analagous statement holds with replaced by any scheme over it. But, from this perspective, Meta-conjecture naturally will involve rigid geometry as this image set is (by construction) the good reduction locus: so the good reduction locus is always a rigid analytic object.

In some sense, this is what the crystalline locus part of Theorem (toy example) was telling us, but it perhaps is a little hard to internalize if you aren’t familiar with the notion of crystalline. So, instead, let me give a much more visually clear illustration of the fact that the good reduction locus in is a rigid-analytic object.

Recall that there is a -invariant map . There is then a famous result that an elliptic curve has (potentially) good reduction if and only if it’s -invariant has norm at most one. In symbols this says

a patently rigid-analytic open subset.

References

[AchingerYoucis] https://arxiv.org/pdf/2410.20500

[AGHM] Andreatta, F., Goren, E., Howard, B. and Madapusi Pera, K., 2018. Faltings heights of abelian varieties with complex multiplication. Annals of Mathematics, 187(2), pp.391-531.

[BST] https://arxiv.org/pdf/2405.12392

[Baez] https://arxiv.org/pdf/2304.08737

[Broshi] Broshi, M., 2013. G-torsors over a Dedekind scheme. Journal of Pure and Applied Algebra, 217(1), pp.11-19.

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[FujiwaraKato] Fujiwara, K. and Kato, F., 2006. Rigid geometry and applications. Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math, 45, pp.327-386.

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[Kottwitz1] Kottwitz, R.E., 1984. Shimura varieties and twisted orbital integrals. Mathematische Annalen, 269(3), pp.287-300.

[Lan] https://www.kwlan.org/articles/intro-sh-ex.pdf

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[Milne2] https://www.jmilne.org/math/articles/1994bP.pdf

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[Milne4] https://www.jmilne.org/math/articles/DT.pdf

[Milne5] Milne, J.S., 1983. The action of an automorphism of C on a Shimura variety and its special points. In Arithmetic and Geometry: Papers Dedicated to IR Shafarevich on the Occasion of His Sixtieth Birthday Volume I Arithmetic (pp. 239-265). Boston, MA: Birkhäuser Boston.

[Milne6] https://www.jmilne.org/math/articles/1990aT.pdf

[Milne7] https://www.jmilne.org/math/xnotes/svh.pdf

[Milne8] https://www.jmilne.org/math/xnotes/Montreal.pdf

[Moonen] https://www.math.ru.nl/~bmoonen/Papers/SMCfinal.pdf

[Morel] https://fhoermann.org/shimura2.pdf

[Rosen] Rosen, M., 1986. Abelian varieties over C. Arithmetic geometry, pp.79-101.

[Serre] https://www.numdam.org/item/SCC_1958-1959__4__A10_0.pdf

[Vasiu] Vasiu, A., 1999. Integral canonical models of Shimura varieties of preabelian type. Asian Journal of Mathematics, 3, pp.401-517.

[Yang] https://arxiv.org/pdf/2304.10751

Shimura varieties and their canonical integral models (Pt. 1)

alexyoucis — Sun, 13 Apr 2025 07:22:18 +0000

This will (hopefully) be the first in a series of four posts based off a lecture series given at the Morningside Center of Mathematics in Beijing. The goal overall is to talk about some recent advances in the -adic geometry of Shimura varieties and their integral canonical models. The goal of this (essentially standalone) post more specifically is to give a sort of broad overview for what Shimura varieties and their integral canonical models are, and why one should care about them.

It is a fool’s errand to try and define Shimura varieties in any sort of rigorous way in such a small amount of time, and I will make zero attempt to do so here. For that I can only point to other references, some of which I list here: [Deligne1], [Deligne2], [GenestierNgo], [Hida], [Hoermann], [Kerr], [Lan], [Milne1], [Milne2], [Moonen], [Morel].

Disclaimer: For the sake of exposition, the below will take many non-trivial liberties, most of which are about ideas for which it’s difficult to even make rigorous sense of. For this reason, I caution the reader for taking anything written here too seriously, and to treat it only as vague motivation. I apologize in advance to any experts who take issue with these inaccuracies.

§0 motives and objects with -structure

Below we will frequently make use of the ideas of ‘motives’ and objects with ‘-structure’. We take a sort of primitivist approach to these notions, not trying too seriously to even describe the desiderata they are meant to satisfy. That said, to help orient the unfamiliar reader, let me say a few words below on each.

§0.1 Motives

(I cannot possibly do justice to the idea of motives here, especially in addressing the subtleties for what such a fantastic theory would entail. For that I can only suggest consulting the much more enlightening references [Milne3] and [Baez] (and especially the references discussed in §5 of the latter).)

For the uninitiated, one can think of the category of of -motives on (for a ring and scheme ) as being the receptacle for an ‘optimal’ cohomology theory

on the category of smooth projective -schemes. Namely, we expect that should be something like an -linear -category. Roughly this means you can -linearly add morphisms in , take tensor products in (in a way that is -linear although I won’t demand the tensor product is ‘over ‘), and that you can make sense of exact sequences in .

The ‘optimality’ here means that for any other ‘reasonable’ cohomology theory , where is (for simplciitly) an exact -linear -category, there should be a factorization

where is some ‘realization functor’. So, any extra structure that the category could contain (Galois action, filtration, Hodge structure,…) should somehow already be captured in , thus making the ‘richest category of cohomological invariants’.

Again, we are taking a very primitivist perspective here, mostly out of necessity. Motives in any naive sense like I am trying to describe them ‘less than exist’: forgetting their existence, there isn’t even a guess of what the category is (or what properties it should satisfy), especially over the type of general bases that I have allowed here. I just want to mention two cases at which one can gesture towards motivic thinking in the way I mean it here.

Remark: For the sake of some readers’ sanity, let me point out that there do exist precisely defined notions of ‘motives’, e.g., Voevodsky motives. The relationship to the idealized vaugery of motives we mean here is largely conjectural, especially over general bases . So, somehow the ‘rigorously defined’ motives considered in, for example, motivic cohomology, don’t directly facet into our discussion.

§0.1.1 Abelian varieties

The category should carry something like a ‘weight grading’ which, in rough terms, should be related to the degree of cohomology, i.e., which , we are taking. For -motives with coefficients in (i.e., ) there is a very rough idea for what the category of -motives should be: abelian schemes.

Namely, there should something like a fully faithful embedding

where the source is the isogeny category of abelian varieties (i.e., the localization of along the class of isogenies. The map

should very roughly send to its Albanese variety.

This idea stands up to some amount of scrutiny via the fact that first cohomology groups of most cohomology theories depend only on the associated Albanese variety:

In other words, one roughly has , in line with the ‘optimality’ we would want to satisfy.

Remark: On the other hand, some readers may also see how abjectly absurd the above is in its oversimplicity. The Albanese variety does not exist for arbitrary bases (although it does exist over reduced varieties over an algebraically closed field as in [Serre]). Moreover, when it exists, the Albanese variety is usually not an abelian scheme itself, but a torsor for an abelian scheme. So, at best, one may believe that (isogeny classes of) abelian varieties do fully faithfully embed into the category of -motives with -coefficients. But even more dire is that the fact that factorizes through the Albanese does not always hold for all bases and all cohomology theories. Again this discussion of motives is meant more for flavor and not so much for precision.

Now, one cannot perform linear algebra operations in like tensor product, i.e., it’s not an exact -linear -category, and this property will be quite important when discussing -objects below. But, as does enjoy these properties, one can form the smallest exact -linear -subcategory of containing . This is called the category of abelian motives over , and is denoted .

§0.1.2 Archimedean Hodge theory

It is a classic observation that comes for Archimedean Hodge theory, that one has a functor

where is the category of (pure) -Hodge structures. In other words, the singular cohomology of the underlying projective complex manifold , together with the Hodge decomposition of , allows one to build from a -Hodge structure. Now, the theory of motives would dictate that there should be a realization functor

such that . Moreover, one can essentially interpret the Hodge conjecture as saying that is fully faithful.

This extends to the relative situation, where for a smooth -variety one can define the category of variations of -Hodge structures, which are like a holomorphically varying family of objects for . One then has a functor

where is the induced morphism of complex manifolds. One also expects from the Hodge conjecture that the realization functor in this relative setting is fully faithful.

Let me emphasize the upshot here. The Hodge conjecture predicts that the category of -motives over a quasi-compact smooth -variety are a full subcategory of (variations of) -Hodge structures. This means that, at least conjecturally, all possible cohomological information that could be contained in a smooth projective -scheme is already contained in its Betti cohomology. Thus, in some sense this shifts our attention from constructing the category of motives in some abstract sense, and instead to singling out the ‘correct’ subcategory of Hodge structures. In particular, this gives a sort of ‘cheating’ way to get at motives: we can instead work with Archimedean Hodge theory as a substitute…at least for smooth -varieties.

To see how this connects with abelian varieties, we recall the following result of Deligne.

Proposition [Deligne3, Rappel (4.4.3)]: Let be a quasi-compact smooth variety over . Then, there is an equivalence of categories

.

Here the target category is the category of variations of -Hodge structures which are

polarizable,
of type –this just means that these are the only pairs of that show up in the Hodge decomposition.

Remark (rem:Deligne-explained): This theorem can be made a lot less daunting in the case , and it further helps explain this polarizability condition. Namely, one can check that if is a finite free -module, then a -Hodge structure of type on it amounts to a complex structure on , in particular the rank of is even, say . One may then consider the quotient which is an abelian variety of dimension …or is it!

In the case of dimension (i.e., ) this truly is the case as all such lattice quotients are elliptic curves. But, for 1" class="latex" /> these are, in general, non-algebraizable complex tori. The exact condition needed to guarantee that they algebraizable is that this complex torus possesses a Riemann form (see [Rosen, §3]) which amounts to the claim that the associated Hodge structure is polarizable.

In particular, Deligne’s theorem tells us that gives us a fully faithful embedding of into . Then, we can apply this ‘cheating’ mindset here. Above we defined the full subcategory of abelian motives, i.e., the -linear -category generated by abelian varieties in . By the Hodge conjecture this should be identified with its image in and agree with the -linear -category generated by in . But, we can actually rigorously define this last category!

To underscore this, if we take , for a smooth -variety , to be the full -linear category generated by in , this is a non-conjectural way of defining what ought to be equivalent to category of (true) abelian motives over . But, of course, the category , unlike , is non-conjectural!

§0.2 Objects with -structure

Recall as above that for a ring an exact -linear -category is one where you can -linearly add morphisms, take tensor products, and that you can make sense of exact sequences.

The key example of this is as follows.

Example: Let be a (finite type, affine, smooth) group scheme over . Then, the category of (finite -projective) representations of is an -linear exact -category in the obvious way.

Definition (defn:H-objects): The category of -objects in is the category of exact (i.e., sends exact sequences to exact sequences) -linear -functors

.

Perhaps the most illuminating example of this idea is the following.

Example (e.g., see [Broshi]): Let be a scheme over a DVR . Then, the category of vector bundles on is an exact -linear -category in the obvious way, and identifies with the category of -torsors on .

In particular, I would implore the reader to think of -objects in (at least up to first approximation) as being something like ‘-torsors in ‘.

§1 A dreamland: moduli of motives

We begin by trying to describe Shimura varieties as they ought to be. As is indicated by the notation, such varieties are built using the input of the data and . The first of these, the more important of the two, is the following.

Definition (defn:shim-data): A Shimura datum is a pair consisting of

a reductive group over ,

,

such that … . Its Hodge cocharacter is defined by for any .

A morphism of Shimura data is a morphism of -groups such that , and is a closed embedding if is.

Let me make some comments on some of the structures appearing in this definition:

The group (i.e., viewed as a real algebraic group) is the so-called Deligne torus. It is the Tannakian fundamental group of the Tannakian category of real Hodge structures (with the usual fiber functor).

So, is naturally identified with (the notation means ‘isomorphism classes’). In particular, one may view the in a Shimura datum as being an isomorphism class of real Hodge structures equipped with -structure.
One has a natural (up to normalization) identification thus the notation on the right-hand side of makes sense.

From the perspective that is an isomorphism class of objects in , one may view as the (conjugacy class of) cocharacter(s) defining the Hodge filtration on these real Hodge structures, whence the name.

We are actually already almost at the naive definition of , and only need one extra piece of notation/terminology.

Definition(defn:reflex-field): The reflex field of the Shimura datum is the field of definition of .

Remark: There is a small subtlety about the phrase ‘field of definition’ for a conjugacy class of cocharacters. Namely, it is possible that the action of on such conjugacy classes fixes as a conjugacy class, but fixes no actual element in . In other words, there needn’t actually be a cocharacter in defined over . That said, this subtlety disappears (see [Kottwitz1, Lemma 1.1.3]) with the mild assumption that is quasi-split (i.e., has a rationally-defined Borel).

So, we are already ready to try and ‘define’ what is. Namely, it is meant to be a smooth quasi-projective -variety, and thus (by Yoneda) to characterize it, it is enough to tell you the value for a smooth quasi-compact -variety .

Expectation (exp:Shim-var): For a smooth quasi-compact -variety ,

To try and explain what the meaning of these extra terms ‘type ‘ and ‘étale -level’ structure means, let us set up some notation. Namely, a -motive on with -structure is an exact -linear -functor . We then obtain a diagram as follows

Here

is the -adic étale realization functor, so the output is an étale -local system,
is the Hodge realization functor, so the output is a variation of -Hodge structure .

We are then defining and to make the diagram commute (i.e., they are -adic and Hodge realization of , respectively).

We can then describe the above-mentioned conditions on as follows:

Type : This means that for every -point of one has that , i.e., that the pullback to of the Hodge realization of is isomorphic to in , i.e., we’re fixing the pointwise isomorphism class of our real Hodge structures to be .

Note that as is defined over , there is an action of on , and so the above condition should be stable under such an action. But, as is defined (roughly) to be the (smallest) field where the isomorphism class of makes sense, this tracks.

Étale -level structure: Here we are fixing a compact open subgroup (where ). Then such a level structure is a global section of the sheaf

Remark (rem:lvl-str): Let us tease this out slightly more. Here is the trivial -local system with -structure: it associates to a representations of the constant local system . One may then think of as the sheaf of ‘simultaneous trivalizations’ of the as varies. This naturally has an action of as the automorphisms of as a collection of -local systems with -structure is . Thus, we can take the quotient sheaf , and an étale -level structure is a global section of this.

This may be a bit hard to grok on first read, but the idea is roughly the following. As each is a -local system one can trivialize it (with -structure) if one works ‘very locally’, e.g., over some masive pro-étale local cover . Now one does not expect any particular (family of) trivialization(s) over to descend to (i.e., you don’t expect to be trivial over ). But, descending a -orbit of such trivializations has such a chance because this means that no particular trivialization on the nose descends, but it does up to ‘blurring’ (i.e., up to changing it by the action of ).

The result of this, roughly, is that this allows only a ‘partial trivialization’ of . For example, as is compact open you know (roughly) that it corresponds to a product of compact open subgroups of as varies, and for almost all it must be something like for almost all . At these ‘hyperspecial’ places (more on this later) one can convince themselves that this partial trivialization says nothing: there is no condition on . For other ‘non-hyperspecial’ , a typical example of such a is , in which case one can roughly think that the ‘partial trivialization’ at is not a full trivialization of but a trivialization of (i.e., the reduction modulo of some -lattice in ). This is not a perfectly rigorous explanation, but it should hopefully give a sense for what’s going on.

Expectation (exp:Shim-var) should immediately indicate to you why Shimura varieties are so important in arithmetic geometry: they are moduli spaces for motives, the object of centralmost importance in the linear algebra study of varieties. That said, this expectation is just that, an expectation:

As mentioned in §0, we don’t have any rigorous definition of , and so we’re foiled at the very first step of trying to make our expectation even a rigorous conjecture. But, there is some hope as gives rise to a smooth -variety , one might apply Archimedean Hodge theory to approach things, as indicated in §0.1.2, especially in special (abelian type!) cases.
The hope of using Archimedean Hodge theory to put Expectation (exp:Shim-var) on more firm footing (as mentioned in 1.) is nice, but, in fact, the expectation itself is not nearly good enough to actually work with Shimura varieties as arithmetic-geometric objects. Namely, it doesnt’t cover the case when is smooth over a -adic field, which would be pivotal for studying (the base change of) Shimura varieties over fields like , a cornerstone of our modern approach to arithmetic geometry.

In this -adic realm Archimedan Hodge theory can do nothing to help us. But, there is still some hope to work -adically: if Archimedean Hodge theory can’t work in the -adic world what about…-adic Hodge theory? More on this later.

Remark (conditions-on-(G,X)): In our ‘definition’ of Shimura datum, I never actually stated the conditions on they are required to satisfy. The reason for this is that they are opaque. But, let me vaguely say their purpose, and take this up again in the next section.

Namely, there is no a priori reason to believe that the functor described in Expectation (exp:Shim-var) should be representable by a scheme, i.e., that it should be algebraic. For example, our definition involves Archimedean Hodge theory, which is an inherently analytic theory–maybe this space should only exist analytically (in an appropriate sense)? Roughly these extra unstated conditions on are to guarantee that this algebraicity holds.

§2 Harsh reality: the lifecycle of a Shimura variety

The dreamland from the last section must now collide with the grim meat-hook reality of Shimura varieties. Namely, while Expectation (exp:Shim-var) is beautiful it is, as we said, not workable in our current understanding of things. That said, Shimura varieties themselves aren’t conjectural: they are flesh-and-blood mathematical objects. So, what do they actually look like?

Remark: The more astute reader will notice that I am not making any assumptions on below. Technically one needs to assume that is a so-called neat subgroup to really get an algebraic variety. This is a technical point though that I will ignore. It also seems well-known among experts that this is somewhat of an illusory problem, and could be remedied by working with stacks in the appropriate categories at all stages. A similar issue (and solution) holds for the distinction between and , for those that know what that means.

The construction, in steps

Let me describe, in the roughest of possible terms, the creation of Shimura varieties in four steps.

Step 1: real manifold

Shimura varieties start out life as something quite far from a smooth quasi-projective algebraic variety over the number field : they start out as real manifolds.

Before we state this precisely, let us observe that as is a -conjugacy class we may write it as for some subgroup . In fact, the (unstated) conditions on force to be the unique (up to conjugacy) maximal compact subgroup of . Regardless, it’s evident that inherits the structure of a real manifold from .

We then set

Here , and is acting diagonally on the product in the first expression whereas is acting only on the factor of the second expression.

Remark: As indicated by the notation , this is the -points of the eventual . In particular, this looks miles away from Expectation (exp:Shim-var), but is slightly closer than these first appearances.

Namely, if is a -motive with -structure as in Expectation (exp:Shim-var) (written there) then you could imagine considering the pair

As we’re over , the étale -level condition is literally just an element of the set

Moreover, determines an object of we know that it must belong to, by the Type condition, the isomorphism class . So, it corresponds to an element of .

As we are interested in isomorphism classes of such , the pair . is in fact only well-defined up to isomorphism of -motives with -structure, i.e., up to the of by . Thus, we see that we only get a well-defined class in

The sort of faith-based part of this then is that the theory of motives should imply that this association does actually form a bijection between this double-quotient set and the set of isomorphism classes appearing in Expectation (exp:Shim-var).

Step 2: complex manifold

All connected components of are isomorphic, and so we fix one . One may then write

where the index set is finite and each is an ‘arithmetic group’ (i.e., the intersection of a compact open subgroup of with ). Up to the finiteness of this index, this is just a simple exercise in writing this double quotient as a union of single quotients.

Thus, to put a complex manifold structure on is tantamount to putting a complex manifold structure on the real manifold . And, this we can specify (and prove) using the (unstated) conditions on in a pleasing way.

Proposition (prop:cmplx-str): There exists a unique complex manifold structure on such that is an object of , i.e., a variation of real Hodge structures.

In other words, we have sort of tailor made to be a period domain (i.e., a moduli space of Hodge structures, see [CMSP]).

Remark: Let me try to make this feel a little less abstract. Fix an element of the conjugacy class . This determines a unique parabolic (e.g., see [Conrad, Theorem 4.1.7]). The associated (partial) flag variety is the -variety

As , this space classifies the parabolics conjugate to . The isomorphism class of this partial flag variety doesn’t depend on the choice of , so we denote it by .

Observe then that we have a natural map of real manifolds

sending to the conjugate parabolic . Then, ultimately, Proposition (prop:cmplx-str) amounts to the claim that there exists a unique complex manifold structure on such that the map is holomorphic (for any choice of ).

Step 3: complex variety

Now comes what, to me, is perhaps the most miraculous part of the construction of Shimura varieties (although this is probably a function of my own mathematical weaknesses).

Theorem (Baily–Borel): There exists a unique smooth quasi-projective -variety with underlying complex manifold such that for any other smooth quasi-projective -variety the natural map

is a bijection.

In words this says that there is a unique way to algebraize (to a smooth quasi-projective -variety) such that any holomorphic map , for a smooth quasi-projective -variety , uniquely algebraizes.

One could perhaps view this through the lens of Expectation (exp:Shim-vars) as saying that the (unstated) conditions on force the type of motivic objects being classified to automatically be algebraizable, e.g., compare this to Deligne’s theorem from §0.1.2.

Remark: Let me try to make this ever so slightly less abstract. For a representation of one can build a holomorphic vector bundle on the complex manifold :

where is acting diagionally on , where it’s action on is through the inclusion , and still only acts on .

(Such a vector bundle is called an automorphic vector bundle. They in fact, as varies, define an object of , i.e., a -torsor on .)

Take for a -dominant weight such that 0" class="latex" />. One can show that

defines a normal, connected, projective -variety (although almost never smooth!), called the Baily–Borel compactification of the Shimura variety. Moreover, it’s not hard to see that there is a holomorphic map

The minor miracle is that this is actually a Zariski open embedding, and so defines a quasi-projective complex variety structure on . Moreover, by a hyperbolicity argument (again related to the unstated conditions on ) this has the desired property as in Baily–Borel’s theorem.

Step 4: descent to reflex field

We now come to the most opaque part of the construction, the one that makes the arithmetic-geometry of Shimura varieties so complicated: the descent from a -variety to a -variety.

Remark: Often one thinks of Shimura varieties as varieties over . From this perspective, the descent to is not part of the definitional structure, but a choice: there could be several models over . Which one do we choose? The one we will describe now is the so-called canonical model. We will see below why this name is justified.

By quasi-projectivity it turns out (see [Milne4]) that giving such a descent is (essentially) equivalent to defining a (‘continuous’) action of the group on .

The definition of this action is ‘done’ in two steps:

Step 4.A:

When is of a very simple type, namely toral type, i.e., for a -torus , one can define this action using class field theory.

Remark: Let me slightly expand on this. As is a point, is a finite disjoint union of copies of . This automatically has a unique model over (something not guaranteed) by taking the corresponding disjoint union of copies of .

We then need to define an action of on this -model. Class field theory gives us a map

As is defined over (by definition!) we get a map

where this second map is the norm map for . Combining these two maps gives us a map

and the action on is now clear.

Step 4.B:

We now bootstrap from Step 4.A. More precisely, we show that there are enough special points, i.e., points in the image of a map of the form

for some toral Shimura variety (see Proposition (prop:canonical) below for how such a map is defined). If you have enough such special points, it’s believable that one can uniquely specify an action of using Step 4.A as input.

Remark: The above description of Step 4.B is an immense oversimplification. First, it’s not even clear what enough means. The reflex fields of these special points will (in general) be bigger than , but one is still somehow hoping that you get enough juice out of the special points as you vary them (i.e., the interesection of their reflex fields is close to ).

Secondly, this may help you determine what the action must be, but it doesn’t help you necessarily prove it exists. Step 4.A forces how acts on special points, and having enough such special points might by some ‘continuity argument’ force the action to be unique if it exists, but how do you extend the action beyond the special points?

It is perhaps telling that this last step was not completed until 1983 by Borovoi and Milne (see [Milne5]), nearly 15 years after the introduction of Shimura varieties. Moreover, the ultimate solution is still quite ‘opaque’–one does not ever really get an explicit description of the action of which is (one of) the culprit(s) for the arithmetic mystery of general Shimura varieties.

The output

Let us now try to explicitly describe what the output of the above procedure is: what actual structure do you get out of the theory of Shimura varieties.

The individual Shimura varieties

To begin with, let us state the obvious. For each Shimura datum and each level structure one gets a smooth quasi-projective variety over the reflex field .

The fact that these models are the right ones, thus justifying the name ‘canonical model’, is that they satisfy the following amazing functoriality property.

Proposition (prop:canonical)[Deligne]: Suppose that is a morphism of Shimura data. Then, for any levels with the map of sets

,

descends uniquely to a map of varieties over the compositum of the reflex fields:

.

Remark: On the one hand, this functoriality is quite amazing as the set-theoretic map underlying this map of -varieties is of an Archimedean-Hodge-theoretic flavor. So the fact that they admit arithmetic models is incredible. On the other hand, it’s not that surprising technically. Namely, it is easy to check by hand that this result holds for toral-type Shimura varieties. This then essentially concludes the proof as descending the map is the same as being equivariant for the action of and this action is determined on special points.

From the perspective of Expectation (exp:Shim-var), it’s clear what this map should do. Given a -motive with -structure on , we get a -motive with structure via the rule:

i.e., if is a representation of , then is a representation of and so it makes sense to apply to it. This is of type by our assumption that , and assuming it’s easy to see that an étale -level structure on induces an étale -level structure on .

The Hecke action

The second, and in some sense equally important, output of our construction of Shimura varieties is the so-called Hecke action. This takes the form of the existence of two types of special maps between the Shimura varieties for at different levels.

The maps

For any element of and level , one can produce an isomorphism

which on -points is given by

One again exploits the canonical model structure to show this map is defined over .

From the perspective of Expectation (exp:Shim-var), the map is the result of using to modify the étale -level structure (to an étale -level structure).

The maps

For any normal containment of levels one builds a a projection map

which on -points is given by

and is a Galois cover with Galois group (with a coset acting as ).

Remark: The phrase ‘Galois cover’ often assumes that the source and target is connected. This is not the case here. So, what I really mean is that is a torsor for the constant group scheme associated to the finite group .

From the perspective of Expectation (exp:Shim-var), the map takes an étale -level structure and applies the ‘blurrification’ obtained by taking the associated étale -level structure, which exactly forgets -worth of information.

The action at infinite level

It is often useful to combine the information of the maps and into one package. Namely, we can consider the -scheme

with the transition maps given by . As these transition maps are finite (and thus affine), this actually does exist as a scheme over . Moreover, as we are limiting over all levels, one actually sees that each element of gives a morphism of -schemes

This defines a continuous action of on in the sense of [Milne6, §II.10].

Given Expectation (exp:Shim-var), one would expect that for a smooth quasi-compact -variety that

From this perspective the map is just modifying this trivialization, and the projection maps

are just sending such a trivialization to its class in .

Hecke correspondences

One often hears of ‘Hecke correspondences’ for Shimura varieties, and so it might be helpful to explain how this facets into the theory. Namely, suppose that we have two different levels and in . Then, there is no direct way to relate them by the maps and , but there is always a relate them via a correspondence. Namely, we obtain the following correspondence

from to , where we have written for simplicity. This is often called a Hecke correspondence.

Remark: The reason that Hecke correspondences are useful is because of their relationship with cohomology, which is of great importance as we will see in §3.2.1 below. Namely, taking in the definition of the Hecke correspondence allows one to build (via standard theory) a morphism of cohomology groups

compatible with -action.

In fact, one can understand this action in another useful way. Namely, the action of action on defines an action of on . Moreover, for any level there is a natural identification

From standard representation theory of locally profinite group this endows the cohomology group with an action of the Hecke algebra . As is well-known, this algebra is generated by indicator functions of the form , and the action of this function on the cohomology group agrees with that coming from the Hecke correspondence above.

Of course, this all works for more general coefficients–so-called, automorphic local systems.

References

[AchingerYoucis] https://arxiv.org/pdf/2410.20500

[AGHM] Andreatta, F., Goren, E., Howard, B. and Madapusi Pera, K., 2018. Faltings heights of abelian varieties with complex multiplication. Annals of Mathematics, 187(2), pp.391-531.

[BST] https://arxiv.org/pdf/2405.12392

[Baez] https://arxiv.org/pdf/2304.08737

[Broshi] Broshi, M., 2013. G-torsors over a Dedekind scheme. Journal of Pure and Applied Algebra, 217(1), pp.11-19.

[Conrad] https://math.stanford.edu/~conrad/papers/luminysga3.pdf

[CMSP] Carlson, J., Müller-Stach, S. and Peters, C., 2017. Period mappings and period domains. Cambridge University Press.

[Deligne1] https://www.numdam.org/item/SB_1970-1971__13__123_0.pdf

[Deligne2] https://publications.ias.edu/sites/default/files/34_VarietesdeShimura.pdf

[Deligne3] https://www.numdam.org/item/PMIHES_1971__40__5_0.pdf

[FujiwaraKato] Fujiwara, K. and Kato, F., 2006. Rigid geometry and applications. Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math, 45, pp.327-386.

[GenestierNgo] https://www.math.uchicago.edu/~ngo/Shimura.pdf

[Hida] Hida, H., 2004. p-adic automorphic forms on Shimura varieties. New York: Springer.

[Hoermann] https://fhoermann.org/shimura2.pdf

[Kerr] https://www.math.wustl.edu/~matkerr/SV.pdf

[Kisin] Kisin, M., 2010. Integral models for Shimura varieties of abelian type. Journal of the American Mathematical Society, 23(4), pp

[Kottwitz1] Kottwitz, R.E., 1984. Shimura varieties and twisted orbital integrals. Mathematische Annalen, 269(3), pp.287-300.

[Lan] https://www.kwlan.org/articles/intro-sh-ex.pdf

[Madapusi] Madapusi Pera, K., 2015. The Tate conjecture for K3 surfaces in odd characteristic. Inventiones mathematicae, 201(2), pp.625-668.

[Milne1] https://www.jmilne.org/math/xnotes/svi.pdf

[Milne2] https://www.jmilne.org/math/articles/1994bP.pdf

[Milne3] https://www.jmilne.org/math/xnotes/MOT.pdf

[Milne4] https://www.jmilne.org/math/articles/DT.pdf

[Milne6] https://www.jmilne.org/math/articles/1990aT.pdf

[Milne7] https://www.jmilne.org/math/xnotes/svh.pdf

[Milne8] https://www.jmilne.org/math/xnotes/Montreal.pdf

[Moonen] https://www.math.ru.nl/~bmoonen/Papers/SMCfinal.pdf

[Morel] https://fhoermann.org/shimura2.pdf

[Rosen] Rosen, M., 1986. Abelian varieties over C. Arithmetic geometry, pp.79-101.

[Serre] https://www.numdam.org/item/SCC_1958-1959__4__A10_0.pdf

[Vasiu] Vasiu, A., 1999. Integral canonical models of Shimura varieties of preabelian type. Asian Journal of Mathematics, 3, pp.401-517.

[Yang] https://arxiv.org/pdf/2304.10751

A taster of rigid geometry (Pt I: the Tate perspective)

alexyoucis — Sat, 12 Sep 2020 04:56:53 +0000

This is the first in a series of 5 posts whose goal is to briefly introduce rigid geometry with a focus on providing a big picture between the interactions between different perspectives of rigid geometry.

Motivation

We shall not go into the motivation for rigid geometry in any serious depth here since this has been done ad infinitum in other sources ([Con1], [Nic], [FK1],…) . But, we would be remiss to not say anything at all.

One of the most fruitful interactions in the semi-classical theory of algebraic geometry is that between the theory of schemes and the theory of complex analytic varieties. Namely, there is an analytification functor

which one can read about, for example, in the book [Nee]. Intuitively, this analytification functor takes a -scheme to which is endowed with the natural structure of a -manifold (it’s really only a manifold if is smooth) by taking the algebraic charts and interpretting them as holomorphic charts.

It turns out that this functor preserves many of the properties of -schemes algebraic geometers are interested in (cf. [SGA1, Exposé XII)). This statement is no better exemplified than in the GAGA results of Serre et al. (e.g. as in [Ser]) as well as the relationship betwen the étale topology of and the analytic topology of exhibited by Grothendieck, Artin, et al. (e.g. see [SGA1, Exposé XII Théorème 5.1] and [SGA4, Exposé XI Théorème 4.4]).

This connection has immense benefits for both the theory of schemes and the theory of complex analytic spaces. For example, one has access to a much wider class of objects in the world of complex analytic spaces that don’t exist in the purely algebro-geometric world. As a simple example of this, in the complex analytic world one can make arguments using discs which are convenient for topological arguments since they’re contractible, whereas such arguments aren’t feasible in the purely scheme theoretic situation. This leads to situations where many results about varieties over characteristic fields are reduced to statements in the complex analytic setting by the so-called Lefschetz principle which are then proved by complex analytic means. For example, for a long period of time the only known proofs of the degeneration of the Hodge spectral sequence (e.g. see this post) and of the Kodaira-Nakano-Akizuki vanishing theorem (cf. loc. cit.) were only known by an application of the Lefschetz principle.

Now, while the the reduction to is useful for proving purely geometric results, the passage from something like to via the Lefschetz principle completely forgets arithmetic. Namely, if is a variety over then the variety has an action of the absolute Galois group which is of great interest to number theorists. Such information is completely lost in the passage from to .

To remedy this one might note that , like , is an ‘analytic field’—it has a natural metric for which it is a complete topological field. So, one might ask whether or not there is some sort of -adic analytification functor which takes varieties over and outputs ‘-adic analytic manifolds’. The hope is that such -adic analytic manifolds would have the benefits of complex analytic manifolds (e.g. access to disks) while retaining the arithmetic information contained in the original variety.

The goal of rigid analytic geometry is, essentially, to create such a theory of -adic analytic manifolds. As it turns out such a theory is fraught with technical difficulties from the start. Most prominently is the fact that the naive -adic disk is totally disconnected which makes naive attempts at defining a theory unwieldy, bordering on usless. Thus, the actual structures involved in the theory of rigid analytic spaces are significantly more nuanced than one first might envisage—they are certainly not just ‘spaces which locally look like open subspaces of ‘ or the like (although there are theories of this form which are useful in contexts other than geometry — e.g. see [Sch]).

In fact, there are (more than) six theories of rigid analytic geometry, and we hope to touch on each of them in these posts. These are Tate’s theory of rigid analytic varieties, Raynaud’s theory of birational geometry of formal schemes, Berkovich’s theory of Berkovich analytic spaces, and Huber’s theory of adic spaces, and Fujiwara–Kato’s theory, and Scholze’s theory of diamonds (although this last one has a slightly different character than the others). In very broad strokes one can think roughly that

Tate’s theory is like the classical theory of varieties over an algebraically closed field, and roughly has the same advantages and pitfalls.
Huber’s theory is like the theory of schemes, and roughly also has the same advantages and pitfalls.
Berkovich’s theory is a middle ground between Tate’s theory and Huber’s theory which has some of the same pitfalls that Tate’s theory has, but with the advantage of being topologically well-behaved (in large generality Berkovich spaces are Hausdorff and arc connected!).
Raynaud’s theory tries to make precise the idea that rigid analytic varieties over should be thought of as the ‘generic fibers’ of formal schemes over . This has the advantage of making available the full theory of formal geometry (which has historically been well-developed) but has the disadvantage of being somewhat difficult to talk about simple objects—many non-isomorphic formal schemes have the same ‘generic fiber’ and so rigid analytic varieties in this perspective are something like equivalence classes of formal schemes.
Fujiwara–Kato’s theory chooses to take Raynaud’s theory to the next level by constructing something akin to the ‘Zariski–Riemann space’ of a formal scheme. Roughly this is something like an inverse limit over all formal schemes which have isomorphic generic fiber.
In the p-adic situation, Scholze’s theory of diamonds chooses to view rigid spaces in terms of their interaction with extremely degenerate, but extremely flexible spaces: the perfectoid spaces. This has the advantage of allowing one to reduce many proofs to arguments with a point-set topology theoretic flavor. The disadvantage (besides the inherent complicatedness of the objects involved) is that the theory of diamonds can only remember ‘topological information’ (e.g. the etale site).

All of these theories are relatively different in their presentation even though they generally capture, at their heart, the same theory (except for the theory of diamonds as already mentioned). In fact, one of the many strengths of rigid analytic geometry is that one can move between these various theories to use their different strengths. For example, there might be some powerful argument utilizing the well-behaved topology of Berkovich spaces that is completely invisible (or certainly much less obvious) from the perspective Huber.

The point of these posts

This series of posts is meant to introduce the reader to rigid geometry in terms of these perspectives: the classical Tate perspective on rigid spaces, the perspective of Raynaud and Fujiwara–Kato in terms of rigid fibers of formal schemes, the perspective of Berkovich spaces, Huber’s perspective in terms of adic spaces, and Scholze’s theory of diamonds.

These posts are mainly meant to be high-level introductions, focusing on giving some major definitions, major results, quirky examples and observations, but no real proofs. In particular, we will (of course) not go into any real detail here. For this, I highly suggest the following sets of notes/books:

For Tate perspective:

Tian’s notes [Tia]. These do a really great job of summarizing the key points of the theory and, in particular, strikes a great balance between terseness and rigor.
Bosch’s book [Bos]. This is the book Tian’s mostly drew from. It is also wonderful, if a bit longer than Tian’s notes. If one wants more detail than what is in [Tian] I suggest looking here.
The book [FvdP] by Fresnel and Van de Put is a pretty commonly cited place to look for this theory. It is a bit hard to read in places, but introduces most of the general theory in a good amount of rigor, tries to give some motivational examples first (by considering subspaces of ), and gets into more advanced topics (e.g. rigid and étale cohomology).
The book [BGR] by Bosch, Guntzer, and Remmert. This is a bit like an EGA for the basic theory of rigid spaces. It’s a great place to look for rigorous statements/proofs of technical results.
The first chapter of [Ayo] is a great place to get a high-level, rapid, but mostly complete introduction to the technical setup of rigid geometry from the Tate perspective.

For Raynaud’s perspective:

I again suggest [Tia], for the same reasons.
I again suggest [Bos], for the same reasons.
The book [FK] is essentially a reimagining/strengthening of the classical perspective of Raynaud. This book is only the first book in a series of books. It is written very, very much in an EGA format.

For Berkovich’s perspective:

I suggest the notes [Jon] as a good place to start.
Of course, the foundational texts [Ber] and [Ber1] are also a great place to look.
Part I and Part II of [DT] is also helpful.

For Huber’s perspective:

I suggest Huber’s papers [Hub1] and [Hub2] as well as Huber’s book [Hub] for much of the rigorous foundations. This is where I first learned most of this material, and remains the only place for many fundamental results (especially considering things about étale cohomology).
The book [SW] has a good introduction to adic spaces in Lecture 2 through Lecture 5.
The notes [Mor] contain many of the foundational results for adic spaces.
The series of lectures notes from [Con+] also contain much foundational material.

For Fujiwara–Kato’s perspective

The only real canonical reference is the first book on this topic by the authors. Namely, [FK].
One can also see the paper [FK1] which is a good conceptual introduction to the ideas which prompt the [FK] theory, and some of its advantages.

For Scholze’s theory of diamonds

The best introduction to these ideas is provided in [SW], where they began to take shape. This lacks many of the deep foundational results though.
These deep foundational results are then instead supplied in [Scho].
A nice overview of the theory is also provided in the first several lectures of Scholze’s course on Fargues–Scholze (see here).

General overview

One can look at the notes [Con1] of Conrad from the 2007 Arizona Winter School. They do a great job of introducing all the key players, but with a special emphasis on the Tate perspective.
One can also look at the notes [Nic] which are similar in scope to [Con1] but have a different set of biases for what to emphasize.
One can also see the text [FK1] which does a good job of discussing a broad swath of the rigid analytic theory.

The Tate perspective

As mentioned above, Tate’s theory of rigid analytic geometry is very similar in spirit to the classical theory of varieties over an algebraically closed fields —namely spaces locally isomorphic to closed subsets of . That said, there is an added twist in the Tate setting since the ‘spaces’ we’ll be considering aren’t actually topological spaces!

For the entirety of this post let us fix to be a non-archimedean complete (but not necessarily locally compact) field with respect to an absolute value . Examples of such fields are

The field of -adic numbers with the usual -adic absolute value (which we will just usually abbreviate to ), or finite extensions thereof (with the unique extension of ).
The field which is the completion of the maximal unramified extension of (with respect to the unique extension of ).
The field which is the completion of (with respect to the unique extension of ).
The field with the usual -adic absolute value , or a finite extension thereof (with the unique extension of the -adic absolute value).
The field with the -adic absolute value.

Let us also fix an element of satisfying (such an element is called a pseudo-uniformizer since it plays the role for such that a uniformizer does for discretely valued fields.

The general setup

Prologue

As mentioned above, Tate’s perspective is the beginning of rigid geometry which, in essence, tries to capture an ‘algebraic theory of -adic manifolds’. By an ‘algebraic theory’ I mean that we want to consider our building blocks to come from ring theory of analytic functions, and our opens to be somewhat algebraic in nature. The big difference between the algebro-geometric setup and the rigid analytic setup is that in the latter instead of just considering opens to essentially be those generated by non-vanishing loci of analytic functions

we also want to consider opens given by inequalities involving analytic functions

(which, by the magic of non-archimedean theory, are open in the topology given by ) and, of course, combinations of these two types of opens (which one might call something like semi-algebraic open subsets).

Remark 1: Note that since this will be an ‘algebraic theory’ of -adic manifolds some sort of surprising things will happen. For example, the closed unit disk over , denoted , will be both connected and quasi-compact. These are necessary steps to make a good algebraic theory work. For example, we would imagine that the global ring of functions on is some ring of analytic functions (convergent power series) and, in particular, an integral domain. So, if is to have a sheaf of analytic functions, the fact that is an integral domain actually forces connectedness (e.g. see the argument in [Stacks, Tag00EC]). This is in stark contrast to the naive -adic manifold

which, with its native -adic topology, is neither connected nor quasi-compact.

Now, just as one can imagine building the theory of varieties in three steps:

(Step 1) Define affine space .
(Step 2) Define affine varieties as closed subsets .
(Step 3) Define general varieties as gluings (along affine opens) of affine varieties.

one can also see Tate’s theory of rigid analytic varieties as following the same three step procedure with some differences. We will explain this progression, roughly, below.

Step 1

The first major difference is that instead of our building blocks being affine space, which intrinsically doesn’t capture our desired notion of inequalities, we instead use closed polydisks which one imagines as

(we will shorten to ) as our building blocks in Step 1.

Of course, the naive guess for what the closed unit polydisk should be is that it’s, roughly, the set

Of course, if is not algebraically closed this has the exact same pitfalls as if one considers to be the ‘affine line’ over . For example, the closed subspace , where , is empty. Thus, as in the case of classical varieties, one first wants to figure things out over an algebraic closure and then ‘descend’ back down.

Now, since is complete we know that there is a unique extension of the absolute value on to the field (e.g. see [Iya, Chapter II Theorem 4.4] or [Bos, Appendix A Theorem 3]). We can thus imagine that, at least over , the ‘closed unit polydisk’ has a clear meaing. Namely, let us make the following definition:

The question then becomes the following: how do we descend back down over ?

Well, if is a finite type -scheme then we can recover (or, at least, the closed points of ) as the quotient (e.g. see [GW, Proposition 5.4]). This suggests, perhaps, that we can make the definition

Of course, this is a bit unwieldy, and so we’d like to give an alternative description of the set .

If we were back in the situation of a finite type affine -scheme then it is true that the closed points of are in bijection with but we can also describe the closed points of as —-the set of maximal ideals of where is thought of as the ring of functions on . Maybe something similarly can be done for .

To begin to do this we need to answer the question “what should the ring of analytic functions on the closed unit polydisk over disk be?” Well the only reasonable possible answer is

where . Thus, a naive hope would be then that the natural map

(which sends to the kernel of the evaluation-at- map ) is a bijection. This, in fact, is the case (e.g. see [BGR, §7.1.1 Proposition 1]).

So, we have that is describable, at least as a set, in algebraic terms—it is. But, of course, we would like to have as more than just as a set—we’d like to have it as some sort of locally ringed space (the correct setting to do ‘modern geometry’).

To this end, let us note that has a topology coming from since itself has a natural topology—that induced by a basis of open subsets of the form

for a positive real number and . This (quotient) topology on is called the canonical topology. Note that in the canonical topology there are many other open sets not of the above form—you can have multiple functions defining your inequalities, non-vanishing loci, strict inequality, ‘circles’ (e.g. see [Bos, §3.3]).

Unfortunately, the canonical topology is far too fine. The classical example is to note that has a natural open cover as the -orbits of the following decomposition of

where is the open disk over and is the unit circle (or analytic -torus). This decomposition shows that with the canonical topology is neither connected nor quasi-compact in contrast to our stated desires. Another way to think about this is that the sheaf theory doesn’t work for this cover. Namely, again we have that

but certainly one can check that

(where is something like the ‘convergent power series on ‘).

Step 2

Before we discuss the fix to this topological problem we actually first discuss the analogue of Step 2 in the theory of rigid spaces. Namely, if we are to think of as then the analogue of Step 2 should be the following.

Let us first define the analogue of finite type -algebras. Namely, let be a -algebra. We say that is an affinoid -algebra if it is a quotient of for some .

The following properties of affinoid algebras show that they are somewhat similar to polynomial rings:

Lemma 2: Let be an affinoid algebra. Then:

Any choice of surjection defines a quotient topology on (using the norm ). This topology is actually independent of presentation.

The ring is Noetherian.

The ring is Jacobson.

A prime ideal of is maximal if and only if is finite-dimensional over .

Every ideal of is closed (in the topology from 1.).

Every homomorphism of -algebras between affinoid -algebras is continuous.

Proof: All of these are contained [Tia, Chapter 1] and mostly in [Tia, §1.3].

Now, for any ideal we get the subset

Of course, one can easily check that we can also identify as the quotient by of the set

In particular, for an affinoid algebra the choice of a surjection identifies concretely in terms of vanishing loci in and thus endows it with a canonical topology.

Note that by Lemma 2 part 4. that if one has a ring map

of affinoid algebras it induces a map

which one can show is continuous with the canonical topology for any presentation of and as quotients of some (e.g. see [Bos, §3.1, Proposition 6]). In particular, we see that the canonical topology on doesn’t depend on the presentation of . We call sets of the form affinoid sets (it will be these sets with more structure that are really the affinoid spaces which form the building blocks of rigid analytic varieties).

Moreover, note that a presentation allows us to easily talk about inequalities in an affinoid set. Namely, if we realize then we get an embedding

and in particular for a set elements we can make senset of the set

Indeed, we can think of the set as a set of functions in and take the -orbit of the set

which is clearly independent of the lifts of along the surjection .

But, there is an instrinic way to describe these sets as well. Namely, for any we know that (again by [Bos, §3.1 Proposition 4]) is a finite extension of and therefore carries a unique absolute value extending that of . One can then define to be the real number given by evaluating this absolute on on the element . One can then check that the set from agrees with the set

(e.g. see [BGR, §7.1.4 Lemma 2]).

Step 1 and Step 2 (cont.)

Of course, the canonical topology on is too fine as in the special case of . So, let’s return to trying to fix this topology or, what amounts to the same thing geometrically, fixing the sheaf theory. By reexamining the classical the classical theory of the structure sheaf on an affine scheme Tate was able to see how to fix the sheaf theory on . Namely, how does the structure sheaf work? Well, one begins by working with the distinguished base of (e.g. see [Vak,§2.5] and [Vak, Exercise 3.5.A]) and defining a presheaf on this base as follows:

One then shows that this is a sheaf on a base (e.g. see [Vak, Theorem 4.1.2]). A pivotal step in this procedure is essentially to use the quasi-compactness of to reduce to showing that a finite distinguished open cover satisfies the sheaf property.

Tate then realized that he should figure out what the analogue of a ‘distinguished open’ inside of is and try to prove that a finite distinguished cover satisfies the sheaf axiom. The analogue of a distinguished open on an affinoid is a so-called rational domain. Namely, for elements such that is the unit ideal one defines the associated rational domain as follows:

which one can pretty easily check are open for the canonical topology on .

Now that we have singled out the correct analogue of for we want to say what the value of our presheaf on these distinguished opens is. The guess is quite clear: the ring should be the ring of analytic functions on that are convergent on . Namely, we could embed inside of and say that is

(where we have implicitly chosen liftings of to but it’s clearly independent of such liftings).

But, a coordinate-free (but still a priori dependent on the choice of the set ) way to describe this ring is as follows:

where

where the topology we are using on (as in Lemma 2 part 1.). In particular, note that is still an affinoid -algebra.

Remark 3: Note that this ring enlarges in more ways than adding in inverses. Namely, in normal algebraic geometry essentially if you have an open subset (let’s say of integral schemes for simplicity) the map allows one to view as an enlargement of by algebraically adjoining inverses of functions not invertible on but invertible on —a sort of generalized localization. In the rigid setting there is not only localization but also completion. The result is that one doesn’t just algebraically adjoin new things but also forces new series to be convergent.

For example has ring of sections . Now, algebraically adjoining does noting (it’s already in the ring) but this ‘completed adjoinment’ forces certain power series in to converge—any series of the form with converges in but needn’t converge in . For a specific example one has that

is in but not in .

Finally, one can show that for an open subset of which is a rational domain—which can be presented as for some set generating the unit ideal—that the ring doesn’t depend on the choice of presentation (i.e. on the set )—see [Tia, Example 1.5.12 part 3.]).

We then have the following theorem of Tate:

Theorem 4(Tate): Let be an affinoid algebra. Then, for any rational domain and any finite covering of by rational domains (cf. [Bos, §3.3 Proposition 17]) the sequence

is exact.

In other words, the presheaf is a sheaf with respect to rational domains and finite coverings by rational domains. The idea of proof is to do explicit computations much as in the case of [Vak, Theorem 4.1.2] (for a full proof see the references later on).

In fact, two stronger versions of this theorem can be shown to hold which are often stated instead of Theorem 4. Namely, one can replace rational domains and finite rational coverings by affinoid domains (resp. admissible opens) and finite affinoid coverings (resp. admissible coverings).

The idea of an affinoid subdomain is that it’s basically the analogue of an ‘affine open subscheme’ in the rigid geometry setting. In particular, using this post one is tempted to define a subset to be an affinoid domain if there exists an affinoid algebra and a map of -algebras such that the induced map has image and for which is universal for such a property: if is a -algebra map of affinoid algebras such that has image landing in then unique factors through . One can show that such are open for the canonical topology (see [Tia, Corollary 1.5.22]) and that the -algebra is unique up to unique isomorphism.

As an example, one can show that rational domains are affinoid domains (e.g. see [Tia, Example 1.5.12 part 3.]). In fact, the super important Gerritzen–Grauert theorem gives a partial converse to this statement: every affinoid domain is a finite union of rational domains. Now, given the unicity of (as in the last paragraph) one can associate to any affinoid domain the ring .

One then has the following strengthening of Theorem 4:

Theorem 5: Let be an affinoid algebra. Then, for any affinoid domain and any finite covering of by affinoid domains (cf. [Bos, §3.3 Proposition 12]) the sequence

is exact.

Proof: See [Tia, Theorem 1.6.3]. The idea is to use the Gerritzen–Grauert theorem to reduce this to rational domains and rational covers, then even simpler types of opens and covers and finally show the sheaf condition explicitly by hand.

The final strengthening allows one to replace affinoid domains , which (again) one should think of as just open affinoid subspaces, with certain open subspaces and certain covers. The idea is that whatever general open subset of we encounter, one shouldn’t be able to produce using another affinoid (not necessarily an affinoid open in and an affinoid cover of which does not have a finite subcover—for this is precisely the case in which we have constructed our theory thusfar to avoid. How does one make this rigorous?

Well, let us call an open subset admissible if there exists an affinoid (in ) cover (possibly infinite) of such that for any map of affinoid -algebras such that the induced map has one has that the affinoid cover of has a refinement (cf. [Bos, Pg. 83]) by a finite affinoid open cover. Similarly, an admissible covering of the admissible open is a collection of admissible open subsets of which cover and have the property that for any -algebra map of -affinoids such that one has that the cover has a refinement by a finite affinoid open cover.

One should interpret non-admissible covers of an affinoid as ‘bad’ and, in particular, should be disregarded.

Example 6: The cover the open unit disk is an admissible open subset of (e.g. see [Tia, Example 2.1.6]) and is admissible (it’s even rational—it’s ) but the cover is not admissible (e.g. see [Con1, Example 2.2.8]). We will later give a good interpretation for this in terms of adic spaces.

We then have the following general strengthening of Theorem 4 and Theorem 5:

Theorem 7:Let be an affinoid algebra. Then, for any admissible open and any admissible covering of by admissible opens the sequence

is exact.

Step 3

Now, we have singled out three (increasingly general) situations where our sheaf theory has worked:

It works for rational domains in and finite rational covers.
It works for affinoid domains in and finite affinoid covers (this is the ‘obvious’ version of 1.—it’s just easier to work with 1. than general finite affinoid covers)
It works for admissible opens in and admissible covers (this is the finest replacement -topology [see below] which ‘acts the same’ as the -topology from 2.—see [BGR, §9.1.2] for more details).

In all three cases we see that we are in the situation of having some set of distinguished open subsets of (rational domains, affinoid domains, admissible opens) and for each distinguished open subset a ‘permissible’ set of covers of that distinguished open subset by distinguished open subsets (finite rational covers, finite affinoid covers, admissible covers). Now, none of these are an acutal topology on (e.g. they aren’t closed under arbitrary unions) but they act similarly enough like a topology to make sheaf theory work. In fact, they are an example of a -topology.

Roughly, a topology on a topological space is

A (usually proper) collection of open subsets of ‘the permissible opens’.
For each permissible open subset a set of open covers of by permissible opens—‘the permissible covers’ (NB: not every open cover by permissible opens is assumed permissible—just a subset of such covers).

One needs to assume that the above colletions have reasonable enough properties to talk about sheaf theory (for a rigorous definition see [Bos, §5.1] and [Bos, §5.2]). Namely, one calls a contravariant functor

(where is the category of abelian groups and the set of permissible opens is considered a category where the maps are the inclusions of permissible open sets) an abelian presheaf on the -space . One calls an abelian presheaf on the -space an abelian sheaf if for all and all the presheaf satisfies the sheaf conditions for the cover— in other words the sequence

is exact. Of course, one can define sheaves with values in any reasonable category (e.g. see [Stacks, 00VL] or [Bos, §5.1] and [Bos, §5.2]).

Remark 8: The reason for the name ‘-topology’ is that a -topology on is, essentially, a Grothendieck subtopology of for those familiar with Grothendieck topologies (e.g. from étale cohomology). This allows one to use all the theory one has learned from the theory of cohomology on sites etc. (e.g. [Stacks, 00UZ] and [Stacks, 01FQ])

Note that what we are calling here ‘permissible opens’ and ‘permissible covers’ is usually called ‘admissible opens’ and ‘admissible covers’, but so as to not confuse these with our explicit admissible covers for affinoid sets we have elected to use this non-standard terminology.

Given a -space and an abelian sheaf on one can define the stalk of at a point . Namely, one can define

where is the set of permissible open subsets of containing .

A locally ringed -space is a -space with a sheaf of rings whose stalks are local rings. If, in addition, is a sheaf of -algebras we call a locally ringed -spaces over . A morphism of locally ringed -spaces and/or locally locally ringed – space over is the obvious one (e.g. again see [Stacks,00VL] or [Bos, §5.1] and [Bos, §5.2]).

Now, from the discussion in the previous subsections we see that for every affinoid algebra one gets two (major) locally ringed -space over given as the pair

where the permissible opens are the admissible opens (resp. affinoid domains), and the permissible covers are the admissible ones (resp. finite affinoid covers). We call this the strong (resp. weak) -topology on . Unless specified otherwise, the -topology on will always taken to be the strong one. We call a locally ringed -space over of the form (with the strong -topology) an affinoid space over . While is really a pair, we shall, in practice conflate and .

This pair satisfied the expected property:

Lemma 9: Let and be affinoid -algebras. Then, the map

given by the map on global sections is an isomorphism. In other words the functor

is fully faithful.

Proof: See [Bos, §5.3 Corollary 7].

Now, given any locally ringed -space over and any open permissible subset one naturally gets a -topology on (e.g. is the set of elements of contained in ) and one can restrict to get a sheaf on so that the pair is also a locally ringed -space over .

So, with all of this setup we can finally defined rigid analytic varieties a la Tate. Namely, a rigid analytic variety is a pair which is a locally ringed -space over such that there is a permissible open cover of such that is isomorphic to an affinoid space. A morphism of rigid spaces is a morphism of locally ringed -spaces over .

One has a natural generalization of Lemma 9:

Lemma 10: Let and be rigid spaces with affinoid. Then, the natural map

is a bijection.

Proof: See [Bos, §5.3 Corollary 7].

Let us finally note that the set of affinoid open subspaces of a rigid space form a distinguished base for (in the sense of [Vak, §2.5]) and thus if one wants to define a sheaf on it suffices to define a sheaf on the distinguished base of affinoid opens (e.g. by [Vak, Theorem 2.5.1]).

The examples

Now with all of this setup we can finally discuss some simple examples of rigid spaces. We shall be introducing some more theory along the way.

The closed unit disk

We start with the basic object which we have already seen many times above. Namely, the closed unit disk (or closed unit ball) over is the rigid analytic variety . Let us start to prove some of the basic properties of .

Topological properties

Perhaps the first thing to check is that , as we envisioned it, is quasi-compact and connected. Of course, since is actually a -space (in particular not a topological space) we need to define what this means. But, of course, it’s the obvious thing. Namely a -space is quasi-compact if every permissible cover of has a finite refinement. And, a -space is connected if there does not exist a permissible cover of such that is empty and

To prove that is quasi-compact and connected we make the following general observation:

Proposition 11: Let be an affinoid -algebra. Then, is quasi-compact. Moreover, is connected if and only if has no non-trivial idempotents.

Proof: We tailor build admissible covers so to force the quasi-compactness. Namely, if is an admissible cover of then letting be the identity map (in the definition of admissible) one sees by definition that must admit a finite affinoid refinement.

To see the second claim we proceed as usual. If with admissible then and so has non-trivial idempotents as soon as and are not the zero ring. To see this note that if is a non-empty rigid space then is non-zero. Indeed, let and let be an affinoid neighborhood of . Then, we have a ring map and, in particular, if is the zero ring this forces to be the zero ring which forces to be empty, which is a contradiction.

Conversely, suppose that has non-trivial idempotents so that with not zero rings. Let and . Then I leave it to the reader to check that the non-vanishing locus of and , called and , are open, non-intersecting, and that is an admissible cover of .

That said, let’s notice our first descrepancy with the scheme picture (although it is consonant with the classical picture of varieties). Namely, recall that if is an integral domain that is more than just connected it’s actually irreducible. This irreducibility, and the existence of generic points, are a key component of what makes scheme-theoretic algebraic geometry tick. This completely fails for objects in the rigid setting. Namely, let us make the following non-standard definition. Namely, let us say that a -space is irreducible if for all admissible opens we have that . Essentially no rigid space is irreducible.

For example, certainly is not irreducible since, for example,

and

are admissible opens (we leave this to the reader, or see [Con1, Example 2.2.8]) which intersect disjointly. If one wants to soup this up to an example consisting only of affinoid opens one can take two non-equal points and let 0" class="latex" />. Then,

are disjoint admissible opens. Thus, is far from being ‘irreducible’.

Remark 12: One should take this reducibility of (most) rigid spaces as both a blessing and curse. Namely, the fact that our spaces are reducible means that they are closer to the normal topological spaces one deals with in complex geometry and therefore we can often times perform natural operations in the rigid setting which are impossible to perform in the scheme setting (e.g. see the discussion on Tate elliptic curves below).

But, on the other hand, we are now robbed of one of the most powerful tools at the disposal of algebraic geometers: the ubiquitous generic point. Since most reasonable schemes we work with are irreducible they have generic points and, in particular, one can often use this generic point to simplify a huge amount of technical assumptions. I didn’t realize how important generic points were to my understanding of scheme theory until I tried to do rigid geometry where such points were suddenly missing.

Let us also note here by that ‘irreducible’ we mean irreducible in the naive sense—irreducible as topological spaces. This, of course, is the type of irreducibility we usually mean in the context of schemes. That said, there is a more ‘algebraic’ notion of irreducibility which does work well in the context of rigid spaces. Essentially, one replaces the usual -topology for a coarser one—the Zariski topology. For this, one can see [Con3].

Functor of points

Let us now contemplate what the functor of points of is. Namely, we have the following observation:

Proposition 13: For any rigid space over is a natural isomorphism of sets

Before we prove this let us define what is. This is a presheaf of -algebras on defined as follows. For an affinoid open we define

where is the ring of power bounded elements of (i.e. those elements such that the set is bounded—see [Bos, §3.1 Theorem 17], [Tia, Proposition 1.4.13], or [BGR, §1.2.5]). Note that this is a sheaf with respect to the distinguished base of affininoid opens (this is easy and left to the reader—use the equivalences given in loc. cit.). Then, as previously remarked, extends uniquely to a sheaf on the -space .

One then sees as the global sections of this sheaf. Intuitively one imagines that is the set of elements of such that (e.g. see [Tia, Proposition 1.4.13] for a rigorous statement) from which Proposition 13 then makes intuitive sense.

Finally we can prove Proposition 13:

Proof( Proposition 13): Note that since and are sheaves for the -topology it suffices to prove this natural isomorphism when is affinoid, say . Note then that by Lemma 9 there is a natural bijection

Now, note that since any -homomorphism is continuous (e.g. see Lemma 38) that such a homomorphism is uniquely determined by where it sends . Namely, if then

Moreover, this also easily shows that is of the form if and only if converges for every sequence in such that . But it’s easy to see that this is equivalent to . The conclusion follows.

In particular, let us note that since is a ring (as we leave to the reader to check) one has that is a ring object in the category of rigid spaces over . This is perhaps not too surprising since is meant to be the rigid geometry avatar of which is itself a ring.

Picard group

Let us also try and say something slightly more interesting about . Namely, let us define a line bundle on a rigid space as a sheaf of -modules which locally (for the -topology) is isomorphic to (cf. [FdvP, Definition 4.5.1]). As per usual the group of (isomorphism classes of) line bundles on forms a group under tensor product which is called the Picard group of and written . What is the Picard group of ?

To give us some intuition, let us first try and understand a closely related question over the complex numbers—we will later see this is actually a bit deceiving, but let’s use it anyways:

Example 14: Let be the open unit disk thought of as a complex manifold. We claim that . Indeed, we have the exponential sequence

Taking the long exact sequence in cohomology, using the fact that

(e.g. see [Wed, Theorem 11.13]), and the fact that is contractible we get a natural isomorphism

and of course we have natural isomorphisms

Thus, it suffices to show that . This last equality can be shown to be true by the Mittag-Leffler theorem.

So, a guess might be that since the open disk and closed disk are not too far apart that the Picard group of is also trivial. This, as it turns out, is correct:

Proposition 15: Let be an affinoid -algebra. Then, there is a natural isomorphism

given by associating a finitely generated projective -module with the sheaf on which associates to an affinoid open the tensor product .

Proof: See [Bos, §6.1 Theorem 4] or [FvdP, Proposition 4.7.2].

In particular, we see that proving that is equivalent to proving that . But, to do this it suffices (e.g. see [Stacks, 0BCH]) to show that is a PID. But, this is well-known (e.g. see [Bos, §2.2 Corollary 10]).

Dimension

Defining the dimension for a rigid space in the classical Tate perspective is necessarily ad-hoc. Namely, one is tempted to try and define it as in the case of schemes—the dimension is the maximal chain of specializations. But, of course, this cannot work well since we have only closed points! One might then try to make an adapatation of what we do in classical variety language (e.g. as in [Har, Chapter 1]) where we use chains of irreducible closed subsets. But, as we saw above (around Remark 12) that irreducible analytic spaces are quite rare. In the Huber and/or Raynaud setting there is a natural fix to these issues, but for now we instead take the following as our definition of dimension. The dimension of a rigid space is the supremum of as ranges over the points of .

So, what is the dimension of ? Of course the answer is , but we need to verify this. But, as observed above, since is a group under addition it suffices to check . What is this ring? Well, almost by definition (since closed subdisks around are clearly a basis for —cf. our definition of stalk) that

In other words, is nothing but the ring of power series convergent in some neighborhood . This is, by inspection, a DVR. Indeed, it’s easy to see that is the unique maximal ideal of (e.g. it’s the same proof as here) from where one can easily deduce the conclusion.

Thus,

as desired.

The punctured closed disk

We now would like to consider the spaces where are any points of . We call such a space a (several times) punctured disk. For sake of simplicity we assume that is algebraically closed (or that the points are -points). This does not change the story in any substantive way, it just allows one to not have to worry about an -point, for finite, splitting in to several points over .

Topological properties

Let us begin with the most basic case: the -analytic variety . There is evidently an automorphism of which takes to . Indeed, this follows from the aforementioned fact that is a ring object over —just take the translation by automorphism of . Thus, we may assume without loss of generality that .

Remark 16: Note that this observation already dispels an easy-to-make misconception about . Namely, if one draws then one has a tendecy to treat points in the ‘boundary’ as being different than points in the interior . But, the above shows that points in really don’t look any different than points in .

For example, a picture of the closed disk lends one to believe that points on the ‘boundary’ don’t have closed disk neighborhoods still contained in the disk. This, of course, is false. If then the set is, in fact, contained in , let alone . This is one of the reasons that notions of ‘boundary’ (e.g. as in Berkovich’s theory of rigid spaces) are more subtle than one might first imagine.

We begin with the following observation:

Proposition 17: The punctured disk is an admissible open subset of . It is connected and not quasi-compact.

Proof: We can essentially kill three birds with one stone. Namely, for each let us set

(this is an example of an annulus). Note that each is an affinoid domain in . In fact, is a rational domain. Indeed, we have that

and evidently generate the unit ideal in since is a unit. The affinoid domain claim then follows from previous comment (cf. [Tia, Example 1.5.12 part 3.]).

Note that is an open cover of because and so if then and thus for some . We claim that is an admissible cover. To do this, let be a morphism such that . Let be the associated map of affinoid algebras and let . Note that since that is a unit in . Indeed, note that we have a factorization of maps of rings

and since is invertible in the invertibility of follows. Let us now consider . Note then that by the maximum modulus principle (e.g. see [BGR, §6.2.1 Proposition 4]) that there exists some 0" class="latex" /> such that for all and so for all . Let be large enough so that . Then, it’s clear by this that . So then evidently has a finite refinement as desired.

Now, to see that is connected it suffices to note that is connected. Indeed this would follow from the obvious -topology analogue of the classical statement “if a topological space has an open cover by connected subspaces all of which intersect then is connected”. To see that is connected we can use Proposition 11 and merely note that

and I leave to the reader to check that is a domain.

Finally, to see that is not quasi-compact it suffices to note that has no finite refinement which is easy and left to the reader.

One thing that follows from the above is the following simple observation:

Corollary 18: The -space is not Noetherian.

By Noetherian we mean that not every admissible open subset is quasi-compact. This is, of course, in stark constrast to the algebraic setting where an affine scheme with Noetherian global sections is Noetherian. This is a subtelty that can often lead one astray when applying algebraic geometry intuition to the rigid setting (we’ll see an example of this in another post).

Functor of points

We can now try and understand the functor of points of . But, this has essentially been observed in the process of proving Proposition 17:

Proposition 19: There is a functorial isomorphism

In particular, we see that is a group object in the category of rigid analytic varieties over .

The multi-punctured disk

Let us now turn our attention to the case of a several times punctured closed disk. Namely, let be an integer and consider .

Topological properties

We would first like to give the analogue of Proposition 17 for these multi-punctured spaces:

Proposition 20: The multi-punctured closed disk is an admissible open subset of . It is connected and not quasi-compact.

Proof: To show that is an admissible open subset which is not quasi-compact, one can proceed essentially as in the proof of Lemma 17 (see [BGR, §9.1.4 Proposition 5] for more details).

To prove connectivity one makes the following general topological observation. Let be a connected topological space and let be a closed point. Suppose that has a connected neighborhood of such that is connected, then is connected. Using this, and the fact that every point of has a neighborhood isomorphic to , one reduces the question (inductively) to the case when . The claim then follows from Proposition 17.

Functor of points

We would now like to compute the functor of points of . But, again, this follows exactly as Proposition 19:

Proposition 21: There is a functorial isomorphism

where note that so that the above makes sense.

The Picard group

Finally, we would like to discuss what the Picard group is. To do this, we assume that is actually spherically complete (also called maximally complete)—for a discussion of spherically complete fields see [BGR, §2.4.4]. Then, one can show that . This is substantially harder for 0" class="latex" /> than the case discussed in the last section. For a proof one can see [FvdP, Theorem 2.7.6].

Remark 22: I’m actually not sure if the Picard group of the multi-punctured disk is non-trivial if is just assumed algebraically closed and/or complete. See Theorem 25 below for why this is not an unreasonable uncertainty.

Dimension

Computing the dimension of the multi-punctured disk is easy. Namely, since is a group object we have, as observed before, that for all points we have that

in particular since for any point the local ring agrees with we deduce that

as one would expect.

The open disk

The next example we want to consider is the open disk . More rigorously, we consider the open subset

which we can think of as an open rigid subspace of .

Topological properties

We start our discussion of as in the case of the closed disk/multi-punctured closed disk:

Proposition 23: The open disk is an admissible open subset of which is connected and not quasi-compact.

Proof: As in the proof of Propsition 17 we can kill all three birds with one stone. Namely, one can readily see that

Let us shorten to . Then it’s easy to see that each is an affinoid open subset of (it’s a rational open). We claim that this is an admissible cover, but this follows exactly as in Proposition 17 (it follows from the Maximum Modulus Principle). Moreover, each is connected (use Proposition 11 and the fact that ). Since this implies that is connected. Finally, since for all (e.g. think about -points) we see that we have given an infinite cover of with no finite refinement.

Functor of points

We can, similar to Proposition 13, describe the functor of points of . Namely, we have the following:

Proposition 24: There is a functorial isomorphism

As in Proposition 13 we need to explain our (non-standard) notation here. Namely, we want to define a sheaf on and to do this it suffices, as mentioned before, to define a sheaf on the Grothendieck topology of affinoid opens and affinoid open covers. So, let be an affinoid open. Then, we set

where is the (additive) group of topologically nilpotent elements of (e.g. the set such that —see [Bos, §3.1 Corollary 18] for more details). One can check that this satisfies the sheaf conditions and thus extends to a sheaf on . Of course, we then have that is the group of global sections of .

Proof(of Proposition 24): Since both sides are sheaves for the -topology on it suffices to check functorial agreeance on affinoids. So, let be affinoid open. Then, let us note that we have an injection

where the last identification is by Proposition 13. So then, we need to figure out what subset this is. But, let us note that if corresponds to a map with image in then (in the language of loc. cit.—see also [Bos, §3.1 Proposition 7]) and so . The converse is also clear since if then defines a map . But, note that since that for some (since ). Then, evidently (given Proposition 13 and the decomposition of in Proposition 13) that and thus deos define a map .

The Picard group

The Picard group of is surprisingly, surprisingly subtle. In particular, our Example 14 was a bit of a ruse. Namely, we have the following:

Theorem 25: The following are equivalent:

.

is spherically complete.

Proof: See [Gru, V, Proposition 2].

In particular, since is not spherically complete we have that ! Maybe I’ll discuss an example in the future—until then I suggest looking at loc. cit.

Dimension

Again, for any we have

so that

as expected.

Affine space (and analytification)

Analytification

We would now like to describe the rigid space which one can think of as the analytification (a la Serre’s GAGA) of the -scheme . In some sense one can think of it as (e.g. see [Bos, §5.4 Lemma 1] for a discussion in this direction), but we shall not do so.

In general, we define the analytification of a locally of finite type -scheme exactly as in complex geometry. Namely, an analytification of is a rigid space and a map of locally ringed -spaces over

(where has the -toplogy given by its usual Zariski topology) such that for any other rigid analytic variety one has that the natural map

is a bijection (here both sets of maps are of locally -ringed spaces over ).

We then have the usual proposition:

Proposition 26: For every locally of finite type -scheme there exists a unique analytifiation.

Proof: See [Bos, §5.4 Definition/Proposition 3].

The idea is somewhat clear though. Namely, since

one can set

where and we open embed by identifying with . In other words, we think of as the union of increasing closed disks. For an affine finite type -scheme we then set

where

which, as one can check, is affinoid.

For a general locally of finite type -scheme one covers by finite type affines, shows that the above definition for affine finite type -schemes turns distinguished open embeddings into affinoid open embeddings, then glues. Again, for more details see [Bos, §5.4 Definition/Proposition 3].

We are in this subsection particularly interested in just but it can’t hurt to list some basic properties of the analytification functor

(it is clearly a functor by the universal property of analytification) in general. Namely we have the following:

Proposition 27: Let and be locally of finite type -schemes and let be a map of -schemes.

There is a canonical bijection of -sets between and .

is connected if and only if is connected.

For The natural map is faithfully flat and, moreover, induces an isomorphism on completions.

is smooth over if and only if is smooth over . More generally, is smooth if and only if is smooth.

is finite if and only if is finite.

.

is reduced if and only if is reduced.

If is any rigid space and is affine, then the natural map is a bijection.

In the above we used three notions which have not been defined: finite morphisms, smooth morphisms, and reduced rigid analytic varieties.

So, let us say that a morphism of rigid analytic varieties is finite if for all affinoid opens we have that is affinoid open, say equal to , and the induced map is finite.

We will say that is smoothly factorizable if we can factorize as where the first map is étale (this means the usual thing—it’s flat and unramified as a map of locally ringed spaces) and the second map is the projection (cf. the definition of smooth given in [Fu, §2.4]) and we say that is smooth if it’s smoothly factorizable locally on source and target. In particular, is smooth over if is regular local for all .

Finally let us say that a rigid analytic variety is reduced if is reduced for all .

Proof (of Proposition 27): See [Con3, Lemma 5.1.1], [Con3, Lemma 5.1.3], [Con3, Theorem 5.1.3], and [Con3, Theorem 5.2.1].

Of course, one also has a GAGA theorem as in the complex analytic setting:

Theorem 28 (Kopf—rigid analytic GAGA): Let be a projective -scheme. Then, there is an equivalence of categories

such which induces natural isomorphisms

for all .

Proof: The original attribution is [Kop] but an alternative proof, in English, can be seen in [Con2].

In the usual way we deduce the following corollary:

Corollary 29: The analytification functor

is fully faithful.

Of course, we remark that the analytification functor is always faithful, but rarely full for non-projective schemes. For example the natural map

is not surjective since the latter contains something like

which converges everywhere on but is not a polynomial. We will see an even more extreme example of this later on with our discussion of Tate curves.

It is also true that the analytification functor is far from essentially surjective. For example we have the following nice proposition:

Proposition 30: Let be a reduced connected finite type -scheme and an affinoid -algebra. Then, there is no non-constant map .

In particular, the above implies that any reduced and connected rigid space which admits a non-constant map to an affinoid (e.g. an affinoid or the open disk or…) is not the analytification of a -scheme.

Proof (of Proposition 30): I don’t know a canonical reference for this. See [HL, Lemma 5.4] or [JV, Proposition 2.8].

Topological properties

Let us now get in to the discussion of proper. We begin with the following:

Proposition 31: The rigid analytic space is connected and not quasi-compact.

Proof: One could use Lemma 27 part 2. to show that is connected, but that’s too complicated. Namely, one can kill both birds with one stone by observing that, by definition, we have that

and . To see that is connected we merely use Proposition 11. Since no is equal to (by definition) we see that is not quasi-compact.

Functor of points

One can cheat and figure out the functor of points by using Proposition 27 part 8. But, we can prove this directly (and thus prove that really is the analytification):

Proposition 32: There is a functorial identification

Proof: It suffices to check that this equality holds for affinoid since both sides are sheaves for the strong -topology. But, then

where we have used the quasi-compactness of to commute this limit (since the image of must land in some ). But, since we have (exactly as in Proposition 13) an identification

but given the definition of our transition maps we then see that

where the union is taken in . We leave it to the reader to verify that

as desired (hint: show both sides are sheaves and so it suffices to show the result on affinoids and then use the definition in [Tia, Proposition 1.4.13], the fact that , and the fact that is submultaplicative).

The Picard group

Let us now discuss what the Picard group of is. One should be leery about guessing that it’s since, in some sense, is like an ‘infinite radius’ version of the open (poly)disk since it’s an infinite increasing union of closed disks. Thanks to Theorem 25 this might make one guess that perhaps the Picard group is non-trivial. Thankfully this is not the case:

Theorem 33: The group is trivial.

Proof: See [Gru, V, Proposition 2] for a nice discussion of this fact.

Remark 34: Just as an interesting side note it’s actually still an open question as to whether or not the ‘Bass-Quillen conjecture’ is true for —in other words whether all vector bundles are trivial. See for instance see [KST] or [Sig].

Dimension

The dimension of is what one would expect. Namely, from Proposition 27 part 6. we have that

as desired. Of course, one can also verify this by hand essentially by showing that each is -dimensional. This follows from generalizing our dimension arguments in the case when from above.

Projective space

We are now interested in studying which, by deifnition for us, is .

Topological properties

We would now like to state the basic topological properties of . Namely, we have the following:

Proposition 34: The space is connected and quasi-compact.

The proof of Proposition 34 can be proven by giving a decomposition of into certain open affinoids. Since this decomposition is independently interesting, we discuss it outside the context of the proof of Proposition 34.

A naive guess is that one can decompose in to open subspace isomorphic to . This is, in fact, true but is not as useful since it’s not a decomposition into affinoids. To decompose in to affinoids we can first remember how the decomposition of into open copies of works. Namely, this decomposition comes by the observation that an element with can be scaled as follows:

One then notes that such a representation is unique and, in fact, the map

(where in the last tuple we omit the entry ) is a bijection.

In the rigid setting we can in fact do better. Namely, let us say that a is unidmodular if for all and for some . Note that by scaling one can assume that every is represented such that is unimodular. One then defines a bijection

given by

(where again we’ve omitted ).

In more rigorously term we have an isomorphism

with (following the nice notation of [Vak, Exercise 4.4.9])

(where again we are omitting ) and we are making the identifications

given by . In fact, one could (perhaps should) take this as our definition of using our description of the functor of points of (using this gluing description as our definition!) to verify it really is an analytification of .

Remark 35: Since we were working at the level of points the above is only really valid over . That said, one can easily check that this decomposition is -invariant and so descends to a decomposition over .

Regardless, this decomposition immediately implies Proposition 34 since each (where we are imprecisely aslo denoting by image under the quotient map ) is connected and quasi-compact, and for all .

Functor of points

The functor of points of is what one expects from its corresponding description in algebraic geometry. Namely:

Theorem 36: There is a functorial bijection

Here we say, as in algebraic geometry, a set of is globally generating if for all in one has that the image of under the map

is a generating set of the -module . Moreover, we say that

if there exists an isomorphism carrying to .

The proof of Proposition 36 is exactly in the scheme theoretic situation, and we leave it to the reader to check this.

Picard group

One can use Rigid Analytic GAGA to see that

and, moreover, that a generator for is given by . Concretely one can think about this as the line bundle obtained on

which is given by the trivial line bundle on each and which is glued via the overlap isomorphisms .

If one doesn’t want to use the full power of Rigid Analytic GAGA one might try to use the isomorphism

(which is true for all locally -ringed space—e.g. see [Stacks, Tag040E] and [Stacks, Tag0A6G]) together with the non-trivial fact that constitutes a Leray cover (i.e. a covering as in [Stacks,03F7]) to deduce that

from which one really can deduce that is a generator by the usual computation. The fact that really is a Leray cover, the fact that has vanishing higher coherent cohomology, is merely a souping up of Theorem 7 (e.g. see [BGR, §8.2.1 Theorem 1]).

Dimension

Given our decomposition

we know that

as one would expect.

The multiplicative group

We now move on the multiplicative group which, by definition for us, is the analytification of the multiplicative group .

Topological properties

We start with the following topological description of :

Theorem 37: The topological space is connected but not quasi-compact.

Again, we might try to apply Proposition 27 part 2., but this is wholly overkill. In fact, both of these properties follow from the following generally useful decomposition of . Namely, recall that we have the decomposition

with each . From this, we see that we have the decomposition

where one can check that . Since each intersects all the other , and each are connected (by Proposition 17) we deduce that is connected. It’s clear that no finite subset of covers and thus we also deduce that is not quasi-compact.

Functor of points

We now move on to the description of the functor of points of . Of course, we expect that it spits out the units of the global sections of a rigid analytic variety and this actually follows from Proposition 27 part 8. But, again, we can see this in a much more down-to-earth fashion:

Theorem 38: There is a functorial bijection

.

Proof: The proof of this is essentially exactly as in Proposition 32. Namely, we see that it suffices to check this for an affinoid space . From there we use the quasi-compactness of to write

to which one then applies Propositition 19 and the decomposition

The Picard group

The Picard group of is again slightly more subtle than one might bargain for. That said, we do have the following result:

Theorem 39: If is spherically complete then .

Proof: This follows from [FvdP, Theorem 2.7.6].

It’s likely that this theorem holds without the assumption that is sphereically complete, but I’m not sure.

Dimension

The dimension of is as one would expect. This follows since each has dimension by our previous discussion about the dimension of multi-punctured disks.

Tate elliptic curves

So, let us consider the meromorphic function field

where more explicitly one can write

0\end{aligned}\right\}" class="latex" />

which is a domain as one can easily check. Let us now fix in with . We can then consider the subset

of which is, in fact, is a subfield as one can easily check.

The realization that Tate had, which was one of the first motivating reasons to want to create a theory of rigid geometry, was the following beautiful result:

Theorem 40 (Tate): There exists a unique elliptic curve over such that . Moreover, one has that

Moreover, one has that there is a functorial isomorphism

for every extension of . Moreover, the map

1\}" class="latex" />

is a bijection with analytic inverse.

Proof: See [Lut, Theorem 1.1] and [Lut, Theorem 1.2] and the references therein.

Now, one is tempted to think that the isomorphism

could be upgraded to an isomorphism

in some sense. Of course, we have to make sense of what this quotient means on the right-hand side.

To do this, we need to give a decomposition of which is shuffled around by the action of . Namely, let us define the following affinoid domains inside of

One note that

and

Both of these spaces are annuli and they have an outer circle and an inner circle defined as further rational domains inside of . We denote the outer circles of by and the inner circles by . Evidently we have that .

Note that for each we have a natural multiplication-by--map

which for an affinoid algebra is just the map

This is an isomorphism of rigid analytic varieties. We note then that one has . Moreover

gives an open cover of .

From this, it seems reasonable to define to be

which which we mean the rigid space obtained by gluing to itself along the isomorphism (gluings along open subspaces clearly exist in the category of rigid analytic spaces).

We then have the following justification for writing this as a quotient

Proposition 41: The obvious map

is -invariant for all and is universal for such.

Proof: Obvious.

The beautiful tie-in with Theorem 40, as we’d hoped, is the following:

Theorem 42(Tate): There is an isomorphism of rigid analytic varieties .

Proof: See [Lut, Theorem 1.2] and the references therein.

We call rigid analytic varieties of the form Tate elliptic curves.

Note that we really needed to be working in the rigid category to have an isomorphism

for two distinct, but equally important, reasons:

The map has infinite fibers. Note that any infinite subset of is dense, and thus one cannot even have a non-constant map from with infinite fibers.
We get from Theorem 41, in particular, a surjective map . Note that any algebraic map (for any elliptic curve over ) is constant since otherwise it would give rise to a surjective map which is impossible by the Riemann–Hurwitz formula (e.g. see [Har, Example 2.5.4]).

Remark 43: As mentioned earlier we see from 2. the striking non-fullness of the map

since the former set is consists only of constant maps but the latter set contains a topological cover!.

Topological properties

The topological properties of Tate elliptic curves are quite simple. Namely, they are quasi-compact and connected. The quasi-compactness comes from the fact that they are a quotient of the compact space

and they are connected since the images of each copy of in are connected and intersect non-trivial.

Functor of points

The functor of points of is not particularly easy to describe since it is, after all, related to the functor of points of which has no explicit description.

The Picard group

By Theorem 28 we know that we have an isomorphism

and thus it suffices to understand the Picard group of . But,

where are the degree line bundles in on . Moreover, we know that

where the first isomorphism has inverse

where is the identity element, and the second isomorphism is from Theorem 40. To understand the composition explicitly, if one fixes in one gets a line bundle on by gluing the trivial line bundles on via the transition map where acts as multiplication by on the trivial bundle. This does descend to a line bundle on . Moreover, this line bundle only depends on up to its class in .

Dimension

The dimension of is obviously one. Indeed, has dimension , and since is obtained by gluing to itself along an open subset, the conclusion follows.

Arithmetic applications

Since one of our main motivations for defining rigid analytic geometry is to study arithmetic geometry we would be remiss to not give some number theoretic applications of the theory of Tate elliptic curves.

Reduction type and Tate’s theorem

To explain these results, it is first helpful refine the results of Theorem 40 and Theorem 42 slightly. To do this, let us first recall some basic theory of elliptic curves over a -adic field . We shall denote the ring of integers of by and the residue field of by (don’t confuse this with our analytic field from above!). We shall denote the valuation on (the unique one extending the -adic valuation on ) by and we shal ldenote the associated absolute value on by .

For the sake of simplicity, we assume that 3" class="latex" />. This will allow us to work with simplified Weierstrass models.

We then recall, following [Con4], that a Weierstrass model of is a relative cubic curve over

together with an isomorphism . Again, note that [Con4] requires that is actually an essentially general cubic, but since 3" class="latex" /> we have that is invertible in and so we can always put our elliptic curve in this form (e.g. see [KM, (2.2.6)]). We define the discriminant of , denoted , as follows:

One can check that is indpendent of the isomorphism class of . One can give a more highfalutin description of the discriminant as in [Con4] where this becomes more obvious.

We then note that is smooth, and thus an elliptic scheme (in the sense of [KM, (2.1.1)]) if and only if . In general there won’t exist a Weierstrass model of which is smooth, and the best thing we can ask is that is a minimal Weierstrass model which means that is minimal amongst Weierstrass models for . A priori if one has two minimal Weierstrass models and one only knows that and, in particular, there is no reason for to be isomorphic to . This is the case though. This can be seen by explicit calculation (as in [Sil, 1.3 in VII]) or by abstract theory (as in [Con4, Corollary 4.6]).

Regardless, we now know that a minimal Weierstrass model for exists and is unique up to isomorphism. We then define the reduction of to be the curve .

Of course, this reduction is a priori just just a cubic curve embeddable in . In particular, there is no reason that should be smooth over . That said, is necessarily very mildly singular (if it’s singular at all):

Proposition 44: Let be any field and let be a singular cubic hypersurface in . Then, has exactly one singular point and is in .

Proof: Suppose first that is algebraically closed. Let and be points of which are both singular. Note then that there exists a line such and both lie in . Note then that by Bézout’s theorem (e.g. see [Liu, Corollary 1.20 in §9.1.2]) we have that the and in particular we see that one of or (using [Lieu, Definition 1.1 §9.1.1]) must be . Without loss of generality we may assume that . But, this implies that is smooth at —intuitively this means that and meet transversally at so that

(where is the tangent space) which then implies that has dimension , which means that is smooth at (for details see [Liu, Proposition 1.8 in §9.1.1]). This is a contradiction.

Suppose now that is not algebraically closed and assume that is a singular point of . Note that if is not a -point of then the preimage of under the map has more than point. Each of these points is necessarily singular which contradicts the first paragraph.

We thus can try to understand the geometric properties of , and thus the properties of , by trying to suss out the properties of the singularity of or, equivalently (perhaps not so obviously) the properties of of the smooth locus which, by Proposition 44, is just for some -point in . Let us note that, as can be checked by hand, the point in is smooth and so, in particular, .

Before we state the marquee classification in this direction, we first note the following beautiful fact:

Proposition 45: There exists the unique structure of a group variety on such that is the identity element. Moreover, is smooth, integral, and dimension . If is not smooth then is affine.

Proof(Sketch): The point is that the chord-tangent construction as in the theory of elliptic curves works here as well.

More explicitly, let be distinct. There exists a unique line such that passes through and . If we set to be the unique line in passing through and which is tangent to at .

Note then that by Bézout’s theorem that for any (possibly with ) we have that is a finite -scheme with -dimensional global sections. If then it’s easy to see that is the disjoint union of three -points of which are and a third point . Moreover, note that necessarily is another point of since the intersection of and neccessarily has multiplicity (cf. see the proof of Proposition 44). If then is the disjoint union of (where for a closed subscheme of we denote by the ideal of induced by ) and an -point . Again, since the multiplicity of at is , we see that is in . In either case we see that we obtain a third point from .

Note then the exact same ideas yield associated to the pair a line and a third point of intersection of the line and which is in . We denote this third -point of by .

The proof that this process gives a group structure on is the same as in [Sil, Proposition 2.2 in §III.2] and the fact that this is algebraic and defined over is the same as in [Sil, Group Law Algorithm 2.3 in §III.2].

Let us now verify the claimed geometric properties of . The fact that it’s smooth and dimension are obvious, and thus it suffices to prove that it’s integral. Since it’s smooth it suffices to prove it’s connected (e.g. by [Stacks, Tag033M]). But, to do this it suffices to show that is irreducible. But, this is clear since

where and are the images of and in . In particular, note that the equation

is evidently irreducible by setting and arguing that any factorization of would have a non-zero term. From this, the claim that is irreducible is clear from basic theory.

Finally we verify the affineness claim. But, this follows from totally general theory (e.g. see Lemma 13 from this post).

Thus, a natural question is what group is. As it turns out there are not many options:

Proposition 46: Let be a finite field and a smooth integral connected group variety of dimension . Then, either is an elliptic curve or it’s isomorphic to one of the following three groups:

The additive group .

The split multiplicative group .

The non-split multiplicative group .

Proof: This follows from Theorem 29 of this post.

Here we using the notation to denote the norm elements of the Weil restriction (e.g. see [Mil1, §2.i]).

Abstractly this is the group scheme which associates to an -algebra the group

where is defined to be the determinant of the -linear map

which makes sense since is a free -module of rank .

Concretely if for then

can be thought about as the subgroup of given as the set of matrices of the form

with the usual matrix multiplication.

So, with this setup we can divide up the elliptic curves over into categories depending on which group is which constitutes four cases:

If is an elliptic curve we say that has good reduction.
If is the additive group we say that has additive reduction.
If is the split multiplicative group we say that has split multiplicative reduction.
If is the non-split multiplicative group we say that has non-split multiplicative reduction.

There is further terminology concerning reduction type that is useful. Namely, if has either split or non-split multiplicative reduction, we say that has multiplicative reduction. If has either good reduction or multiplicative reduction then is said to have semistable reduction. If has either good or additive reduction it is said to have potentially good reduction. The reason for this naming scheme will soon be clear.

As we said before, we can also understand the reduction type of in terms of the singularity of at its singular point (if it has one) as follows:

Proposition 47: Let be an elliptic curve over and assume that . Then, the following are equivalent:

has good reduction.

.

If one writes as the projective closure of , then is separable.

So are equivalent the following:

has additive reduction.

The completed local ring is isomorphic as -algebra to .

0" class="latex" /> and .

If one writes as the projective closure of , then has a repeated root of multiplicity .

So are equivalent the following:

has split multiplicative reduction.

The completed local ring is as a -algebra isomorphic to .

, , and is a square in .

If one writes as the projective closure of , then can be factorized in in the form for some .

So are equivalent the following:

has non-split multiplicative reduction.

The completed local ring is isomorphic as a -algebra to where is an irreducible homogenous polynomial of degree .

, , and is not a square in .

If one writes as the projective closure of , then is irreducible in .

Proof: All of these are fairly straightforward and are covered in [Sil, Proposition 5.1 in §VII.5] except the statements concerning the completed local rings, but these can be checked elementarily given the explicit description of the polynomials cutting out in as explained in each part.

Using these characterizations, especially those involving the valuation of , it’s not super hard to deduce the following:

Corollary 48 (Semistable reduction theorem): Let be an elliptic curve over . Then, if has multiplicative reduction then so does for every finite extension . If has good reduction, then so does for every finite extension . If has additive reduction, then there exists a finite extension such that has good reduction.

Proof: See [Sil, Proposition 5.4 in §VII.5]. .

Remark 49: Be careful to not make the following mistake. It seems tempting to think the above theorem is wrong, for examples that additive reduction becomes good reduction over an extension, given Proposition 47. Namely, if one assumes that the minimal Weierstrass model of is . That said, this claim only necessarily holds true if is unramified (i.e. that is étale). So, the extension which changes an additive reduction curve to good reduction is necessarily ramified.

Another useful thing that follows from Proposition 47 is the following:

Corollary 50: An elliptic curve over has multiplicative reduction if and only if 1" class="latex" />. Consequently, has potentially good reduction if and only if .

Proof: For example, see [Sil, Proposition 5.5 in §VII.5].

We can then understand Tate’s theorem as follows:

Theorem (Tate) 51: The map

given by is a bijection.

Proof: See, for example, [Sil1, Theorem 5.3 in §V.5].

This makes intuitive sense since one might imagine that should have the result of since projects to in , and this looks quite a bit like should have split multiplicative reduction. Of course, this is clearly missing part of the picture since, for example, we don’t see where the singularity of is entering into the picture. We will revisit this discussion when we discuss Raynaud’s perspective of rigid geometry.

Arithmetic applications

Now that we have a more refined sense of which elliptic curves are Tate elliptic curves, they are precisely the ones with split multiplicative reduction, we can now more properly state the arithmetic applications of the theory.

Tate modules

The first application is to the study of the -adic Tate module of an elliptic curve over . Namely, recall that if is an elliptic curve then its -adic Tate module is the representation given by

where is the closed subgroup scheme of given by the -torsion and the transition maps are the multiplication-by- maps (e.g. see [Sai, §1.3] or [Sil, §III.7]). We also have the rational -adic Tate module which is the -representation given by

Note that as an abstract -module one has that is isomorphic to and thus is isomorphic abstractly to as a -vector space (e.g. see [Sai, Proposition 1.19] or [Sil, Proposition 7.1 in §III.7]).

Remark 52: More abstract one has identifications of Galois representations

where these objects are the étale cohomology groups of , but we won’t really use or need this fact. See this post for more details.

Using the theory of Tate elliptic curves we deduce the following:

Proposition 53: Let be an elliptic curve over with split multiplicative reduction. Then, for every one has an extension of Galois representations

which is non-split for sufficiently large and therefore one has a non-split extensions

and

Here we are using the notation to denote the -representation given by and by the representation with the transition maps being the -power maps. We are also using and to denote the trivial Galois representation on these groups.

Proof (of Proposition 53): Let us make an identification for some . Let us also fix for each a primitive root of unity in and a -th root of . Let’s assume that we’ve made this choice compatabily so that

for all .

Let us then note that from the isomorphism of

we easily see that

as Galois representations. But, evidently we then have the subrepresentation given by and generated by . Note moreover that for each in one has that we can write for a unique . Thus, we see that the quotient is isomorphic to with the trivial action as desired.

The fact that this sequence doesn’t split for sufficiently large is clear. Indeed, such a section for the projection map must send to for some and for this section to be Galois equivariant we need that this element is Galois fixed. This implies that is in , and evidently if is sufficiently large this is impossible.

The -adic statement follows by passing to the limit. Checking non-splitting at the rational level requires a small argument, but we leave this to the reader.

In fact, we can elaborate on this result even further. Namely, note that one has

where the first object is the set of isomorphism classes of extensions in the category of Galois -representations, and the second equality follows from Kummer theory (e.g. see this post). We can then understand the above as saying that the association of an elliptic curve with split multiplicative reduction to the extension class of in is the map sending to the class of in . The non-splitness is then clear since is not in for sufficiently large . The non-splitting of the rational Tate module is then clear since the natural map

is injective.

Of course, one might be interested in understanding what happens in the case of non-split multiplicative reduction. But, this is quite simple as the following shows:

Proposition 54: Let and over any field and suppose that becomes isomorphic to upon base extension to the quadratic extension . Then, there is an isomorphism of Galois representations

where is the quadratic character associated to .

Proof: For example this follows from [Sil1, Lemma 5.2(c) in §V.5].

Here the quadratic character associated to is the unique non-trivial homomorphism factoring through . By we mean the twist of by which means the same underlying module of but with the twisted Galois action. Namely, for and if denotes the action of on in , then the action of on in is .

Note that one interesting consequence of the above discussion is that -adic Tate modules (and thus -adic étale cohomology groups by Remark 52) needn’t be semisimple Galois representations.

Bounding torsion

One of the first major theorems that one proves about the arithmetic of elliptic curves is the Mordell–Weil theorem which states that if is an elliptic curve over a number field , then is finitely generate. This result requires quite a bit of sophistication in the form of the weak Mordell–Weil theorem (e.g. see this post) and techniques using height functions (see [Sil, §VIII.4-§VIII.6] for details).

One consequence of this is that the group of torsion points of is finite. That said, one can get a significantly easier proof of this, at least assuming the Tate theory, of elliptic curves. The point of this is not necessarily to claim that this is a ‘more elementary’ or ‘simpler’ proof than the usual proof of the Mordell–Weil theorem (or, of course, this can be deduced more simply by using Nagell–Lutz and some theory about reduction—for example see [Mil2, §II.4]) but to illustrate that we can get a purely arithmetic-geometric result by analytic methods.

Regardless, we see the following result quite easily:

Proposition 55: Let be an elliptic curve over a number field . Suppose that has a prime such that has multiplicative reduction. Then, is finite.

Proof: Set to either be , if has split multiplicative reduction over , or the quadratic extension of that splits if has non-split multiplicative reduction. Note then that we have an injection

and we know from Tate’s theorem that

for some with . That said, is clasically isomorphic as a group to

where is the set of roots of unity in (which is finite) and is a finite free -module (e.g. see [Lor, §25.4]). But, this group has the property that all of its quotients have finite torsion from where the claim follows.

Of course, by Corollary 50 we know that has

Endomorphism rings

As a last example, we borrow the following result from [Sil1, §V.6] for which a -adic proof using the theory of Tate elliptic curves is due to Serre.

Namely, let us recall that if is a characteristic field and is an elliptic curve over then we denote by the ring of endomorphisms of -group scheme . As it turns out, there are not many options for the ring . Namely, this ring must either be or an order in an imaginary quadratic extension of (e.g. see [Sil, Corollary 9.4 in §III.9]). We say that has geometric complex multiplication if is an order in a quadratic imaginary extension of

Despite first appearances, curves with geometric complex multiplication play a profound role not only in the theory of elliptic curves. Namely, they are certain ‘special’ (or ‘exceptional’) types of curves that are often exceptions for many of the deep theorems in the arithmetic of elliptic curves (cf. [Sil, Theorem 4.9 in §V.4] or [Sil, Corollary 6.3 in §IV.6]) but also serve as the conceptual basis for the theory of canonical models of Shimura varieties (e.g. see [Mil3, §10]).

For this reason, any statement which helps understand elliptic curves with geometric complex multiplication are incredibly significant from the point of view of arithmetic. It is then interesting that Serre was able to prove the following theorem using the theory of -adic Tate curves:

Proposition 56: Let be a number field and an elliptic curve over . If has geometric complex multiplication then .

Proof: For a proof using the theory of Tate elliptic curves see [Sil1, §V.6].

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Objects with no rational models

alexyoucis — Fri, 04 Sep 2020 00:09:41 +0000

The goal of this post, which is just a bit of fun between more serious posts/projects, is to discuss some examples of algebro-geometric objects over which have no models over smaller subfields and explain how moduli theory can help clarify their discovery in certain situations.

Motivation

This post was born out of a very simple question I had at the beginning of my second year of graduate school:

Main question: “What is an explicit example of a variety over which has no model over ?”

Let us be slightly more rigorous about what we mean. Namely, we say that a variety over is defined over if there exists a variety over and an isomorphism .

This sounds like a fairly simple problem. For example, one is tempted to make the claim that

is not defined over since its defining equation in involves non-rational coefficients. But, of course, obviously as a -variety one has that

by the isomorphism and the latter is obviously defined over .

So, one is quickly led to trying more complicated examples involving equations of a more sophisticated form. For example, one might try equations of the form

where or, in other words, elliptic curves. This seems promising since if and are both irrational then there’s no obvious way to ‘fix this’. Indeed, if we try to ‘suck’ the into , as we sucked the into in our previous example, we see that we’ll be forced to complicate the coefficients of and . It’s then not at all clear that somehow you can make a change of variables that simultaneously makes all coefficients rational.

Of course, this is not a proof, and some complicated and surprising isomorphisms exist. For example, certainly looks like it should not be defined over at first glance but, in fact, one has an isomorphism

and the latter, as complicated as it is, is evidently defined over .

Given the above, it then seems less obvious how to actually find an example of a variety over which provably has no model, let alone an example where satisfies specified properties (e.g. being some fixed dimension, smoothness, affineness, etc.).

The goal of this post is to try and explain a general geometric methodology by which one might try to answer the main question. We will then give some additional examples of objects without models which doesn’t fit quite as nicely into this geometric framework, but which are still interesting.

Moduli theory

We now develop a geometric framework in which finding examples of varieties with no models can be made to be methodological.

Moduli problems

It is first useful to widen our gaze and consider the general situation in which our main question naturally sits. Namely, let us consider the structure that the objects in our original have. Namely, we had

A family of objects over (varieties over )
A family of objects over (varieties over ).
A notion of isomorphism between objects in either situation.
A pullback map which associates to an object over an object over (the fiber product of curves).

Of course, the structure above was just part of the more genral structure where instead of objects only over or we have objects over for any -scheme (namely -schemes).

We see then that the framework in which to discuss the generalization of our main question is likely the following. Let us define a moduli problem over to be a contravariant functor

where

is the category of schemes over .
is the category of groupoids.

Here by a groupoid we mean a small category (i.e. a category whose collection of objects is a set) where every morphism in is an isomorphism. A morphism of groupoids is just a functor between the groupoids.

Let’s unpack this definition a bit. The data of a moduli problem over is the following:

For every -scheme a groupoid , which is just a collection of objects ‘over ‘ together with a notion of isomorphism between them.
For every map of -schemes a functor (which we call the pullback functor)..

This is precisely the structure we observed our main question having above.

Remark 1: It is probably useful to put our definition into the context of more rigorous mathematics. Namely, what we’re calling a moduli problem over is what is known (e.g. as in [Olsson, Chapter 3]) as a split category fibered in groupoids over . These are simpler objects to understand than general categories fibered in groupoids as explained in [Olsson, §3.4] but, as seen in loc. cit., essentially every category fibered in groupoids is equivalent to a split one (although it is not true that it itself must be split). Thus, we shall always confuse the two notions. We will see what this means more concretely below in Remark 3.

Let us give some examples of moduli problems over

Example 2: We start with the moduli problem that was underlying our main question. Namely, let us define

where we set to be the category of -schemes with morphisms being isomorphisms of -schemes. The pullback functor is the usual fiber product of schemes/morphisms. More explicitly, if we have a morphism of -schemes then the functor

is defined on objects by the fibered product and on morphisms by the pullback of morphisms to the fibered product.

Remark 3: We see in the above where, in reality, it is more natural to work with categories fibered in groupoids with no particular splitting as mentioned in Remark 1. Namely, the fiber product is only defined up to isomorphism. So, to really make a moduli problem in our sense, one has to choose for all morphisms and -schemes over an explicitly chosen fibered product in a way that is compatible with composition of morphisms in . Again, we will ignore this subtelty both here and in the future examples.

Example 4: Let and be integers. For a -scheme we call a morphism a genus curve over if it is smooth, proper, and for every point of the -variety is a geometrically connected curve of genus . By an -marked curve of genus over we mean a genus curve over together with -marked sections of . By a morphism of -marked

we mean a morphism of genus curves over such that for all .

We can then define a moduli problem over

where for an -scheme we set to be the groupoid of -marked genus curves over with morphisms being isomorphisms. For a morphism we define the pullback map

to be as expected:

where

is on the -factor and on the -factor.

Example 5: Let us define

to associate to a -scheme the groupoid of affine -schemes with morphisms being usual isomorphisms of -schemes. The pullback morphisms is the usual fiber product of affine schemes.

Example 6: Let

be a presheaf on . Then, evidently is a moduli problem over since any set can be thought of as a groupoid where the only morphisms are the identity morphisms.

Example 7: We give an even more special case of the last example. Namely, if is any -scheme, then we can think of as a moduli problem over by identifying with its contravariant functor

which is reasonable to do by Yoneda’s lemma (e.g. see [Stacks, Tag001P])

Example 8: Let be an integer. We then define the modular curve with full level structure to be the moduli problem

where is the category of pairs where is an elliptic scheme (which is a relative version of elliptic curves—see [KM, Chapter 2] or [Hida1, Chapter 2]) and

is an isomorphism of group schemes which we call a trivialization. The isomorphisms in this category are isomorphisms of elliptic schemes that preserve the trivialization in the obvious way. The pullback map is the obvious one. Let us note, as essentially shown in [KM, Chapter 2] and [Hida1,Chapter 2], one has that is essentially identified with the moduli problem from Example 4.

There is a natural generalization of to higher abelian varieties denoted by for any integers and so that . To define this would take us too far astray, but see cite [Hida2, §6.3.2] for details (note that in the notation of [Hida2, §6.3.2] our notation is , , and ).

Example 9: We denote by

the moduli problem which associates a -scheme the category or finite type affine group schemes over with morphisms being isomorphisms of -group schemes. The pullback map is the obvious one. We also have a submoduli problem of consisting of those affine group schemes which are reductive (e.g. see [Conrad1, Definition 3.1.1]).

Example 10: Let be a (connected) reductive group over (e.g. see [Milne, §19.b]). We then define a presheaf

which assigns to a -scheme the set of maximal tori in (e.g. see [Conrad1, §3.2]).

We define a morphism of moduli problems over to just be a a natural transformation of functors . There is the obvious notion of an isomorphism then between moduli problems over which means that for all -schemes the functor is an isomorphism of categories. We call an equivalence if is an equivalence of categories for all .

We end this subsection by making a definition that is clearly useful given that our main question is about isomorphism classes. Namely, note that if is a groupoid then the set

is a set which we call the connected components of — aptly named since if we think of as being a directed multigraph then is the set of connected components in the graph theoretic sense.

We note then that associated to any moduli problem over is a presheaf

which we call the component presheaf of . Let us note that if if is a morphism of moduli problems over then we get an induced morphism . Moreover, if is a presheaf then since has no non-identity morphsims for all , we see that . Combining these two, we see that any morphism , where the former is a moduli problem over and the latter is a presheaf, factors uniquely through the obvious map .

The generalized main question and fine moduli scheme

So, we have formalized a generalization of the objects in the main problem in the last section but, as of now, this all seems like empty verbiage. What is the point of the above if it doesn’t help us solve the main question? Well, it does. Or, perhaps more to the point, it more easily allows us to create a framework in which our main problem can be systematically solved.

Before we do that though, let us try and formalize what the extension of the main question to the general situation would be. A moment’s thought shows that, with our above notational choices, we can formulate it as follows:

Generalized main question: Let be a moduli problem over . Then, is the map

surjective?

Let’s warm-up for our approach to understanding the generalized main question by thinking about the problem in a very specific case. Namely, as we saw in Example 7 it’s true that any scheme over over defines a moduli problem over . It turns out that we can leverage geometry then to answer the generalized main question quite easily for (assuming minor conditions).

Namely:

Proposition 11: Let be a scheme locally of finite type over which is of positive dimension. Then, the map is not surjective.

Proof: Let be any affine open subset of . Evidently then one has that the map

is surjective. Since has an affine open subscheme of the same dimension, we may assume without loss of generality that is affine.

By Noether normalization (e.g. see [Stacks, Tag00OY]) we know that there exists a finite surjective map where 0" class="latex" />. Note then that we have a commuting diagram

Now, the right vertical map is surjective since is surjective and, by assumption, the top horizonal map is surjective. Evidently this then implies that the bottom horizontal map is surjective. But, this is a contradiction.

Of course, this gives a fairly definitive answer to the generalized main question in the case when is a scheme. In fact, it’s to our benefit to put a little more thought into this statement. Namely, let us say that a moduli problem over has a fine moduli scheme if there exists a scheme over and an equivalence . Note that by Yoneda’s lemma such a scheme is unique up to isomorphism.

We can then deduce from Proposition 11 the following:

Corollary 12: Let be a moduli problem over which posseses a fine moduli scheme which is locally of finite type over and of positive dimension. Then, the map

is not surjective.

We now consider how this interacts with the examples from above by discussing those which possess a fine moduli scheme:

Example 13: Let and be integers. It is then a theorem of Mumford (e.g. see [Hida2, Theorem 6.20]) that , the moduli problem from Example 8, has a fine moduli scheme which is quasi-projective, smooth, and of dimension (for this latter fact see the description of the complex points of after (A4) on [Hida2, Pg. 277]). From this and Corollary 12 we deduce a negative answer to the generalized main question for the moduli problem for .

Example 14: Let be a reductive group over . Then, the moduli problem classifying maximal tori in , as in Example 10, has a fine moduli scheme space which is smooth, quasi-affine, and of positive dimension of is not a torus. Indeed, by [Conrad1, Theorem 3.2.6] the fine moduli scheme exists and is isomorphic to for any maximal torus of . Now, if were zero-dimensional it must be trivial since is smooth (by loc. cit.) and geometrically connected (since is, see [Milne, Theorem 2.37(c)]). But, this then implies that and, since is connected, that . But, this implies that (e.g. combine [Milne, Corollary 17.39(a)] and [Milne, 17.84]). Thus, combining this with Corollary 12 we see that we have a negative answer to the generalized main problem for .

Of course, the problem is that ‘most’ interesting moduli problems are not going to have a fine moduli scheme. In fact, there is a good way to think about why this is a case. One can easily note two things about schemes which makes them special in the pantheon of moduli problems over

They have ‘gluing properties’ that a general moduli problem over needn’t have. More rigorously, any presheaf which has a fine moduli scheme is a sheaf for the étale or fppf topologies on (e.g. see [Olsson, §2.1-§2.2] and [Olsson, Theorem 4.1.2]).
They have no non-trivial automorphisms. Namely, for any presheaf on and any -scheme we have that is a set and so, in particular, for any object of one has that is trivial (where this automorphism group is in terms of morphisms in the category ).

In particular, from 2. we see that any moduli problem over which has objects with non-identity automorphisms is bound to not have a fine moduli scheme.

Remark 15: What we’re describing above might be closer to what is called an ‘algebraic space’. See Remark 17 below.

That said, one might hope to ‘rigidify’ a moduli problem over so that it actually obtains a fine moduli scheme or, at least, so that it possesses property 2. from above. Namely, if one has an object then for any extra structure on the set of automorphisms preserving this extra structure certainly is going to be smaller than the set of all automorphisms of . So, adding extra structure (i.e. rigidifying) should cut down on the automorphism groups of objects in our moduli problem.

For example, has no fine moduli scheme since, for example, condition 2. fails for —an elliptic curve has non-trivial automorphisms (e.g. the inversion map ). But, one can think of for as a rigidification of since it’s essentially the data of an object of and an extra piece of information given by the isomorphism . Now, for (as implicitly stated in Example 13) one has that the rigidified objects of have no automorphisms and, in fact, possess fine moduli schemes.

Of course, this is not particularly useful for solving our main question since even if one can rigidify a moduli problem over so that it has a fine moduli scheme, all that will tell us is that there are objects of which don’t have models over with the extra rigidifying structure. Of course, this doesn’t preclude that these objects do have models over without this extra structure.

Coarse moduli spaces

So, we saw in the last section that one has a fairly concrete way of showing that the generalized main question has a negative answer if our moduli problem has a fine moduli scheme. We also saw that at the end of the last section that even if we can somehow modify a moduli problem to have a fine moduli scheme that this is not, in general, good enough to answer the genearlized main question. So, what do we do?

One obvious guess is that instead of trying to force Proposition 11 to apply directly to a moduli problem by assuming that has a fine moduli scheme, that we should try to adapt Proposition 11 to work in more generality.

Of course, we used geometry in the proof of Proposition 11 and so we will need to have some sort of theory of geometry for moduli spaces. But, as it stands, moduli spaces are so general as to make this not really tenable. So, a first step would be single out those moduli problems that are sufficiently geometric. Doing this, and carrying out the development of geometry in those cases, is precisely the modern theory of (algebraic/Artin) stacks as initiated by Deligne and Mumford in [DM] and further developed by many people.

We shall not need the theory of algebraic stacks explicitly but we mention very briefly the two extra major conditions that one imposes on a moduli problem to make them an algebraic stack. The first is that is a a ‘sheaf’ for the étale (equivalently, for algebraic stacks, the fppf topology—see [LMB, Corollaire 10.7]). Of course, since has values in groupoids, not sets, this is more complicated than the idea of being a presheaf being a sheaf for the étale topology and in fact making sense of what the sheaf condition means in this context is exactly what the ‘stack’ portion of the moniker ‘algebraic stack’ is referring to—see [Olsson, Chapter 4] for details. The second condition is that while is not a scheme, it at least needs to be covered by a scheme in a sense we shall not make precise here.

Remark 16: Several remarks are in order here. First, we only stated the two extra ‘major’ conditions to be an algebraic stack. There is one extra condition which is necessary—see [Olsson, Definition 8.1.4] for details. For more information on stacks see [Olsson], [LMB], and [Stacks, Tag0ELS] for general theory and [Bertin] for a more leisurely introduction.

Now the main reason that the theory of algebraic stacks is not explicitly useful for us here is that in fact (at least to the author’s knowledge) while it is incredibly rich theory, it is not sufficient to repeat the argument in Proposition 11.

This brings us to the other obvious guess for trying to escape the yoke of fine moduli schemes. Namely, for a moduli probelm we can ask whether or not there exists a scheme which may not be equivalent to , but ‘best approximates’ (amongst schemes) in a precise sense.

Let us try to be more rigorous. Namely, suppose that is a moduli problem over . We then say that a -scheme is a coarse moduli scheme for if there exists a morphism of moduli problems

such that

The map is initial amongst maps to -schemes.
The map is a bijection whenever is an algebraically closed field.

Note that by condition 2. the space is uniquely determined up to isomorphism by these conditions.

Remark 17: A remark is in order about the relationship between the above, which is non-standard terminology, and the notion of coarse moduli space which exists in the literature (e.g. [Olsson, Definition 11.1.1]).

The category of -schemes is not very well-behaved within the category of moduli spaces over (e.g. a presheaf which étale locally is isomorphic to a scheme need not be isomorphic to a scheme). For this reason one generally enlarges the category of schemes to the category of algebraic spaces. Intuitively these are presheaves which are étale locally schemes (e.g. see [Stacks, Tag0BGQ]) or they are also algebraic stacks with no automorphisms (e.g. see [Stacks, Tag04SZ]).

One then defines a coarse moduli space by replacing every instance of ‘scheme’ with ‘algebraic space’ in the definition of coarse moduli scheme. Note that if is a coarse moduli space of a moduli problem which is also a scheme, then it is a coarse moduli scheme in the sense above. The converse is not clear to me (is there a non-scheme algebraic space which has an initial map to schemes?), but we will never really contend with this issue here.

One definitive advantage of working with coarse moduli spaces opposed to coarse moduli schemes is the huge generality in which they exist. This is typified by the Keel–Mori theorem (e.g. see [Olsson, Theorem 11.1.2] and [Conrad2]).

Note that since is a scheme it’s a presheaf and thus as mentioned before we have a factorization

and we denote this factorized arrow by as well.

Now, as intimated the existence of a coarse moduli scheme is actually a strong enough to answer the generalized main question in the negative. More rigorously:

Proposition 18: Suppose that is a moduli problem over which admits a coarse moduli scheme which is a positive-dimensional scheme locally of finite type over . Then, the map

is not surjective.

Proof: Let us note that we have a commuting diagram

Since the right vertical map is bijective since is a coarse moduli scheme of , we see that if the top horizontal map is surjective then so is the bottom horizontal map. But, this then contradicts Proposition 11. The exact same argument works to show that the map is not surjective.

Remark 19: One can replace Proposition 18 by a generalization where one only assumes that admits a coarse moduli space (in the sense of Remark 17) which has a positive dimensional open subscheme of finite type over . This is, I believe, automatically the case if is separated, locally of finite type, and is of positive dimension (in the sense of [Stacks,Tag04N6])—indeed one uses [Stacks, Tag0BAS] and the definition of dimension.

So, we see that the existence of a coarse moduli schemes (which is sufficiently well-behaved) is enough to guarntee a negative answer to our generalized main question. Of course, this is only useful if, unlike the case of fine modulie schemes, more moduli problems have coarse moduli schemes. The main example of this will be discussed in the next section.

But, before we go on we note that we can actually make our solution to the generalized main question for a moduli problem with coarse moduli scheme more constructive. Namely, let us say that an element of is -rational if is in . Then, evidently we see that if is in the image of the map then is -rational. So, any which is not -rational cannot be in the image. This will be the main technique we will use to find objects with no model—find objects which are not -rational.

The main question for curves

We now try to leverage our discussion of coarse moduli schemes from the last section to answer more explicitly the generalized main question for the moduli problems . It turns out that the nature of this discussion varies greatly depending on whether , , or . So, we handle these cases separately.

The case of genus one

We start with the case of genus because it has both an interesting answer, but also is likely to be more familiar to those with a basic background in algebraic geometry. The thing that makes the case of genus slightly more tricky than the case of genus greater than is that doesn’t apparently have a coarse moduli scheme if one looks through the literature. The reason for this is quite simple: is not an algebraic stack, and thus it’s not common to talk about coarse moduli spaces for such objects. But, we’ll see here that this doesn’t matter much.

We begin by noting that while , the moduli problem of genus curves, doesn’t apparently have a coarse moduli scheme, the closely related moduli problem of elliptic curves does:

Proposition 20: The moduli problem has as a coarse moduli space.

Proof: For a full rigorous proof see [Olsson, Theorem 13.1.15].

In fact, the map

realizing is the -invariant. See loc. cit. for more details, but let us suffice it to say that if one has an extension of then every elliptic curve over is isomorphic to one of the form

for some such that (e.g. see [Silverman, §III.1] and [Silverman, §III.3]). Then,

(see again [Silverman, §III.1] and compare this with [Olsson, Theorem 13.1.15]).

From this, we deduce the following:

Corollary 21: Let be an elliptic curve over . If has a model over as an elliptic curve, then .

In fact, one can show that

if and

so that the composition

is surjective for all field extensions of . Using this, one can actually upgrade the only if statement in Corollary 21 to an if and only if statement.

So, how does this answer our question about ? Well, I claim that we can use the -invariant to build a coarse moduli scheme for . How? Well, the key observation is that we actually have a contravariant morphism of moduli problems

given on a -scheme as the functor

sending a curve of genus over to its Jacobian which is, by definition, the identity component (e.g. see [BLR, Chapter 8] and [BLR, Theorem 1 in §9.3]). The Jacobian is evidently an element of since it’s a smooth group scheme over whose fibers are cruves of genus (as can be seen from [Silverman, Proposition 3.4 in §III.3]) and which is proper (e.g. see [EGAIV Troisième partie, Corollaire 15.7.10]).

Now, what makes the map so useful is the following:

Proposition 22: For any algebraically closed extension over the induced map

is a bijection.

Proof: Suppose that and are two genus curves over . Choose base points (which is possible since is algebraically closed). Note then that there are isomorphisms of group schemes (e.g. see [Silverman, Proposition 3.4 in §III.3]). Thus, if then . Thus, is injective.

To see that the map is surjective we again note that for any elliptic curve over one has an isomorphism of group varieties (again by [Silverman, Proposition 3.4 in §III.3]).

From this, we deduce the following:

Proposition 23: The composition

is a coarse moduli scheme.

Proof: To see that this map is initial, note that we have a natural map of moduli problems

which just forgets about the marked point. Suppose that we have a map of moduli problems

for a -scheme . Note then that by Proposition 20 there exists a unique morphism such that

We then claim that

Indeed, since the map to factors through the component presheaf it suffices to note that for any genus curve over one has that

But, let us note that since is smooth, there exists an étale cover such that has a section. Note then that this implies that is isomorphic to as follows from [KM, Theorem 2.1.2]. So, then we see that has image in given by

since is an elliptic curve. But, this implies that and have the same image under the map which implies they’re equal since this map is injective since is a sheaf.

The claim that induces a bijection on -points for an algebraically closed extension of follows by combining Proposition 20 and Proposition 22.

Thus, combining everything above, we have the following corollary:

Corollary 24: Let be a genus curve over . Then, is defined over if and only if is rational.

Thus we have answered our generalized mai question for , and we will have answered our main question as soon as we exhibit a genus curve over with irrational -invariant. But, for example, follows from our discussion above we have that the genus curve we know that the curve

has -invariant and thus is not defined over .

The case of genus

We now move onto the case of genus curves. This is, in some sense, simpler than the genus case since there is a well-known explicit description of the coarse moduli scheme for in the literature (whereas, as we remarked in the last section, there is none for since it’s not an algebraic stack and thus not usually considered). But, it is also harder than the genus case since the coarse moduli scheme is more complicated.

In particular, we start with the following:

Theorem 25 ([Igusa, Theorem 2]): There is a morphism

where is given degree . Moreover, this morphism realizes as the coarse moduli scheme of .

Let us explain in more detail what this coarse moduli space is. Namely, we are thinking of as a graded ring where each is given degree . With this definition one sees that is homogenous of degree and thus we can form the quotient ring

which has an inherited grading. We can then talk about the proj construction of this graded ring. We then note that since is homogenous of degree and thus we can consider the non-vanishing locus in

which is an affine -variety with coordinate ring

(i.e. the degree elements of the localization of at ).

Let us denote this scheme which is the coarse moduli scheme by for notation convenience. The coordinates of are called the Igusa invariants. Let us note that, by definition, we can realize as a locally closed subscheme of the weighted projective space:

In particular, we see that we can describe as the set of equivalence classes of quintuples such that

Each is in .
One has that .
One has that .

where is equivalent to if there exists some in such that for all .

Thus, since is evidently finite type over and of positive dimension we know that the answer is negative for the generalized main question for .

In fact, to find a genus curve over not defined over we need only find a such that quintuple

of Igusa invariants is not equivalent to rational qunituple. Unfortunately, the actual definitions of the Igusa invariants is pretty complicated (e.g. see [BSSVY, §4.1]). That said, there is an implimentation of the Igusa in Magma (e.g. see [Magma]) and using this one can show that if is the normalization of

then

which one can quickly check is not equivalent to a rational quintuple. Thus, is not defined over .

The case of genus greater than

The case of 2" class="latex" /> is essentially the same as in the case but is much less explicit. Namely, we have the following well-known result:

Theorem 26: Let be an integer. Then, has a coarse moduli space which is a smooth affine -scheme of dimension .

Proof: See [MFK, Proposition 5.4].

From this, we deduce from Proposition 18 that for each there exists a genus curve over which is not defined over . Of course, it’s more difficult to give an explicit example in general since the explicit descriptions of , and thus the -rational objects in are more difficult to describe.

The case of genus

We end with the case of genus curves. This case, for our particular generalized main question, is not so interesting. Namely, we recall the following basic theorem:

Proposition 27: Let be a field and a smooth proper geometrically connected curve over of genus . If , then .

Proof: See [Liu, Proposition 4.1 in §7.4.1].

From this we see that is a singleton and thus the answer is trivially affirmative to the generalized main question for .

In particular, this would suggest that would, if it has a coarse moduli space, have one of dimension . Indeed, one can in fact show that is a coarse moduli space (e.g. see [Hartshorne, Proposition 25.1]).

Miscellanea

In this final section we’d like to sew up the cases of the generalized main question for the examples we discussed above. Note that Example 2, Example 4, Example 7, Example 8, Example 10 are covered by the discussion above. Thus, it reall suffices to discuss, Example 9, Example 5, and .

Example 5

We would like to give at least one example of an affine variety over which has no model over . There is no moduli theoretic technique I am aware of for the production of such an example (I’m not even aware any well-behaved moduli spaces of affine varieties with no extra structure!), so our construction will be more ad hoc.

The key fact is that there exists smooth proper geometrically connected curves over such that is trivial. In fact, there exist explicit examples of such curves of genus for each (e.g. see [Poonen2]).

Remark 28: One can extend the construction below to the case of genus curves with a little more work, using the objects constructed in [Poonen3]. We leave it to the reader to flesh this out.

Let be such a curve and let be an element of not contained in . Then, is an affine open subscheme of (e.g. see [Youcis, Lemma 13] for a proof of this well-known result).

We then have the following claim:

Proposition 29: The smooth integral affine curve does not have a model over .

Proof: Suppose that has a model over . Let be a smooth compactification of . We claim that is isomorphic to . Indeed, since is projective one knows that the set of isomorphism classes of varieties over such that is classified by the cohomology set

(e.g. see [Poonen1, Theorem 4.5.2]). But, since is trivial this implies that any such must be isomorphic to over . Since is a compactification of we know that . Thus, .

Note then that becomes isomorphic to over and so consists of a degree effective divisor or, equivalently, for some . Note now that the isomorphism

extends to an isomorphism

such that maps to . But, since is trivial we know that is the identity, and so . This is a contradiction.

Thus, we see that the generalized main question has a negative answer for the moduli problem from Example 5.

Remark 30: The argument given above is somewhat persnickety and, in particular, it’s not clear (to me at least) how to generalize it to produce examples in higher dimensions.

Example 6

Finding examples of presheaves that have an affirmative answer to the generalized main questions are trivial. For example, you can take any constant presheaf. Thus, one either has an affirmative or negative answer to the generalized main question for the moduli problem in Example 6 depending on the specifically chosen presheaf.

Example 9

It is not obvious but it is true that generalized main question has an affirmative answer for the moduli problem from Example 9. The main ingredients are as follows. Every reductive group over is split (in the sense of [Milne, §21.a]). It is then a non-trivial theorem that has a model over . Essentially, one constructs from and the choice of a maximal torus of a root datum (e.g. see [Milne, §21.c]). One then uses the Existence Theorem (e.g. see [Milne, Theorem 23,55]) to say that there exists a split reductive group over which has this associated root datum. One then uses the Isomorphism Theorem (e.g. see [Milne, Theorem 23.25]) to deduce that .

Interestingly though, the generalized main question for the larger moduli problem of all affine finite type group schemes has a negative answer. Indeed, by [Milne, Theorem 14.37(b)] it suffices to show that there exist nilpotent lie algebras over which don’t have models over . Examples of such objects were originally constructed by Malcev in [Malcev]. But, since this article seems difficult to obtain the reader may also see [Scheuneman, §3 Proposition 2].

References

[Bertin] Bertin, J., 2013. Algebraic stacks with a view toward moduli stacks of covers. In Arithmetic and Geometry Around Galois Theory (pp. 1-148). Birkhäuser, Basel.

[BSSVY] Booker, A.R., Sijsling, J., Sutherland, A.V., Voight, J. and Yasaki, D., 2016. A database of genus-2 curves over the rational numbers. LMS Journal of Computation and Mathematics, 19(A), pp.235-254.

[BLR] Bosch, S., Lütkebohmert, W. and Raynaud, M., 2012. Néron models (Vol. 21). Springer Science & Business Media.

[Conrad1] Conrad, B., 2014. Reductive group schemes. Autour des schémas en groupes, 1, pp.93-444.

[Conrad2] Conrad, B., 2005. Keel–Mori theorem via stacks. preprint. http://math.stanford.edu/~conrad/papers/coarsespace.pdf

[EGAIV Troisième partie] Grothendieck, Alexander. Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie. Publications Mathématiques de l’IHÉS, Volume 28 (1966) , pp. 5-255. http://www.numdam.org/item/PMIHES_1966__28__5_0/

[Hartshorne] Hartshorne, R., 2009. Deformation theory (Vol. 257). Springer Science & Business Media.

[Hida1] Hida, H., 2011. Geometric modular forms and elliptic curves. World Scientific.

[Hida2] Hida, H., 2012. p-adic automorphic forms on Shimura varieties. Springer Science & Business Media.

[Igusa] Igusa, J.I., 1960. Arithmetic variety of moduli for genus two. Annals of Mathematics, pp.612-649.

[LMB] Laumon, G. and Moret-Bailly, L., 2018. Champs algébriques (Vol. 39). Springer.

[KM] Katz, N.M. and Mazur, B., 1985. Arithmetic moduli of elliptic curves (No. 108). Princeton University Press.

[Magma] https://magma.maths.usyd.edu.au/magma/handbook/text/1523

[Malcev] Mal’cev, A.I., 1951. On a class of homogeneous spaces, Translations Amer. Math. Soc.(2), 39, pp.276-307.

[McNinch] McNinch, G.J., 2010. Levi decompositions of a linear algebraic group. Transformation Groups, 15(4), pp.937-964.

[Milne] Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.

[MFK] Mumford, D., Fogarty, J. and Kirwan, F., 1994. Geometric invariant theory (Vol. 34). Springer Science & Business Media.

[Olsson] Olsson, M., 2016. Algebraic spaces and stacks (Vol. 62). American Mathematical Soc..

[Poonen1] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc..

[Poonen2] Poonen, B., 2000. Varieties without extra automorphisms~ I: curves. Mathematical Research Letters, 7(1), pp.67-76.

[Poonen3] Poonen, B., 2000. Varieties without extra automorphisms II: Hyperelliptic curves. Mathematical Research Letters, 7(1), pp.77-82.

[Scheuneman] Scheuneman, J., 1967. Two-step nilpotent Lie algebras. Journal of Algebra, 7(2), pp.152-159.

[Stacks] Stacks Project Authors, 2020. Stacks project. https://stacks.math.columbia.edu/

[Youcis] Youcis, Alexander. CLASSIFYING ONE DIMENSIONAL GROUPS (II). https://ayoucis.wordpress.com/2019/11/19/classifying-one-dimensional-groups-ii/

A new paper draft

alexyoucis — Mon, 10 Feb 2020 14:40:57 +0000

I have not been able to post of late as I’ve been quite busy working on several projects.

I wanted to make a post though discussing a new draft with my collaborator A. Bertoloni Meli that I’m quite excited about. In it we discuss a method for characterizing the local Langlands conjecture for certain groups as in Scholze’s paper [Sch]. Namely we show that for certain classes of groups an equation like that in the Scholze–Shin conjecture (see [Conjecture 7.1, SS]) is enough to characterize the local Langlands conjecture (for supercuspidal parameters) at least if one is willing to assume that other expected properties of the local Langlands conjecture hold.

The main original idea of this paper is the realization that while the Langlands–Kottwitz–Scholze method only deals with Hecke operators at integral level (e.g. see the introduction to [Sch]) that one can circumvent the difficult questions this raises (e.g. see [Question 7.5,SS]) if one is willing to not only consider the local Langlands conjecture for in isolation, but also the local Langlands conjecture for certain groups closely related to (so-called elliptic hyperendoscopic groups). Another nice byproduct of this approach is that while the Scholze–Shin conjecture is stated as a set of equations for all endoscopic triples for our paper shows that one needs only consider the trivial endoscopic situation (for elliptic hyperendoscopic groups of ).

This paper is closely related to the paper mentioned in this previous post where me and A. Bertoloni Meli discuss the proof of the Scholze–Shin conjecture for unramified unitary groups in the trivial endoscopic triple setting.

References

[Sch] Scholze, Peter. The Local Langlands Correspondence for GL_n over -adic fields, Invent. Math. 192 (2013), no. 3, 663–715.

[SS] Scholze, P. and Shin, S., 2013. On the cohomology of compact unitary group Shimura varieties at ramified split places. Journal of the American Mathematical Society, 26(1), pp.261-294.

Punctured disks and punctured curves and cohomology, oh my! (Part III: the affine curve case)

alexyoucis — Wed, 27 Nov 2019 19:46:50 +0000

This is a continuation of this post.

The affine case

Restating and applying purity

Let us now finally return to computing to proving Proposition 2 in the affine case. Let us first write it more concretely:

Proposition 31: Let be an algebraically closed field and a smooth connected projective curve over of genus . Let and . Set . Then, for every prime invertible in and integer

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Given our discussion of cohomology with supports this is fairly easy. In fact, it’s laughably easy if we take the Purity Theorem for granted. Namely, let us begin by noting that by Proposition 23 we have a long exact sequence

and thus from Proposition 4 it really suffices to compute . If we assume the Purity Theorem (in the form of Corollary 30) this is trivial since is a disjoint union of points. Namely, we see

from which deducing Proposition 31 is a triviality.

Giving a more hands-on proof

But, the above feels a little bit like cheating to me. Namely, we don’t need the elephant gun that is the Purity Theorem to do this computation. In fact, this computation really is just a basic case of the Purity Theorem that we can do by hand. So, let’s do it.

Note that by iteration it suffices to deal with the case when is a single point (but is now allowed to be affine—-this won’t change the computation by the Excision Theorem). Or, if this iteration argument doesn’t make you happy convince yourself of the easy fact that since that

which, either way, reduces us to the computation of for a single closed point .

Now, the key point for this computation is then that by Lemma 26 we can reduce this to a really explicit computation. Namely, Lemma 26 tells us that (since our schemes are Noetherian) that

This may not seem very helpful but, in fact, the scheme is something incredibly explicit!

Namely, we have the following

Lemma 32: Let the notation be as above. Then, there is an isomorphism of schemes

and can be described as the integral closure of inside (NB: since is algebraically closed there is no difference here between henselization and strict henselization).

Proof: The first claim follows from the fact that there exists an open subset of in and an etale map with . Indeed, just take any etale map . Then, we know that will map to some point in which, by etaleness, and the fact that is algebraically closed closed implies that . One can then just postcompose with the translation isomorphism . We now claim that this pointed etale map induces an isomorphism

but since is an etale neighborhood this is almost by the definition of the henselization/strict henselization.

It remains to justify why has the claimed form. There are several ways to justify this. For example, one can see [Mil, Example I.4.10 (a)] and/or [Mil, Example I.4.10 (b)] and the references therein. One can also see [Nag, Page 180].

So, we see that (as hinted at in the idea of proof for the Purity Theorem) one can use the fact that our calculation is etale local nature and the fact that smooth things all locally look the same to reduce this to an explicit computation with a specific scheme. Namely, by the above it’s clear that we need to compute the group . But, how do we do this?

Well, using Proposition 23 and the fact that strictly Henselian local rings have vanishing higher cohomology (e.g. see [Fu, Proposition 5.7.3] or [Stacks, Tag09AY]) it’s easy to see that

and for 1" class="latex" />

where here we are denoting by the scheme (the punctured spectrum).

Now has pretty simple structure in that it’s a Henselian DVR (e.g. combine [Stacks, Tag06DI] with [Stacks, Tag06LK]). In particular, it’s easy to see that where . From this it’s easy to conclude that

and that

Thus, we are reduced to understanding the Galois cohomology of . But, this is surprisingly simple in our case:

Lemma 33: Let be a Henselian DVR of residue characteristic . Let be a uniformizer of and set and . Assume that is algebraically closed. Finally, let be a finite Galois extension of degree with . Then, .

Note that the reason this Lemma is of interest to us is that it evidently implies that

Corollary 34: In particular, if denotes the pro-prime-to- completion of then (non-canonically)

This corollary also makes complete geometric sense. We are to think of as being something like a ‘small etale disk around in ‘ Then, of course, we should think of as being something like a ‘punctured disk’ around and, consequently, it would make sense that

should be something like . Corollary 34 validates this belief up to the ever-present concerns about wild ramification.

We also see that Corollary 34 immediately implies that

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So, summarizing the above discussion we get that for 0" class="latex" /> that

which, combining with our trivial result that gives us

This is as already told to us by the Purity Theorem and which, we’ve already observed, easily implies Proposition 31.

Thus, all that remains to finish our hands-on proof of Proposition 31 is the proof of Lemma 33:

Proof (Lemma 33): Let be any finite Galois extension of of order which is prime . Since is Henselian there is at a unique extension of the valuation on to a valuation on (e.g. see [EP, Lemma 4.1.1] and [EP, Theorem 4.1.3]). Let be the valuation ring of and let denote a uniformizer of . Note that since (e.g. see [EP, Theorem 3.3.5]) and (since the residue field of is algebraically closed) we must have that is totally ramified. Thus, where .

But, note that has a solution in the residue field of (since it’s algebraically closed) and since is coprime to the residue characteristic of we see from the fact that is Henselian (e.g. this follows from [EP, Lemma 4.1.1]) that has an -root in . Thus, we see that, by replacing with an -root of , we can assume . Then by standard theory (e.g. see [Sut, Theorem 11.5] and note that the proof doesn’t use completeness, but only Henselianess).

To summarize the proof in the affine case : we used the theory of cohomology with supports to break the cohomology into two parts. The first part was the classes on that come from . The second part was the classses on introduced by the punctures we made at the points . Then, using that cohomology with supports is insensitive to shrinking in the etale topology we were able to focus in on the loops around each puncture individually and, using the fact that smooth curves all etale locally look , we were able to essentially understand this as a computation involving the punctured spectrum of the henselian local ring at which, again, we intuited as a punctured disk. This space was concrete enough/well-behaved enough that we could compute its cohomology by hand and got (ignoring wild ramification issues) that its cohomology agrees with what we expected (a punctured disk).

Two bonus computations

Even though our main goal (at least locally) was to prove Proposition 2 (which we have now done) our computations relied so heavily on smoothness hypotheses that one might be scared that the methods we have developed are ill-equipped to deal with singular cases. This is not at all the case. Namely, using our theory of cohomology with supports we can compute this cohomology much like we did for smooth curves, but in reverse.

Namely, let be our singular curve and let be its smooth locus and its finite discrete singular locus. We are now interested in not in the cohomology of but in the cohomology of . But, again, by Proposition 23 this comes down to studying the cohomology of (which we know by Proposition 2/Proposition 31) and the cohomology of with supports at . The latter can, as in the smooth case, be studied by studying etale local geometry of at which, again, allows us to make explicit computations involving the singularity type of .

The two examples we compute arefortunately/unfortunately (depending on your bend) the obvious ones: the cuspidal and nodal cubics.

The nodal cubic

As before let us fix an algebraically closed field of characteristic not or (for sanity’s sake) and a prime invertible in . Let us define to be the projective nodal cubic curve given by

this has a singular locus (where we write ) and smooth locus . The structure of itself is quite simple. Namely, we claim that . Indeed, this follows from the fact that the normalization of is the map

(e.g. see [Mac]) which has the property that consists of two points so that isomorphically maps to .

Now, by Proposition 23 we get a long exact sequence

Now, since we know from Proposition 2/Proposition 31 that

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and thus we really only need to compute .

To compute this we use Lemma 26 (since our schemes are Noetherian) to say that

But, the point is that while the global geometry of is somewhat daunting the etale local geometry at is quite simple. Namely, we have the following:

Lemma 35: Consider . Then, there is an isomorphism

where .

Proof: It suffices to find an etale neighborhood of such that . To do this we let be the unique double cover of .

Abstractly one can describe it as follows. As is well-known (e.g. see [Har, Exercise 6.9]). Let be the line bundle on corresponding to . Then, (it’s the unique such line bundle). Then, using this isomorphism one can put an algebra structure on and we can set .

A less highfalutin way to build is to have it be the projectivization of the map

corresponding to the -algebra map with

Then, one can show that this map is a finite etale cover (of degree ). Moreover, the point of maps to . Moreover one can explicitly check that from where the conclusion follows.

In words, if we’re only interested in etale local geometry (which is the case due the fact that cohomology with supports is insensitive to shrinking in the etale topoogy) we can eschew the global complicated nature of the nodal cubic when focusing on its singular point and see it as, essentially, the intersection of two lines.

Thus, to compute

we need really only compute

But, just as before in the smooth curve case we can use Proposition 23 and the vanishing of higher cohomology of strictly local rings to see that if we set then

and for 1" class="latex" />

But, what is in this case? It is no longer the spectrum of the fraction field of a Henselian DVR. Indeed, it’s actually the disjoint union of two such objects. The idea is that

(where means the non-vanishing locus of in ) and, as one can check, and are both (essentially canonically) the spectra of the fraction field of .

We can then use the results of the previous section (namely Corollary 34) to thus deduce that

(don’t forget that that has two components!). In fact we can geometrically intepret this non-vanishing. Remember that (roughly) classifies -torsors on that become trivialized on . If is normal this can never hapen (as we saw by the vanishing of ) essentially because if you have a connected finite etale cover one has that is actually integral (e.g. see [Stacks, Tag0BQL]) and so is still connected so that cannot be trivial. But, for the nodal cubic the covers can fail to have this property. In fact note that, geometrically, the nodal curve looks like a pinched torus. Its covers then look like necklaces of wedges of spheres (e.g. see the discussion here and here) where its exactly the wedge points (the ‘kissing points’) that map to the node . So, for such a cover one has that is certainly not trivial, but just looks like a bunch of disjoint copies of –the trivial torsor over !

Let us return to the rigorous computation. For 1" class="latex" />

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(where we have used that is a disjoint union of the spectra of two Henselian discretely valued fields and Corollary 34). From this we then deduce that

as we would expect topologically.

The cuspidal cubic

Finally, let’s flip our method on its head by computing the etale cohomology of an directly and then using this to say something non-trivial about the cohomology groups .

First, let’s fix our set up. Again, let be an algebraically closed field of characteristic not or and let be a prime invertible in . Let us consider the projective cuspidal cubic:

this has singular locus (where we abbreviate to ) and smooth locus .

The cheeky way to compute the cohomology of is then to employ the following observation:

Lemma 36: Let notation be as above. Then, the normalization of is and the normalization map is a universal homeomorphism.

Proof: Note that is evidently finite (or even easier is that it’s proper) and surjective and thus universally closed and universally surjective. Thus, it suffices to check that is universally injective. It suffices by standard theory (e.g. see the discussion in [Stacks, Tag01S2]) to show that for every point that is purely inseparably. For the generic point this is clear since is birational so it suffices to check for closed points of or, equivalently, for the points lying over the cusp. This is done in [Mac] explicitly for the cusp.

Now, one deduces from the topological invariance of the etale site (e.g. see [Stacks, Tag03SI]) that and have the same cohomology. In particular, we see that

Note though that from Lemma 36 we also deduce that and thus its cohomology is known from Proposition 2/Proposition 31:

Using Proposition 23 we deduce the following:

Let us now consider the objects and and see what the above computation tells us.

To start, interestingly enough, from the equality (again we’re implicitly using Lemma 26 and thus Noetherianity, but I assume this needn’t be said at this point)we actually deduce that

is surjective and thus is connected. Surprisingly, this is essentially equivalent to Lemma 36. Namely, Lemma 36 roughly says that is (geometrically) unibranch (e.g. see [Stacks Tag0BPZ]), which since is normal away from , is really saying that is geometrically unibranch. But, being geometrically unibranch is actually equivalent to the claim that is connected (e.g. see [Stacks, Tag0BQ4]).

Let us in fact notice that being connected implies that, in fact, is a field. Indeed, note that is reduced (e.g. see [Stacks, Tag06DH]) and thus it will be domainif and only if it’s irreducible. But, note that is dimension (e.g. see [Stacks, Tag06LK]) and thus there are no intermediary primes lying between the minimal primes of and . Thus, is a discrete set on the minimal primes of and since is connected, this implies that there is a unique minimal prime of . Thus, is a domain. Moreover, since it’s of dimension it’s easy to see that where .

Note that this field is the fraction field of a Henselian local domain but one which is not a DVR. Thus, the results like Corollary 34 don’t apply. In fact, I don’t actually know how to compute directly. That said, from our sideways calculation of . Indeed, using the usual isomorphism for 1" class="latex" />

we deduce that

Again, this is quite different from our other calculation and quite strange. In our previous calculations we calculated a global group by studying the local group. Here we had a local group that, a priori, we didn’t know how to explicitly compute but once we globalized it we were able to exploit the global picture to get the desired result.

I have a gut feeling that there is some topological invariance trick that works locally for computing the cohomology of similar to how we computed the cohomology for the cuspidal cubic. I don’t currently see it though. Please feel free to let me know if you see an approach!

References

[Con] Conrad, B., 2015. Abelian varieties. https://stuff.mit.edu/people/fengt/249C.pdf

[EP] Engler, A.J. and Prestel, A., 2005. Valued fields. Springer Science & Business Media.

[Fu] Fu, L., 2011. Etale cohomology theory (Vol. 13). World Scientific.

[Har] Hartshorne, R., 2013. Algebraic geometry (Vol. 52). Springer Science & Business Media.

[Lit] Litt, Daniel. Cohomology of curves. https://www.daniellitt.com/s/cohomologyofcurves.pdf

[Mac] Mack-Crane, Sander. Normalization of Algebraic and Arithmetic Curves. https://algebrateahousejmath.wordpress.com/2016/11/27/normalization-of-algebraic-and-arithmetic-curves/

[Mil] Milne, J.S. and Milne, J.S., 1980. Etale cohomology (PMS-33) (No. 33). Princeton university press.

[Nag] Nagata, M., 1962. Local rings. Interscience Tracts in Pure and Appl. Math.

[Ols] Olsson, M., 2016. Algebraic spaces and stacks (Vol. 62). American Mathematical Soc.

[Poo] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc.

[SGA 1] Grothendieck, A. and Raynaud, M., 2002. Revêtementset groupe fondamental (SGA 1). arXiv preprint math/0206203.

[Stacks] The Stacks Project Authors, The Stacks Project, https://stacks.math.columbia.edu

[Star] Starr, Jason. Brauer groups and Galois cohomology of function
fields of varieties. https://www.math.stonybrook.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf

[Sut] Sutherland, Andrew. Totally ramified extensions and Krasner’s lemma. http://math.mit.edu/classes/18.785/2016fa/LectureNotes11.pdf