{{< include macros.qmd >}}

Let $A$ be an abelian group.

::: {#def-}

An $A$-valued is a function

satisfying, for all $r$, $s$ and $t$ in $\PP_1(\QQ)$,

1. $m\{r\to s\} = -m\{s\to r\}$,

2. $m\{r\to s\} + m\{s\to t\} = m\{r\to t\}$.

Denote by $\cM(A)$ the abelian group of all $A$-valued modular symbols.

We will also write $\cM=\cM(\CC)$.

:::

The group $\GL_2(\QQ)$ acts on $\cM(A)$ on the left, by the rule

We are interested in modular symbols invariant under a congruence

subgroup $\Gamma\subset SL_2(\ZZ)$ and, to simplify the exposition, we

will concentrate on $\Gamma=\Gamma_0(N)$. The most important examples of

modular symbols will arise from integrating modular forms. Let

$f\in S_2(\Gamma_0(N))$ be a *newform*\index{newform}, and define

Note that since $f$ is a cusp form the above integrals converge.

Moreover, they can be explicitly computed: choose some $\tau\in\HH$ and

write

If $r=\infty$ then the integral from $x$ to $\tau$ can be calculated

with the formula

Otherwise, choose a matrix $\gamma\in\SL_2(\ZZ)$ with

$\gamma \infty = r$ and reduce to the case above, using the change of

variables

A priori the modular symbol $\lambda_f$ belongs to $\cM(\CC)$, although

a deep theorem of Shimura gives a much more precise description of its

values. Define the plus-minus symbols

::: {#thm-}

Let $f\in S_2(\Gamma_0(N))$ be a newform such that

$$f(q)=\sum_{n=1}^\infty a_n q^n,\quad a_1=1, a_n\in\ZZ.$$ There exists

$\Omega_f^+\in\RR$ and $\Omega_f^-\in i\RR$ such that

$$\lambda_f^\pm\{r\to s\}\in \Omega_f^\pm\ZZ.$$ Therefore

$\frac{1}{\Omega_f^\pm}\lambda_f^\pm\in \cM(\ZZ)$.

:::

A crucial property of $\lambda_f$ and thus of $\lambda_f^\pm$ is their

invariance with respect to $\Gamma_0(N)$:

::: {#prp-}

We have, for all $\gamma\in\Gamma_0(N)$,

:::

::: {.proof}

Write $\omega_f = 2\pi i f(z)dz$, and note that:

:::

In the next section we will study the space of $\Gamma_0(N)$-invariant

modular symbols in more detail.

Write $\cM^{\Gamma_0(N)}$ for the space of $\Gamma_0(N)$-invariant

modular symbols. It is equipped with an action of the Hecke operators

$T_p$ with $p\nmid N$, via the formula

::: {#prp-}

The map $f\mapsto \lambda_f$ is an injective, $\CC$-linear

Hecke-equivariant map.

:::

::: {.proof}

Assuming that $\lambda_f=0$, define the following holomorphic

function on $\HH\cup \PP^1(\QQ)$:

$$F(\tau) = \int_\infty^\tau 2\pi i f(z)dz.$$ Note that

$F(\gamma\tau)-F(\tau) = \lambda_f\{r\to\gamma r\}$ for any choice of

$r\in\PP^1(\QQ)$. Since by assumption $\lambda_f\{r\to\gamma r\}$ is

zero, we get that $F$ is $\Gamma_0(N)$-invariant. Therefore $F$ is

bounded on $\HH$, and hence is constant by Liouville's theorem.

Therefore $F'(\tau)=0$. But note that by the fundamental theorem of

Calculus $F'(\tau)=2\pi i f(\tau)$. Hence $f=0$.

:::

In order to investigate the image of $f\mapsto \lambda_f$, we first need

to know the dimension of $\cM^{\Gamma_0(N)}$. Let

$g=\dim S_2(\Gamma_0(N))$ and let $s$ be the number of cusps of

$\Gamma_0(N)$.

::: {#thm-}

The space $\cM^{\Gamma_0(N)}$ has dimension $2g+s-1$.

:::

Therefore the map $f\mapsto \lambda_f$ cannot be surjective, and in fact

it will fail to be surjective in two ways. First, complex conjugation

gives a natural action on $\cM^{\Gamma_0(N)}$, by

$$\ol m\{r\to s\} = \ol{m\{r\to s\}}.$$ However $\ol{\lambda}_f$ is the

modular symbol attached to

$\ol{2\pi i f(z)dz} = -2\pi i \bar f(z) d\bar z$, which we didn't

consider. Therefore we get a new homomorphism

which is still injective and its image has thus dimension $2g$ inside

the $2g+s-1$-dimensional space $\cM^{\Gamma_0(N)}$.

Secondly, we need to consider the so-called .

::: {#def-}

A $\Gamma_0(N)$-invariant modular symbol $m$ is called *Eisenstein*\index{Eisenstein} if

there exists a $\Gamma_0(N)$-invariant function

$M\colon \PP^1(\QQ)\to \CC$ such that

:::

The space of Eisenstein modular symbols has dimension $s-1$ and is

linearly disjoint from the image of $\lambda$ above. This gives a

complete description of $\cM^{\Gamma_0(N)}$.

::: {#thm-}

The map $\lambda$ gives a Hecke-equivariant isomorphism

:::

One important feature of modular symbols is that they are computable.

That is, we can calculate the space $\cM^{\Gamma_0(N)}$ without using

the Eichler--Shimura isomorphism and thus avoiding the computation of

path integrals. The key to making this possible consists in noticing

that a modular symbol $m$ is determined by "a few" of its values

$m\{r\to s\}$.

::: {#def-}

Two elements $a/b$ and $c/d$ in $\PP^1(\QQ)$ are if $ad-bc=\pm 1$. Here,

we use the convention that these fractions are in reduced terms, and

$\infty = 1/0$.

:::

The following lemma is crucial in the algorithms for computing with

modular symbols.

::: {#lem-}

Any two elements $a/b$ and $c/d$ in $\PP^1(\QQ)$ can be joined by a

succession of paths between adjacent cusps.

:::

::: {.proof}

It is enough to see how to join $a/b$ to $\infty$. We will find

$t/a'\in\PP_1(\QQ)$ such that:

$$\{a/b\to \infty\} = \{a/b\to t/a'\}+\{t/a'\to \infty\}.$$ Choose $a'$

satisfying $$a'a\equiv 1\pmod{b},\quad |a'|\leq b/2.$$ Next, choose $t$

such that $$aa'-bt = 1.$$ Then $\{a/b\to t/a'\}$ is a path joining

adjacent cusps, and we reduced to a problem of smaller size, since

$|a'|\leq b/2$. One can see how to adapt the Euclidean algorithm that

computes the greatest common divisor of $a$ and $b$ to perform the above

calculation.

:::

::: {#exm-}

Consider $a/b = 2/3$ and $c/d = 1/0=\infty$. Then these are not

adjacent, but note that $2/3$ is adjacent to $1/2$, that $1/2$ is

adjacent to $1/1$, and $1/1$ is adjacent to $1/0$. Therefore we have

have joined the cusps $2/3$ and $\infty$ with a chain of adjacent cusps.

:::

Using the first defining property of modular symbols, the above

proposition says that a modular symbol is determined by the values

$m\{r\to s\}$ where $r$ and $s$ are adjacent. That is, a modular symbol

is completely determined by its values on

To study this set, define the projective line over $\ZZ/N\ZZ$ as

where $(x:y)\sim (x':y')$ if and only if there is

$u\in(\ZZ/N\ZZ)^\times$ such that $x'=ux$ and $y'=uy$.

::: {#lem-}

The set

$\Gamma_0(N)\backslash \left\{\left(\frac ab,\frac cd\right) ~|~ ad-bc = 1\right\}$

is in natural bijection with $\PP^1(\ZZ/N\ZZ)$.

:::

::: {.proof}

First, note that the set

$\left\{\left(\frac ab,\frac cd\right) ~|~ ad-bc = 1\right\}$ is in

bijection with $\SL_2(\ZZ)$ via $(a/b,c/d)\mapsto \smtx acbd$. So to

conclude the proof we need to show that the map

$\smtx abcd\mapsto (c\colon d)$ induces a bijection

$$\Gamma_0(N)\backslash \SL_2(\ZZ)\to \PP^1(\ZZ/N\ZZ).$$ To see this,

note that the map is surjective, since given $(c:d)\in \PP^1(\ZZ/N\ZZ)$

we can find a matrix in $\SL_2(\ZZ/N\ZZ)$ whose second row is $(c,d)$.

Using that $\SL_2(\ZZ)\to \SL_2(\ZZ/N\ZZ)$ is surjective we can lift

this matrix to $\SL_2(\ZZ)$. Secondly, if two matrices in $\SL_2(\ZZ)$

map to the same element in $\PP^1(\ZZ/N\ZZ)$ then modulo $N$ these

matrices are of the form

Then note that the product $\gamma_1\gamma_2^{-1}$ is

$\gamma_1\gamma_2^{-1} \equiv \smtx 1001\pmod{N}$, and hence the

matrices in $\SL_2(\ZZ)$ are in the same coset for $\Gamma_0(N)$.

:::

Therefore a modular symbol $m$ is determined by the function

In particular, the dimension of $\cM^{\Gamma_0(N)}$ is finite, bounded

by $\#\PP^1(\ZZ/N\ZZ)$.

Note, however, that not all functions $\PP^1(\ZZ/N\ZZ)\to\CC$ represent

a modular symbol. In fact, for such a function to be a modular symbol it

has to satisfy some linear relations coming from the two axioms defining

modular symbols.

::: {#prp-msym-relations}

A function

$\varphi\colon \PP^1(\ZZ/N\ZZ)\to \CC$ satisfies $\varphi=[\cdot]_m$ for

some modular symbol $m\in\cM^{\Gamma_0(N)}$ if and only if

1. $\varphi(x) = -\varphi(\frac{-1}{x})$, for all

$x\in\PP^1(\ZZ/N\ZZ)$.

2. $\varphi(x) = \varphi(\frac{x}{x+1}) + \varphi(x+1)$, for all

$x\in \PP^1(\ZZ/N\ZZ)$.

:::

::: {.proof}

Suppose that $\varphi=[\cdot]_m$ for some modular symbol

$m\in\cM^{\Gamma_0(N)}$, and let $x=[b\colon d]\in \PP^1(\ZZ/N\ZZ)$.

Then

\begin{align*}

\varphi(x)&=\varphi(b\colon d) = [b\colon d]_m = m\left\{\frac ab\to \frac cd\right\}\\

&= -m\left\{\frac{-c}{-d}\to \frac ab\right\} = -[-d\colon b]_m=-\varphi(-1/x).

\end{align*}

Similarly, we compute

\begin{align*}

\varphi(x)&=\varphi(b\colon d) = [b\colon d]_m = m\left\{\frac ab\to \frac cd\right\} = m\left\{\frac ab\to\frac{a+c}{b+d}\right\} + m\left\{\frac{a+c}{b+d}\to \frac cd\right\}\\

&=[b\colon b+d]_m+[b+d\colon d]_m = \varphi\left(\frac{x}{x+1}\right) + \varphi(x+1).

\end{align*}

:::

The above proposition allows for an algorithm that computes the space

$\cM^{\Gamma_0(N)}$, by solving the linear system of equations for

$\varphi$. Moreover, the Hecke action is also computable on this

resulting representation. The details of this were worked out for the

first time in [@cremona-book].

We compute the space of modular symbols for $\Gamma_0(11)$. First we

enumerate the elements of $\PP^1(\ZZ/11\ZZ)$:

$$\PP^1(\ZZ/11\ZZ) = \{\infty,0,1,\ldots,10\}.$$ Using the two-term

relations

of @prp-msym-relations we find that if

$\varphi\in \mathbb{M}(\Gamma_0(11))$ then:

$$\varphi(\infty) = -\varphi(-1/\infty) = -\varphi(0).$$ Similarly, we

find: \begin{align*}

\varphi(1)&=-\varphi(10)\\

\varphi(2)&=-\varphi(5)\\

\varphi(3)&=-\varphi(7)\\

\varphi(4)&=-\varphi(8)\\

\varphi(6)&=-\varphi(9).\\

\end{align*} Therefore an M-symbol $\varphi$ is determined by its

values on $0,1,2,3,4,6$. Now we find the $3$-term relations:

$x$ $\infty$ 0 1 2 3 4 5 6 7 8 9 10

$x+1$ $\infty$ 1 2 3 4 5 6 7 8 9 10 0

$\frac{x}{x+1}$ 1 0 6 8 9 3 10 4 5 7 2 $\infty$

The table above is to be read as follows. For example, the first column

says $\varphi(\infty)=\varphi(\infty) + \varphi(1)$. The last column

implies, in turn, $\varphi(10)=\varphi(0)+\varphi(\infty)$. We see from

the first column that $\varphi(1)=0$ (and thus $\varphi(10)=0$). Column

3 gives then that $\varphi(6)=-\varphi(2)$, and Column 4 gives

$\varphi(4)=\varphi(3)-\varphi(2)$. All the other columns are redundant,

and so any modular symbol $\varphi$ is (freely) determined by its values on $0$, $2$

and $3$. We can write down a basis $\{f,g,h\}$ for $\mathbb{M}(\Gamma_0(11))$

::: {.centering}

$\infty$ 0 1 2 3 4 5 6 7 8 9 10

f -1 **1** 0 **0** **0** 0 0 0 0 0 0 0

g 0 **0** 0 **1** **0** -1 -1 -1 0 1 1 0

h 0 **0** 0 **0** **1** 1 0 0 -1 -1 0 0

:::

Next we calculate $T_2$ acting on the basis $\{f,g,h\}$. Since we have

only given the definition of $T_p$ on modular symbols, we will need to

relate the M-symbols $\{f,g,h\}$ to their corresponding modular symbols.

We will abuse notation and use the same notation for those. Each element

of $\PP^1(\ZZ/N\ZZ)$ can be lifted to a matrix in $\SL_2(\ZZ)$. In fact,

we can write the following table:

::: {.centering}

$x=(c\colon d)\in\PP^1(\ZZ/N\ZZ)$ $\smtx abcd\in\SL_2(\ZZ)$ $\frac{a}{c}\to\frac{b}{d}$

$0$ $\smtx 1001$ $\infty \to 0$

$2$ $\smtx {-1}{-1}{2}{1}$ $-1/2\to -1$

$3$ $\smtx {-2}{-1}{3}{1}$ $-2/3\to -1$

:::

Let us write $T_2 f = af+bg+ch$, with $a,b,c$ to be determined. Note

that $a=(T_2f)(0)$, and thus we compute:

\begin{align*}

[0]_{T_2(m)} &= (T_2m)\{\infty \to 0\}\\

&= m\{\frac 20\to \frac 01\}+m\{\frac{\infty + 0}{2}\to \frac{0+0}{2}\}+m\{\frac{\infty+1}{2}\to \frac{0+1}{2}\}\\

&=m\{\infty\to 0\} + m\{\infty \to 0\} + m\{\infty \to \frac 12\}\\

&=2m\{\infty\to 0\} + m\{\infty\to 1\}+m\{1\to\frac 12\}\\

&=2[0]_m + [(0\colon 1)]_m + [1\colon 2]_m\\

&=3[0]_m + [1/2]_m = 3[0]_m + [6]_m.

\end{align*}

Analogous computations give

Note that we could express the resulting values in other ways using the

relations for M-symbols, so the above equations are not unique. In any

way, this allows us to find that

$$T_2f = 3f,\quad T_2g = -f-2g,\quad T_2h = -2h.$$ We find then that the

matrix of $T_2$ in the basis $\{f,g,h\}$ is

$$[T_2] = \mat{3&-1&0\\0&-2&0\\0&0&-2},$$ whose eigenvalues are $3$ and

$-2$ (the eigenvalue $-2$ with multiplicity $2$). Since we have a

decomposition

$\cM(\Gamma_0(11))\cong \cE\oplus S_2(\Gamma_0(11))\oplus \ol{S_2(\Gamma_0(11))}$,

we deduce that $\dim S_2(\Gamma_0(11))=1$ (and also $\dim\cE = 1$).

Moreover, if $F\in S_2(\Gamma_0(11))$ is any nonzero cusp form, then we

know that $T_2F = -2F$, so $a_2(F)=-2$.

Similar computations would give us the Hecke eigenvalues for all $T_p$

operators (with $p\neq 11$). By the Eichler--Shimura construction, these

numbers are telling us the number of points of a certain elliptic curve.

In fact, let $E$ be the elliptic curve of conductor $11$ given by the

equation $$E_{/\QQ} \colon y^2+y=x^3-x^2-10x-20.$$ When reduced modulo

$2$, we get $\ol E$: $$\ol E_{\FF_2} \colon y^2+y=x^3+x^2.$$ Note that

$$\#\ol E(\FF_2)=\#\{\infty, (0,0),(0,1),(1,0),(1,1)\} =5,$$ which

matches with the prediction from the modular symbols computation: we

expected $p+1-\#E(\FF_p) = a_p$ and, in fact: $2+1-5 = -2$.