7 Modular symbols
7.1 First definitions
Let \(A\) be an abelian group.
Definition 7.1 An \(A\)-valued is a function \[m\colon {\mathbb{P}}_1({\mathbb{Q}})\times {\mathbb{P}}_1({\mathbb{Q}}) \longrightarrow A, \quad (r,s)\mapsto m\{r\longrightarrow s\}\] satisfying, for all \(r\), \(s\) and \(t\) in \({\mathbb{P}}_1({\mathbb{Q}})\),
\(m\{r\longrightarrow s\} = -m\{s\longrightarrow r\}\),
\(m\{r\longrightarrow s\} + m\{s\longrightarrow t\} = m\{r\longrightarrow t\}\).
Denote by \({\mathcal{M}}(A)\) the abelian group of all \(A\)-valued modular symbols. We will also write \({\mathcal{M}}={\mathcal{M}}({\mathbb{C}})\).
The group \(\operatorname{GL}_2({\mathbb{Q}})\) acts on \({\mathcal{M}}(A)\) on the left, by the rule \[(\gamma m)\{r\longrightarrow s\} = m\{\gamma^{-1} r\longrightarrow\gamma^{-1} s\}.\]
We are interested in modular symbols invariant under a congruence subgroup \(\Gamma\subset SL_2({\mathbb{Z}})\) 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, and define \[\lambda_f \{r\longrightarrow s\} = \int_r^s 2\pi i f(z) dz.\]
Note that since \(f\) is a cusp form the above integrals converge. Moreover, they can be explicitly computed: choose some \(\tau\in{\mathbb{H}}\) and write \[\int_r^s 2\pi i f(z)dz = \int_r^\tau 2\pi i f(z)dz + \int_\tau^s 2\pi i f(z)dz.\] If \(r=\infty\) then the integral from \(x\) to \(\tau\) can be calculated with the formula \[\int_\infty^\tau 2\pi i f(z)dz=\sum_{n=1}^\infty \frac{a_n}{n} e^{2\pi i n \tau}.\] Otherwise, choose a matrix \(\gamma\in{\operatorname{SL}}_2({\mathbb{Z}})\) with \(\gamma \infty = r\) and reduce to the case above, using the change of variables \[\int_r^\tau 2\pi i f(z)dz = \int_\infty^{\gamma^{-1}\tau} 2\pi if(\gamma z)d(\gamma z)=\int_\infty^{\gamma^{-1}\tau} 2\pi i (f|_2\gamma)(z)dz.\] A priori the modular symbol \(\lambda_f\) belongs to \({\mathcal{M}}({\mathbb{C}})\), although a deep theorem of Shimura gives a much more precise description of its values. Define the plus-minus symbols \[\lambda_f^\pm \{r\longrightarrow s\} = 2\pi i \left(\int_r^s f(z) dz \pm \int_{-r}^{-s} f(z)dz\right).\]
Theorem 7.1 Let \(f\in S_2(\Gamma_0(N))\) be a newform such that \[f(q)=\sum_{n=1}^\infty a_n q^n,\quad a_1=1, a_n\in{\mathbb{Z}}.\] There exists \(\Omega_f^+\in{\mathbb{R}}\) and \(\Omega_f^-\in i{\mathbb{R}}\) such that \[\lambda_f^\pm\{r\longrightarrow s\}\in \Omega_f^\pm{\mathbb{Z}}.\] Therefore \(\frac{1}{\Omega_f^\pm}\lambda_f^\pm\in {\mathcal{M}}({\mathbb{Z}})\).
A crucial property of \(\lambda_f\) and thus of \(\lambda_f^\pm\) is their invariance with respect to \(\Gamma_0(N)\):
Proposition 7.1 We have, for all \(\gamma\in\Gamma_0(N)\), \[\lambda_f\{\gamma r\longrightarrow\gamma s\} = \lambda_f\{r\longrightarrow s\}.\]
Proof. Write \(\omega_f = 2\pi i f(z)dz\), and note that: \[\lambda_f\{\gamma r\longrightarrow\gamma s\}=\int_{\gamma r}^{\gamma s}\omega_f = \int_{r}^{s} \omega_f|_2\gamma = \int_r^s \omega_f=\lambda_f\{r\longrightarrow s\}.\]
In the next section we will study the space of \(\Gamma_0(N)\)-invariant modular symbols in more detail.
7.2 The Eichler–Shimura isomorphism
Write \({\mathcal{M}}^{\Gamma_0(N)}\) 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 \[(T_p m)\{r\longrightarrow s\} = m\{pr\longrightarrow ps\} + \sum_{j=0}^{p-1}m\left\{\frac{r+j}{p}\longrightarrow\frac{s+j}{p}\right\}.\]
Proposition 7.2 The map \(f\mapsto \lambda_f\) is an injective, \({\mathbb{C}}\)-linear Hecke-equivariant map.
Proof. Assuming that \(\lambda_f=0\), define the following holomorphic function on \({\mathbb{H}}\cup {\mathbb{P}}^1({\mathbb{Q}})\): \[F(\tau) = \int_\infty^\tau 2\pi i f(z)dz.\] Note that \(F(\gamma\tau)-F(\tau) = \lambda_f\{r\longrightarrow\gamma r\}\) for any choice of \(r\in{\mathbb{P}}^1({\mathbb{Q}})\). Since by assumption \(\lambda_f\{r\longrightarrow\gamma r\}\) is zero, we get that \(F\) is \(\Gamma_0(N)\)-invariant. Therefore \(F\) is bounded on \({\mathbb{H}}\), 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 \({\mathcal{M}}^{\Gamma_0(N)}\). Let \(g=\dim S_2(\Gamma_0(N))\) and let \(s\) be the number of cusps of \(\Gamma_0(N)\).
Theorem 7.2 The space \({\mathcal{M}}^{\Gamma_0(N)}\) 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 \({\mathcal{M}}^{\Gamma_0(N)}\), by \[\overline m\{r\longrightarrow s\} = \overline{m\{r\longrightarrow s\}}.\] However \(\overline{\lambda}_f\) is the modular symbol attached to \(\overline{2\pi i f(z)dz} = -2\pi i \bar f(z) d\bar z\), which we didn’t consider. Therefore we get a new homomorphism \[\lambda\colon S_2(\Gamma_0(N))\oplus\overline{S_2(\Gamma_0(N))} \longrightarrow{\mathcal{M}}^{\Gamma_0(N)},\] which is still injective and its image has thus dimension \(2g\) inside the \(2g+s-1\)-dimensional space \({\mathcal{M}}^{\Gamma_0(N)}\).
Secondly, we need to consider the so-called .
Definition 7.2 A \(\Gamma_0(N)\)-invariant modular symbol \(m\) is called Eisenstein if there exists a \(\Gamma_0(N)\)-invariant function \(M\colon {\mathbb{P}}^1({\mathbb{Q}})\longrightarrow{\mathbb{C}}\) such that \[m\{r\longrightarrow s\} = M(s)-M(r),\quad r,s\in {\mathbb{P}}^1({\mathbb{Q}}).\]
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 \({\mathcal{M}}^{\Gamma_0(N)}\).
Theorem 7.3 The map \(\lambda\) gives a Hecke-equivariant isomorphism \[M_2(\Gamma_0(N))\oplus \overline{S_2(\Gamma_0(N))} \longrightarrow{\mathcal{M}}^{\Gamma_0(N)}.\]
7.3 Computation of modular symbols
One important feature of modular symbols is that they are computable. That is, we can calculate the space \({\mathcal{M}}^{\Gamma_0(N)}\) 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 \(m\{r\longrightarrow s\}\).
Definition 7.3 Two elements \(a/b\) and \(c/d\) in \({\mathbb{P}}^1({\mathbb{Q}})\) 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.
Lemma 7.1 Any two elements \(a/b\) and \(c/d\) in \({\mathbb{P}}^1({\mathbb{Q}})\) 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 \(t/a'\in{\mathbb{P}}_1({\mathbb{Q}})\) such that: \[\{a/b\longrightarrow\infty\} = \{a/b\longrightarrow t/a'\}+\{t/a'\longrightarrow\infty\}.\] Choose \(a'\) satisfying \[a'a\equiv 1\pmod{b},\quad |a'|\leq b/2.\] Next, choose \(t\) such that \[aa'-bt = 1.\] Then \(\{a/b\longrightarrow t/a'\}\) 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.
Example 7.1 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. \[2/3\sim 1/2\sim 1/0 \sim 1/0\]
Using the first defining property of modular symbols, the above proposition says that a modular symbol is determined by the values \(m\{r\longrightarrow s\}\) where \(r\) and \(s\) are adjacent. That is, a modular symbol is completely determined by its values on \[\Gamma_0(N)\backslash \left\{\left(\frac ab,\frac cd\right) ~|~ ad-bc = 1\right\}.\] To study this set, define the projective line over \({\mathbb{Z}}/N{\mathbb{Z}}\) as \[{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}) = \{(x:y)\in ({\mathbb{Z}}/N{\mathbb{Z}})^2 ~|~ \gcd(x,y,N)=1\}/\sim,\] where \((x:y)\sim (x':y')\) if and only if there is \(u\in({\mathbb{Z}}/N{\mathbb{Z}})^\times\) such that \(x'=ux\) and \(y'=uy\).
Lemma 7.2 The set \(\Gamma_0(N)\backslash \left\{\left(\frac ab,\frac cd\right) ~|~ ad-bc = 1\right\}\) is in natural bijection with \({\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\).
Proof. First, note that the set \(\left\{\left(\frac ab,\frac cd\right) ~|~ ad-bc = 1\right\}\) is in bijection with \({\operatorname{SL}}_2({\mathbb{Z}})\) via \((a/b,c/d)\mapsto \left(\begin{smallmatrix}a&c\\b&d\end{smallmatrix}\right)\). So to conclude the proof we need to show that the map \(\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\mapsto (c\colon d)\) induces a bijection \[\Gamma_0(N)\backslash {\operatorname{SL}}_2({\mathbb{Z}})\longrightarrow{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}).\] To see this, note that the map is surjective, since given \((c:d)\in {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\) we can find a matrix in \({\operatorname{SL}}_2({\mathbb{Z}}/N{\mathbb{Z}})\) whose second row is \((c,d)\). Using that \({\operatorname{SL}}_2({\mathbb{Z}})\longrightarrow{\operatorname{SL}}_2({\mathbb{Z}}/N{\mathbb{Z}})\) is surjective we can lift this matrix to \({\operatorname{SL}}_2({\mathbb{Z}})\). Secondly, if two matrices in \({\operatorname{SL}}_2({\mathbb{Z}})\) map to the same element in \({\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\) then modulo \(N\) these matrices are of the form \[\gamma_1\equiv \left(\begin{matrix}a&b\\c&d\end{matrix}\right)\pmod{N},\quad \gamma_2 \equiv \left(\begin{matrix}au^{-1}&bu^{-1}\\cu&du\end{matrix}\right)\pmod{N}.\] Then note that the product \(\gamma_1\gamma_2^{-1}\) is \(\gamma_1\gamma_2^{-1} \equiv \left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)\pmod{N}\), and hence the matrices in \({\operatorname{SL}}_2({\mathbb{Z}})\) are in the same coset for \(\Gamma_0(N)\).
Therefore a modular symbol \(m\) is determined by the function \[[\cdot]_m\colon {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\longrightarrow{\mathbb{C}},\quad [b\colon d]_m =m\{a/b\longrightarrow c/d\},\quad ad-bc = 1.\] In particular, the dimension of \({\mathcal{M}}^{\Gamma_0(N)}\) is finite, bounded by \(\#{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\).
Note, however, that not all functions \({\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\longrightarrow{\mathbb{C}}\) 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.
Proposition 7.3 A function \(\varphi\colon {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\longrightarrow{\mathbb{C}}\) satisfies \(\varphi=[\cdot]_m\) for some modular symbol \(m\in{\mathcal{M}}^{\Gamma_0(N)}\) if and only if
\(\varphi(x) = -\varphi(\frac{-1}{x})\), for all \(x\in{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\).
\(\varphi(x) = \varphi(\frac{x}{x+1}) + \varphi(x+1)\), for all \(x\in {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\).
Proof. Suppose that \(\varphi=[\cdot]_m\) for some modular symbol \(m\in{\mathcal{M}}^{\Gamma_0(N)}\), and let \(x=[b\colon d]\in {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\). Then \[\begin{align*} \varphi(x)&=\varphi(b\colon d) = [b\colon d]_m = m\left\{\frac ab\longrightarrow\frac cd\right\}\\ &= -m\left\{\frac{-c}{-d}\longrightarrow\frac ab\right\} = -[-d\colon b]_m=-\varphi(-1/x). \end{align*}\] Similarly, we compute \[\begin{align*} \varphi(x)&=\varphi(b\colon d) = [b\colon d]_m = m\left\{\frac ab\longrightarrow\frac cd\right\} = m\left\{\frac ab\longrightarrow\frac{a+c}{b+d}\right\} + m\left\{\frac{a+c}{b+d}\longrightarrow\frac cd\right\}\\ &=[b\colon b+d]_m+[b+d\colon d]_m = \varphi\left(\frac{x}{x+1}\right) + \varphi(x+1). \end{align*}\]
The above proposition allows for an algorithm that computes the space \({\mathcal{M}}^{\Gamma_0(N)}\), 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 1997).
7.4 A worked out example
We compute the space of modular symbols for \(\Gamma_0(11)\). First we enumerate the elements of \({\mathbb{P}}^1({\mathbb{Z}}/11{\mathbb{Z}})\): \[{\mathbb{P}}^1({\mathbb{Z}}/11{\mathbb{Z}}) = \{\infty,0,1,\ldots,10\}.\] Using the two-term relations of Proposition 7.3 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 \(\mathbb{M}(\Gamma_0(11))\)
| \(\infty\) | 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 \({\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\) can be lifted to a matrix in \({\operatorname{SL}}_2({\mathbb{Z}})\). In fact, we can write the following table:
| \(x=(c\colon d)\in{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})\) | \(\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in{\operatorname{SL}}_2({\mathbb{Z}})\) | \(\frac{a}{c}\longrightarrow\frac{b}{d}\) |
|---|---|---|
| \(0\) | \(\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)\) | \(\infty \longrightarrow 0\) |
| \(2\) | \(\left(\begin{smallmatrix}-1&-1\\2&1\end{smallmatrix}\right)\) | \(-1/2\longrightarrow-1\) |
| \(3\) | \(\left(\begin{smallmatrix}-2&-1\\3&1\end{smallmatrix}\right)\) | \(-2/3\longrightarrow-1\) |
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: \[\begin{align*} [0]_{T_2(m)} &= (T_2m)\{\infty \longrightarrow 0\}\\ &= m\{\frac 20\longrightarrow\frac 01\}+m\{\frac{\infty + 0}{2}\longrightarrow\frac{0+0}{2}\}+m\{\frac{\infty+1}{2}\longrightarrow\frac{0+1}{2}\}\\ &=m\{\infty\longrightarrow 0\} + m\{\infty \longrightarrow 0\} + m\{\infty \longrightarrow\frac 12\}\\ &=2m\{\infty\longrightarrow 0\} + m\{\infty\longrightarrow 1\}+m\{1\longrightarrow\frac 12\}\\ &=2[0]_m + [(0\colon 1)]_m + [1\colon 2]_m\\ &=3[0]_m + [1/2]_m = 3[0]_m + [6]_m. \end{align*}\] Analogous computations give \[[2]_{T_2(m)} = [1]_m+[4]_m+[5]_m+[7]_m,\quad [3]_{T_2(m)} = [1]_m+[6]_m+[7]_m+[8]_m\] 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 \[[T_2] = \left(\begin{matrix}3&-1&0\\0&-2&0\\0&0&-2\end{matrix}\right),\] whose eigenvalues are \(3\) and \(-2\) (the eigenvalue \(-2\) with multiplicity \(2\)). Since we have a decomposition \({\mathcal{M}}(\Gamma_0(11))\cong {\mathcal{E}}\oplus S_2(\Gamma_0(11))\oplus \overline{S_2(\Gamma_0(11))}\), we deduce that \(\dim S_2(\Gamma_0(11))=1\) (and also \(\dim{\mathcal{E}}= 1\)). 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 \[E_{/{\mathbb{Q}}} \colon y^2+y=x^3-x^2-10x-20.\] When reduced modulo \(2\), we get \(\overline E\): \[\overline E_{{\mathbb{F}}_2} \colon y^2+y=x^3+x^2.\] Note that \[\#\overline E({\mathbb{F}}_2)=\#\{\infty, (0,0),(0,1),(1,0),(1,1)\} =5,\] which matches with the prediction from the modular symbols computation: we expected \(p+1-\#E({\mathbb{F}}_p) = a_p\) and, in fact: \(2+1-5 = -2\).