Modular Forms of Weight 2¶

We saw in Chapter Modular Forms of Level 1 (especially Section Structure Theorem for Level 1 Modular Forms) that we can compute each space $M_k(\SL_2(\Z))$ explicitly. This involves computing Eisenstein series $E_4$ and $E_6$ to some precision, then forming the basis $\{E_4^a E_6^b : 4a+6b = k, 0\leq a,b \in \Z\}$ for $M_k(\SL_2(\Z))$ . In this chapter we consider the more general problem of computing $S_2(\Gamma_0(N))$ , for any positive integer $N$ . Again we have a decomposition

$M_2(\Gamma_0(N)) = S_2(\Gamma_0(N)) \oplus \Eis_2(\Gamma_0(N)),$

where $\Eis_2(\Gamma_0(N))$ is spanned by generalized Eisenstein series and $S_2(\Gamma_0(N))$ is the space of cusp forms, i.e., elements of $M_2(\Gamma_0(N))$ that vanish at all cusps.

In Chapter Eisenstein Series and Bernoulli Numbers we compute the space $\Eis_2(\Gamma_0(N))$ in a similar way to how we computed $M_k(\SL_2(\Z))$ . On the other hand, elements of $S_2(\Gamma_0(N))$ often cannot be written as sums or products of generalized Eisenstein series. In fact, the structure of $M_2(\Gamma_0(N))$ is, in general, much more complicated than that of $M_k(\SL_2(\Z))$ . For example, when $p$ is a prime, $\Eis_2(\Gamma_0(p))$ has dimension $1$ , whereas $S_2(\Gamma_0(p))$ has dimension about $p/12$ .

Fortunately an idea of Birch, which he called modular symbols, provides a method for computing $S_2(\Gamma_0(N))$ and indeed for much more that is relevant to understanding special values of $L$ -functions. Modular symbols are also a powerful theoretical tool. In this chapter, we explain how $S_2(\Gamma_0(N))$ is related to modular symbols and how to use this relationship to explicitly compute a basis for $S_2(\Gamma_0(N))$ . In Chapter General Modular Symbols we will introduce more general modular symbols and explain how to use them to compute $S_k(\Gamma_0(N))$ , $S_k(\Gamma_1(N))$ and $S_k(N,\eps)$ for any integers $k\geq 2$ and $N$ and character $\eps$ .

Section Hecke Operators contains a very brief summary of basic facts about modular forms of weight $2$ , modular curves, Hecke operators, and integral homology. Section Modular Symbols introduces modular symbols and describes how to compute with them. In Section Computing the Boundary Map we talk about how to cut out the subspace of modular symbols corresponding to cusp forms using the boundary map. Section Computing a Basis for is about a straightforward method to compute a basis for $S_2(\Gamma_0(N))$ using modular symbols, and Section Computing Using Eigenvectors outlines a more sophisticated algorithm for computing newforms that uses Atkin-Lehner theory.

Before reading this chapter, you should have read Chapter Modular Forms and Chapter Modular Forms of Level 1. We also assume familiarity with algebraic curves, Riemann surfaces, and homology groups of compact Riemann surfaces.

Hecke Operators¶

Recall from Chapter Modular Forms that the group $\Gamma_0(N)$ acts on $\h^*=\h\cup \P^1(\Q)$ by linear fractional transformations. The quotient $\Gamma_0(N)\backslash \h^*$ is a Riemann surface, which we denote by $X_0(N)$ . See [DS05, Ch. 2] for a detailed description of the topology on $X_0(N)$ . The Rieman surface $X_0(N)$ also has a canonical structure of algebraic curve over $\Q$ , as is explained in [DS05, Ch. 7] (see also [Shi94, Section 6.7]).

Recall from Section Modular Forms of Any Level that a cusp form of weight $2$ for $\Gamma_0(N)$ is a function $f$ on $\h$ such that $f(z)dz$ defines a holomorphic differential on $X_0(N)$ . Equivalently, a cusp form is a holomorphic function $f$ on $\h$ such that

the expression $f(z)dz$ is invariant under replacing $z$ by $\gamma(z)$ for each $\gamma\in \Gamma_0(N)$ and
$f(z)$ vanishes at every cusp for $\Gamma_0(N)$ .

The space $S_2(\Gamma_0(N))$ of weight $2$ cusp forms on $\Gamma_0(N)$ is a finite-dimensional complex vector space, of dimension equal to the genus $g$ of $X_0(N)$ . The space $X_0(N)(\C)$ is a compact oriented Riemann surface, so it is a $2$ -dimensional oriented real manifold, i.e., $X_0(N)(\C)$ is a $g$ -holed torus (see Figure 3.1).

Condition (b) in the definition of $f$ means that $f$ has a Fourier expansion about each element of $\P^1(\Q)$ . Thus, at $\infty$ we have

$f(z) &= a_1 e^{2\pi i z} + a_2 e^{2 \pi i 2z} + a_3 e^{2 \pi i 3z} + \cdots\\ &= a_1 q + a_2 q^2 + a_3 q^3 + \cdots,$

where, for brevity, we write $q=q(z)=e^{2\pi i z}$ .

Example 3.1

Let $E$ be the elliptic curve defined by the equation $y^2 + xy = x^3 + x^2 - 4x - 5$ . Let $a_p = p+1-\#\tilde{E}(\F_p)$ , where $\tilde{E}$ is the reduction of $E$ mod $p$ (note that for the primes that divide the conductor of $E$ we have $a_3=-1$ , $a_{13}=1$ ).2 For $n$ composite, define $a_n$ using the relations at the end of Section Computing Using Eigenvectors.

Then the Shimura-Taniyama conjecture asserts that

$f&=q + a_2 q^2 + a_3 q^3 + a_4 q^4 + a_5 q^5 + \cdots\\ &= q + q^2 - q^3 -q^4 + 2q^5 + \cdots$

is the $q$ -expansion of an element of $S_2(\Gamma_0(39))$ . This conjecture, which is now a theorem (see [BCDT01]), asserts that any $q$ -expansion constructed as above from an elliptic curve over $\Q$ is a modular form. This conjecture was mostly proved first by Wiles [Wil95] as a key step in the proof of Fermat’s last theorem.

Just as is the case for level $1$ modular forms (see Section Hecke Operators) there are commuting Hecke operators $T_1, T_2, T_3, \ldots$ that act on $S_2(\Gamma_0(N))$ . To define them conceptually, we introduce an interpretation of the modular curve $X_0(N)$ as an object whose points parameterize elliptic curves with extra structure.

Proposition 3.2

The complex points of $Y_0(N)=\Gamma_0(N)\backslash \h$ are in natural bijection with isomorphism classes of pairs $(E,C)$ , where $E$ is an elliptic curve over $\C$ and $C$ is a cyclic subgroup of $E(\C)$ of order $N$ . The class of the point $\lambda \in \h$ corresponds to the pair

$\left(\C/(\Z + \Z \lambda), \quad \left(\frac{1}{N}\Z + \Z\lambda\right)/(\Z + \Z \lambda)\right).$

Proof

See Exercise 3.1.

Suppose $n$ and $N$ are coprime positive integers. There are two natural maps $\pi_1$ and $\pi_2$ from $Y_0(n\cdot N)$ to $Y_0(N)$ ; the first, $\pi_1$ , sends $(E,C)\in Y_0(n\cdot N)(\C)$ to $(E,C')$ , where $C'$ is the unique cyclic subgroup of $C$ of order $N$ , and the second, $\pi_2$ , sends $(E,C)$ to $(E/D, C/D)$ , where $D$ is the unique cyclic subgroup of $C$ of order $n$ . These maps extend in a unique way to algebraic maps from $X_0(n\cdot N)$ to $X_0(N)$ :

(1) $\xymatrix{ & X_0(n\cdot N) \ar[dl]_{\pi_2}\ar[dr]^{\pi_1}\\ X_0(N) & & X_0(N).}$

The $n^{th}$ Hecke operator $T_n$ is ${\pi_1}_* \circ \pi_2^*$ , where $\pi_2^*$ and ${\pi_1}_*$ denote pullback and pushforward of differentials, respectively. (There is a similar definition of $T_n$ when $\gcd(n,N)\neq 1$ .) Using our interpretation of $S_2(\Gamma_0(N))$ as differentials on $X_0(N)$ , this gives an action of Hecke operators on $S_2(\Gamma_0(N))$ . One can show that these induce the maps of Proposition 2.31 on $q$ -expansions.

Example 3.3

There is a basis of $S_2(39)$ so that

$T_2 = \mthree{\hfill 1}{\hfill 1}{\hfill 0}{-2}{-3}{-2}{\hfill 0}{\hfill 0}{\hfill 1} \quad\text{and}\quad T_5 = \mthree{-4}{-2}{-6}{\hfill4}{\hfill4}{\hfill4}{\hfill0}{\hfill0}{\hfill2}.$

Notice that these matrices commute. Also, the characteristic polynomial of $T_2$ is $(x - 1) \cdot (x^{2} + 2x - 1)$ .

Homology¶

The first homology group $\H_1(X_0(N),\Z)$ is the group of closed $1$ -cycles modulo boundaries of $2$ -cycles (formal sums of images of $2$ -simplexes). Topologically $X_0(N)$ is a $g$ -holed torus, where $g$ is the genus of $X_0(N)$ . Thus $\H_1(X_0(N),\Z)$ is a free abelian group of rank $2g$ (see, e.g., [Gre81, Ex. 19.30] and [DS05, Section 6.1]), with two generators corresponding to each hole, as illustrated in the case $N=39$ in Figure 3.1.

Figure 3.1

The homology of $X_0(39)$

$\H_1(X_0(39),\Z) \isom \Z\cross\Z\cross\Z\cross\Z\cross\Z\cross\Z$

The homology of $X_0(N)$ is closely related to modular forms, since the Hecke operators $T_n$ also act on $\H_1(X_0(N),\Z)$ . The action is by pullback of homology classes by $\pi_2$ followed by taking the image under $\pi_1$ , where $\pi_1$ and $\pi_2$ are as in (1).

Integration defines a pairing

(2) $\langle\,,\,\rangle: S_2(\Gamma_0(N)) \cross \H_1(X_0(N),\Z) \ra \C.$

Explicitly, for a path $x$ ,

$\langle f , x \rangle = 2\pi i \cdot \int_{x} f(z) dz.$

Theorem 3.4

The pairing (2) is nondegenerate and Hecke equivariant in the sense that for every Hecke operator $T_n$ , we have $\langle f T_n, x\rangle = \langle f, T_n x\rangle$ . Moreover, it induces a perfect pairing

$\langle\,,\,\rangle: S_2(\Gamma_0(N)) \cross \H_1(X_0(N),\R) \ra \C.$

This is a special case of the results in Section Pairing Modular Symbols and Modular Forms.

As we will see, modular symbols allow us to make explicit the action of the Hecke operators on $\H_1(X_0(N),\Z)$ ; the above pairing then translates this into a wealth of information about cusp forms.

We will also consider the relative homology group $\H_1(X_0(N),\Z;\{\text{cusps}\})$ of $X_0(N)$ relative to the cusps; it is the same as usual homology, but i n addition we allow paths with endpoints in the cusps instead of restricting to closed loops. Modular symbols provide a “combinatorial” presentation of $\H_1(X_0(N),\Z)$ in terms of paths between elements of $\P^1(\Q)$ .

Modular Symbols¶

Let $\sM_2$ be the free abelian group with basis the set of symbols $\{\alpha,\beta\}$ with $\alpha,\beta\in\P^1(\Q)$ modulo the 3-term relations

$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0$

above and modulo any torsion. Since $\sM_2$ is torsion-free, we have

$\{\alpha,\alpha\} = 0\qquad\text{and}\qquad \{\alpha,\beta\} = -\{\beta,\alpha\}.$

Warning

The symbols $\{\alpha,\beta\}$ satisfy the relations $\{\alpha,\beta\} = -\{\beta,\alpha\}$ , so order matters. The notation $\{\alpha,\beta\}$ looks like the set containing two elements, which strongly (and incorrectly) suggests that the order does not matter. This is the standard notation in the literature.

Figure 3.2

The modular symbols $\{\alpha,\beta\}$ and $\{0,\infty\}$

As illustrated in Figure 3.2, we “think of” this modular symbol as the homology class, relative to the cusps, of a path from $\alpha$ to $\beta$ in $\h^*$ .

Define a left action of $\GL_2(\Q)$ on $\sM_2$ by letting $g\in\GL_2(\Q)$ act by

$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$

and $g$ acts on $\alpha$ and $\beta$ via the corresponding linear fractional transformation. The space $\sM_2(\Gamma_0(N))$ of modular symbols for `\Gamma_0(N)` is the quotient of $\sM_2$ by the submodule generated by the infinitely many elements of the form $x - g(x)$ , for $x$ in $\sM_2$ and $g$ in $\Gamma_0(N)$ , and modulo any torsion. A modular symbol for $\Gamma_0(N)$ is an element of this space. We frequently denote the equivalence class of a modular symbol by giving a representative element.

Example 3.6

Some modular symbols are $0$ no matter what the level $N$ is! For example, since $\gamma=\abcd{1}{1}{0}{1}\in \Gamma_0(N)$ , we have

$\{\infty,0\} = \{\gamma(\infty),\gamma(0)\} = \{\infty,1\},$

so

$0 = \{\infty,1\}-\{\infty,0\} = \{\infty,1\} + \{0,\infty\} = \{0,\infty\} + \{\infty,1\} = \{0,1\}.$

See Exercise 3.2 for a generalization of this observation.

There is a natural homomorphism

(3) $\varphi: \sM_2(\Gamma_0(N)) \to \H_1(X_0(N),\{\text{cusps}\},\Z)$

that sends a formal linear combination of geodesic paths in the upper half plane to their image as paths on $X_0(N)$ . In [Man72] Manin proved that (3) is an isomorphism (this is a fairly involved topological argument).

Manin identified the subspace of $\sM_2(\Gamma_0(N))$ that is sent isomorphically onto $\H_1(X_0(N),\Z)$ . Let $\sB_2(\Gamma_0(N))$ denote the free abelian group whose basis is the finite set $C(\Gamma_0(N)) = \Gamma_0(N)\backslash \P^1(\Q)$ of cusps for $\Gamma_0(N)$ . The boundary map

$\delta: \sM_2(\Gamma_0(N))\ra \sB_2(\Gamma_0(N))$

sends $\{\alpha,\beta\}$ to $\{\beta\}-\{\alpha\}$ , where $\{\beta\}$ denotes the basis element of $\sB_2(\Gamma_0(N))$ corresponding to $\beta\in\P^1(\Q)$ . The kernel $\sS_2(\Gamma_0(N))$ of $\delta$ is the subspace of cuspidal modular symbols. Thus an element of $\sS_2(\Gamma_0(N))$ can be thought of as a linear combination of paths in $\h^*$ whose endpoints are cusps and whose images in $X_0(N)$ are homologous to a $\Z$ -linear combination of closed paths.

Theorem 3.7

The map $\varphi$ above induces a canonical isomorphism

$\sS_2(\Gamma_0(N)) \isom \H_1(X_0(N),\Z).$

Proof

This is [Man72, Thm. 1.9].

For any (commutative) ring $R$ let

$\sM_2(\Gamma_0(N), R) = \sM_2(\Gamma_0(N)) \tensor_\Z R$

and

$\sS_2(\Gamma_0(N), R) = \sS_2(\Gamma_0(N)) \tensor_\Z R.$

Proposition 3.8

We have

$\dim_\C\sS_2(\Gamma_0(N),\C) = 2\dim_\C S_2(\Gamma_0(N)).$

Proof

We have

$\dim_\C\sS_2(\Gamma_0(N),\C) = \rank_\Z \sS_2(\Gamma_0(N)) = \rank_\Z \H_1(X_0(N),\Z) = 2g.$

Example 3.9

We illustrate modular symbols in the case when $N=11$ . Using Sage (below), which implements the algorithm that we describe below over $\Q$ , we find that $\sM_2(\Gamma_0(11);\Q)$ has basis $\{\infty,0\}$ , $\{-1/8,0\}$ , $\{-1/9,0\}$ . A basis for the integral homology $\H_1(X_0(11),\Z)$ is the subgroup generated by $\{-1/8,0\}$ and $\{-1/9,0\}$ .

sage: set_modsym_print_mode ('modular')
sage: M = ModularSymbols(11, 2)
sage: M.basis()
({Infinity,0}, {-1/8,0}, {-1/9,0})
sage: S = M.cuspidal_submodule()
sage: S.integral_basis()     # basis over ZZ.
({-1/8,0}, {-1/9,0})
sage: set_modsym_print_mode ('manin')    # set it back

Computing with Modular Symbols¶

In this section, we describe a trick of Manin that we will use to prove that spaces of modular symbols are computable.

By Exercise 1.6 the group $\Gamma_0(N)$ has finite index in $\SL_2(\Z)$ . Fix right coset representatives $r_0, r_1, \ldots, r_m$ for $\Gamma_0(N)$ in $\SL_2(\Z)$ , so that

$\SL_2(\Z) = \Gamma_0(N)r_0 \, \union\, \Gamma_0(N)r_1 \, \union\, \cdots \, \union\, \Gamma_0(N)r_m,$

where the union is disjoint. For example, when $N$ is prime, a list of coset representatives is

$\mtwo{1}{0}{0}{1}, \mtwo{1}{0}{1}{1}, \mtwo{1}{0}{2}{1}, \mtwo{1}{0}{3}{1}, \ldots, \mtwo{\hfill 1}{0}{N-1}{1}, \mtwo{0}{-1}{1}{\hfill 0}.$

Let

(4) $\P^1(\Z/N\Z)= \{(a:b) \,: \, a,b\in\Z/N\Z, \, \,\gcd(a,b,N)=1 \,\}/\sim$

where $(a:b)\sim (a':b')$ if there is $u\in(\Z/N\Z)^*$ such that $a=ua', b=ub'$ .

Proposition 3.10

There is a bijection between $\P^1(\Z/N\Z)$ and the right cosets of $\Gamma_0(N)$ in $\SL_2(\Z)$ , which sends a coset representative $\abcd{a}{b}{c}{d}$ to the class of $(c:d)$ in $\P^1(\Z/N\Z)$ .

Proof

See Exercise 3.3.

See Proposition 1.27 for the analogous statement for $\Gamma_1(N)$ .

We now describe an observation of Manin (see [Man72, Section 1.5]) that is crucial to making $\sM_2(\Gamma_0(N))$ computable. It allows us to write any modular symbol $\{\alpha, \beta\}$ as a $\Z$ -linear combination of symbols of the form $r_i\{0,\infty\}$ , where the $r_i\in \SL_2(\Z)$ are coset representatives as above. In particular, the finitely many symbols $r_0\{0,\infty\}, \ldots, r_m\{0,\infty\}$ generate $\sM_2(\Gamma_0(N))$ .

Proposition 3.11

[Manin] Let $N$ be a positive integer and $r_0,\ldots, r_m$ a set of right coset representatives for $\Gamma_0(N)$ in $\SL_2(\Z)$ . Every $\{\alpha, \beta\} \in \sM_2(\Gamma_0(N))$ is a $\Z$ -linear combination of $r_0\{0,\infty\}, \ldots, r_m\{0,\infty\}$ .

We give two proofs of the proposition. The first is useful for computation (see [Cre97a, Section 2.1.6]); the second (see [MTT86, Section 2]) is easier to understand conceptually since it does not require any knowledge of continued fractions.

Proof

Since

$\{\alpha,\beta\}=\{0,\beta\}-\{0,\alpha\},$

it suffices to consider modular symbols of the form $\{0,b/a\}$ , where the rational number $b/a$ is in lowest terms. Expand $b/a$ as a continued fraction and consider the successive convergents in lowest terms:

$\frac{b_{-2}}{a_{-2}} = \frac{0}{1},\quad \frac{b_{-1}}{a_{-1}} = \frac{1}{0},\quad \frac{b_0}{a_0} = \frac{b_0}{1},\, \ldots,\quad \frac{b_{n-1}}{a_{n-1}},\quad \frac{b_n}{a_n}= \frac{b}{a}$

where the first two are included formally. Then

$b_k a_{k-1} - b_{k-1} a_k = (-1)^{k-1},$

so that

$g_k = \mtwo{b_k}{\hfill (-1)^{k-1}b_{k-1}}{a_k}{(-1)^{k-1}a_{k-1}}\in \SL_2(\Z).$

Hence

$\left\{\frac{b_{k-1}}{a_{k-1}}, \frac{b_k}{a_k}\right\} = g_k \{ 0, \infty\} = r_i \{0,\infty\},$

for some $i$ , is of the required special form. Since

$\{0,b/a\} = \{0,\infty\} + \{\infty,b_0\} + \left\{\frac{b_0}{1}, \frac{b_1}{a_1}\right\} + \cdots + \left\{\frac{b_{n-1}}{a_{n-1}}, \frac{b_n}{a_n}\right\},$

this completes the proof.

Proof

As in the first proof it suffices to prove the proposition for any symbol $\{0,b/a\}$ , where $b/a$ is in lowest terms. We will induct on $a\in\Z_{\geq 0}$ . If $a=0$ , then the symbol is $\{0,\infty\}$ , which corresponds to the identity coset, so assume that $a>0$ . Find $a'\in\Z$ such that

$ba'\con 1\pmod{a};$

then $b' = (b a' - 1)/a \in\Z$ so the matrix

$\delta = \mtwo{b}{b'}{a}{a'}$

is an element of $\SL_2(\Z)$ . Thus $\delta = \gamma \cdot r_j$ for some right coset representative $r_j$ and $\gamma \in \Gamma_0(N)$ . Then

$\{0, b/a\} - \{0, b'/a'\} = \{b'/a', b/a\} = \mtwo{b}{b'}{a}{a'}\cdot \{0,\infty\} = r_j \{0,\infty\},$

as elements of $\M_2(\Gamma_0(N))$ . By induction, $\{0, b'/a'\}$ is a linear combination of symbols of the form $r_k\{0,\infty\}$ , which completes the proof.

Example 3.12

Let $N=11$ , and consider the modular symbol $\{0,4/7\}$ . We have

$\frac{4}{7} = 0 + \frac{1}{1+\frac{1}{1+\frac{1}{3}}},$

so the partial convergents are

$\frac{b_{-2}}{a_{-2}} = \frac{0}{1},\quad \frac{b_{-1}}{a_{-1}} = \frac{1}{0},\quad \frac{b_0}{a_0} = \frac{0}{1},\quad \frac{b_1}{a_1} = \frac{1}{1},\quad \frac{b_2}{a_2} = \frac{1}{2},\quad \frac{b_3}{a_3} = \frac{4}{7}.$

Thus, noting as in Example 3.6 that $\{0,1\}=0$ , we have

$\begin{eqnarray*} \{0,4/7\} &=& \{0,\infty\} +\{\infty,0\} + \{0,1\}+ \{1,1/2\} + \{1/2,4/7\}\\ &=& \mtwo{1}{-1}{2}{-1}\{0,\infty\} + \mtwo{4}{1}{7}{2}\{0,\infty\}\\ &=& \mtwo{1}{0}{9}{1}\{0,\infty\} + \mtwo{1}{0}{9}{1}\{0,\infty\}\\ &=& 2 \cdot \left[\mtwo{1}{0}{9}{1}\{0,\infty\}\right]. \end{eqnarray*}$

We compute the convergents of $4/7$ in Sage as follows (note that $0$ and $\infty$ are excluded):

sage: convergents(4/7)
[0, 1, 1/2, 4/7]

Manin Symbols¶

As above, fix coset representatives $r_0,\ldots,r_m$ for $\Gamma_0(N)$ in $\SL_2(\Z)$ . Consider formal symbols $[r_i]'$ for $i=0,\ldots, m$ . Let $[r_i]$ be the modular symbol $r_i\{0,\infty\} = \{r_i(0), r_i(\infty)\}$ . We equip the symbols $[r_0]',\ldots,[r_m]'$ with a right action of $\SL_2(\Z)$ , which is given by $[r_i]'.g = [r_j]'$ , where $\Gamma_0(N) r_j = \Gamma_0(N) r_i g$ . We extend the notation by writing $[\gamma]' = [\Gamma_0(N)\gamma]' = [r_i]'$ , where $\gamma \in \Gamma_0(N) r_i$ . Then the right action of $\Gamma_0(N)$ is simply $[\gamma]'.g = [\gamma g]'$ .

Theorem 1.2 implies that $\SL_2(\Z)$ is generated by the two matrices $\sigma = \abcd{0}{-1}{1}{\hfill0}$ and $\tau=\abcd{1}{-1}{1}{\hfill0}$ . Note that $\sigma=S$ from Theorem 1.2 and $\tau = TS$ , so $T = \tau\sigma \in \langle \sigma, \tau\rangle.$

The following theorem provides us with a finite presentation for the space $\M_2(\Gamma_0(N))$ of modular symbols.

Theorem 3.13

Consider the quotient $M$ of the free abelian group on Manin symbols $[r_0]',\ldots,[r_m]'$ by the subgroup generated by the elements (for all $i$ ):

${[r_i]'} + [r_i]'\sigma \qquad\text{ and }\qquad {[r_i]'} + [r_i]'\tau + [r_i]'\tau^2 ,$

and modulo any torsion. Then there is an isomorphism

$\Psi: M \xrightarrow{\,\sim\,} \sM_2(\Gamma_0(N))$

given by $[r_i]' \mapsto [r_i] = r_i \{0,\infty\}$ .

Proof

We will only prove that $\Psi$ is surjective; the proof that $\Psi$ is injective requires much more work and will be omitted from this book (see [Man72, Section 1.7] for a complete proof).

Proposition 3.11 implies that $\Psi$ is surjective, assuming that $\Psi$ is well defined. We next verify that $\Psi$ is well defined, i.e., that the listed 2-term and 3-term relations hold in the image. To see that the first relation holds, note that

${[r_i]} + [r_i]\sigma &= \{r_i(0),r_i(\infty)\} + \{r_i\sigma(0), r_i\sigma(\infty)\}\\ &= \{r_i(0),r_i(\infty)\} + \{r_i(\infty), r_i(0)\} \\ & = 0.$
For the second relation we have

${[r_i]} + [r_i]\tau + [r_i]\tau^2 &= \{r_i(0),r_i(\infty)\} + \{r_i\tau(0), r_i\tau(\infty)\} + \{r_i\tau^2(0), r_i\tau^2(\infty)\} \\ &= \{r_i(0),r_i(\infty)\} + \{r_i(\infty), r_i(1)\} + \{r_i(1), r_i(0)\} \\ &= 0.$

Example 3.14

By default Sage computes modular symbols spaces over $\Q$ , i.e., $\sM_2(\Gamma_0(N);\Q) \isom \sM_2(\Gamma_0(N))\tensor\Q$ . Sage represents (weight $2$ ) Manin symbols as pairs $(c,d)$ . Here $c,d$ are integers that satisfy $0\leq c,d < N$ ; they define a point $(c:d) \in \P^1(\Z/N\Z)$ , hence a right coset of $\Gamma_0(N)$ in $\SL_2(\Z)$ (see Proposition 3.10).

Create $\sM_2(\Gamma_0(N);\Q)$ in Sage by typing ModularSymbols(N, 2). We then use the Sage command manin_generators to enumerate a list of generators $[r_0], \ldots, [r_n]$ as in Theorem 3.13 for several spaces of modular symbols.

sage: M = ModularSymbols(2,2)
sage: M
Modular Symbols space of dimension 1 for Gamma_0(2)
of weight 2 with sign 0 over Rational Field
sage: M.manin_generators()
[(0,1), (1,0), (1,1)]

sage: M = ModularSymbols(3,2)
sage: M.manin_generators()
[(0,1), (1,0), (1,1), (1,2)]

sage: M = ModularSymbols(6,2)
sage: M.manin_generators()
[(0,1), (1,0), (1,1), (1,2), (1,3), (1,4), (1,5), (2,1),
(2,3), (2,5), (3,1), (3,2)]

Given x=(c,d), the command x.lift_to_sl2z(N) computes an element of $\SL_2(\Z)$ whose lower two entries are congruent to $(c,d)$ modulo $N$ .

sage: M = ModularSymbols(2,2)
sage: [x.lift_to_sl2z(2) for x in M.manin_generators()]
[[1, 0, 0, 1], [0, -1, 1, 0], [0, -1, 1, 1]]
sage: M = ModularSymbols(6,2)
sage: x = M.manin_generators()[9]
sage: x
(2,5)
sage: x.lift_to_sl2z(6)
[1, 2, 2, 5]

The manin_basis command returns a list of indices into the Manin generator list such that the corresponding symbols form a basis for the quotient of the $\Q$ -vector space spanned by Manin symbols modulo the $2$ -term and $3$ -term relations of Theorem 3.13.

sage: M = ModularSymbols(2,2)
sage: M.manin_basis()
[1]
sage: [M.manin_generators()[i] for i in M.manin_basis()]
[(1,0)]
sage: M = ModularSymbols(6,2)
sage: M.manin_basis()
[1, 10, 11]
sage: [M.manin_generators()[i] for i in M.manin_basis()]
[(1,0), (3,1), (3,2)]

Thus, e.g., every element of $\sM_2(\Gamma_0(6))$ is a $\Q$ -linear combination of the three symbols $[(1,0)]$ , $[(3,1)]$ , and $[(3,2)]$ . We can write each of these as a modular symbol using the modular_symbol_rep function.

sage: M.basis()
((1,0), (3,1), (3,2))
sage: [x.modular_symbol_rep() for x in M.basis()]
[{Infinity,0}, {0,1/3}, {-1/2,-1/3}]

The manin_gens_to_basis function returns a matrix whose rows express each Manin symbol generator in terms of the subset of Manin symbols that forms a basis (as returned by manin_basis).

sage: M = ModularSymbols(2,2)
sage: M.manin_gens_to_basis()
[-1]
[ 1]
[ 0]

Since the basis is $(1,0)$ , this means that in $\sM_2(\Gamma_0(2);\Q)$ , we have $[(0,1)] = -[(1,0)]$ and $[(1,1)] = 0$ . (Since no denominators are involved, we have in fact computed a presentation of $\sM_2(\Gamma_0(2);\Z)$ .)

To convert a Manin symbol $x=(c,d)$ to an element of a modular symbols space $M$ , use M(x):

sage: M = ModularSymbols(2,2)
sage: x = (1,0); M(x)
(1,0)

Next consider $\sM_2(\Gamma_0(6);\Q)$ :

sage: M = ModularSymbols(6,2)
sage: M.manin_gens_to_basis()
[-1  0  0]
[ 1  0  0]
[ 0  0  0]
[ 0 -1  1]
[ 0 -1  0]
[ 0 -1  1]
[ 0  0  0]
[ 0  1 -1]
[ 0  0 -1]
[ 0  1 -1]
[ 0  1  0]
[ 0  0  1]

Recall that our choice of basis for $\sM_2(\Gamma_0(6);\Q)$ is $[(1,0)], [(3,1)], [(3,2)]$ . Thus, e.g., the first row of this matrix says that $[(0,1)] = -[(1,0)]$ , and the fourth row asserts that $[(1,2)] = -[(3,1)] + [(3,2)]$ .

sage: M = ModularSymbols(6,2)
sage: M((0,1))
-(1,0)
sage: M((1,2))
-(3,1) + (3,2)

Hecke Operators¶

Hecke Operators on Modular Symbols¶

When $p$ is a prime not dividing $N$ , define

$T_p(\{\alpha,\beta\}) = \mtwo{p}{0}{0}{1}\{\alpha,\beta\} + \sum_{r \hspace{-.6em}\mod p} \mtwo{1}{r}{0}{p} \{\alpha,\beta\}.$

The Hecke operators are compatible with the integration pairing $\langle\, , \, \rangle$ of Section Hecke Operators, in the sense that $\langle f T_p, x\rangle = \langle f, T_p x \rangle$ . When $p\mid N$ , the definition is the same, except that the matrix $\abcd{p}{0}{0}{1}$ is not included in the sum (see Theorem 1.44). There is a similar definition of $T_n$ for $n$ composite (see Section General Definition of Hecke Operators).

Example 3.15

For example, when $N=11$ , we have

$\begin{eqnarray*} T_2\{0,1/5\} &=& \{0,2/5\} + \{0,1/10\} + \{1/2,3/5\}\\ &=& -2\{0,1/5\}. \end{eqnarray*}$

See Figure 3.3.

Figure 3.3

Image of $\{0,1/5\}$ under $T_2$

Hecke Operators on Manin Symbols¶

In [Mer94], L. Merel gives a description of the action of $T_p$ directly on Manin symbols $[r_i]$ (see Section Hecke Operators on Manin Symbols for details). For example, when $p=2$ and $N$ is odd, we have

(5) $T_2([r_i]) = [r_i]\mtwo{1}{0}{0}{2} + [r_i]\mtwo{2}{0}{0}{1} + [r_i]\mtwo{2}{1}{0}{1} + [r_i]\mtwo{1}{0}{1}{2}.$

For any prime, let $C_p$ be the set of matrices constructed using the following algorithm (see [Cre97a, Section 2.4]):

Algorithm 3.16

Given a prime $p$ , this algorithm outputs a list of $2\times 2$ matrices of determinant $p$ that can be used to compute the Hecke operator $T_p$ .

Output $\mtwo{1}{0}{0}{p}$ .
For :
1. Let $x_1 = p$ , $x_2 = -r$ , $y_1 = 0$ , $y_2 = 1$ , $a=-p$ , $b=r$ .
2. Output $\mtwo{x_1}{x_2}{y_1}{y_2}$ .
3. As long as , do the following:
  1. Let $q$ be the integer closest to $a/b$ (if $a/b$ is a half integer, round away from $0$ ).
  2. Let $c=a-bq$ , $a=-b$ , $b=c$ .
  3. Set $x_3=qx_2 - x_1,\, x_1 = x_2,\, x_2 = x_3$ , and\ $y_3=qy_2 - y_1,\, y_1 = y_2,\, y_2 = y_3$ .
  4. Output $\mtwo{x_1}{x_2}{y_1}{y_2}$ .

Proposition 3.17

Let $C_p$ be as above. Then for $p\nmid N$ and $[x]\in \M_2(\Gamma_0(N))$ a Manin symbol, we have

$T_p([x]) = \sum_{g \in C_p} [xg].$

Proof

See Proposition~2.4.1 of [Cre97a].

There are other lists of matrices, due to Merel, that work even when $p\mid N$ (see Section Hecke Operators on Manin Symbols).

The command HeilbronnCremonaList(p), for $p$ prime, outputs the list of matrices from Algorithm 3.16.

  sage: HeilbronnCremonaList(2)
  [[1, 0, 0, 2], [2, 0, 0, 1], [2, 1, 0, 1], [1, 0, 1, 2]]
  sage: HeilbronnCremonaList(3)
  [[1, 0, 0, 3], [3, 1, 0, 1], [1, 0, 1, 3], [3, 0, 0, 1],
  [3, -1, 0, 1], [-1, 0, 1, -3]]
  sage: HeilbronnCremonaList(5)
  [[1, 0, 0, 5], [5, 2, 0, 1], [2, 1, 1, 3], [1, 0, 3, 5],
  [5, 1, 0, 1], [1, 0, 1, 5], [5, 0, 0, 1], [5, -1, 0, 1],
  [-1, 0, 1, -5], [5, -2, 0, 1], [-2, 1, 1, -3],
  [1, 0, -3, 5]]
  sage: len(HeilbronnCremonaList(37))
  128
  sage: len(HeilbronnCremonaList(389))
  1892
  sage: len(HeilbronnCremonaList(2003))
  11662

.. index::
   pair: Sage; Hecke operator `T_2`

Example 3.18

We compute the matrix of $T_2$ on $\sM_2(\Gamma_0(2))$ :

sage: M = ModularSymbols(2,2)
sage: M.T(2).matrix()
[1]

Example 3.19

We compute some Hecke operators on $\sM_2(\Gamma_0(6))$ :

sage: M = ModularSymbols(6, 2)
sage: M.T(2).matrix()
[ 2  1 -1]
[-1  0  1]
[-1 -1  2]
sage: M.T(3).matrix()
[3 2 0]
[0 1 0]
[2 2 1]
sage: M.T(3).fcp()  # factored characteristic polynomial
(x - 3) * (x - 1)^2

For $p\geq 5$ we have $T_p = p+1$ , since $M_2(\Gamma_0(6))$ is spanned by generalized Eisenstein series (see Chapter Eisenstein Series and Bernoulli Numbers).

Example 3.20

We compute the Hecke operators on $\sM_2(\Gamma_0(39))$ :

sage: M = ModularSymbols(39, 2)
sage: T2 = M.T(2)
sage: T2.matrix()
[ 3  0 -1  0  0  1  1 -1  0]
[ 0  0  2  0 -1  1  0  1 -1]
[ 0  1  0 -1  1  1  0  1 -1]
[ 0  0  1  0  0  1  0  1 -1]
[ 0 -1  2  0  0  1  0  1 -1]
[ 0  0  1  1  0  1  1 -1  0]
[ 0  0  0 -1  0  1  1  2  0]
[ 0  0  0  1  0  0  2  0  1]
[ 0  0 -1  0  0  0  1  0  2]
sage: T2.fcp()     # factored characteristic polynomial
(x - 3)^3 * (x - 1)^2 * (x^2 + 2*x - 1)^2

The Hecke operators commute, so their eigenspace structures are related.

sage: T2 = M.T(2).matrix()
sage: T5 = M.T(5).matrix()
sage: T2*T5 - T5*T2 == 0
True
sage: T5.charpoly().factor()
(x^2 - 8)^2 * (x - 6)^3 * (x - 2)^2

The decomposition of $T_2$ is a list of the kernels of $(f^e)(T_2)$ , where $f$ runs through the irreducible factors of the characteristic polynomial of $T_2$ and $f^e$ exactly divides this characteristic polynomial. Using Sage, we find them:

sage: M = ModularSymbols(39, 2)
sage: M.T(2).decomposition()
[
Modular Symbols subspace of dimension 3 of Modular
Symbols space of dimension 9 for Gamma_0(39) of weight
2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 2 of Modular
Symbols space of dimension 9 for Gamma_0(39) of weight
2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular
Symbols space of dimension 9 for Gamma_0(39) of weight
2 with sign 0 over Rational Field
]

Computing the Boundary Map¶

In Section Modular Symbols we defined a map $\delta: \sM_2(\Gamma_0(N)) \to \sB_2(\Gamma_0(N))$ . The kernel of this map is the space $\sS_2(\Gamma_0(N))$ of cuspidal modular symbols. This kernel will be important in computing cusp forms in Section Computing Using Eigenvectors below.

To compute the boundary map on $[\gamma]$ , note that $[\gamma] = \{\gamma(0),\gamma(\infty)\}$ , so if $\gamma=\abcd{a}{b}{c}{d}$ , then

$\delta([\gamma]) = \{\gamma(\infty)\} - \{\gamma(0)\} = \{a/c\} - \{b/d\}.$

Computing this boundary map would appear to first require an algorithm to compute the set $C(\Gamma_0(N)) = \Gamma_0(N) \backslash \P^1(\Q)$ of cusps for $\Gamma_0(N)$ . In fact, there is a trick that computes the set of cusps in the course of running the algorithm. First, give an algorithm for deciding whether or not two elements of $\P^1(\Q)$ are equivalent modulo the action of $\Gamma_0(N)$ . Then simply construct $C(\Gamma_0(N))$ in the course of computing the boundary map, i.e., keep a list of cusps found so far, and whenever a new cusp class is discovered, add it to the list. The following proposition, which is proved in [Cre97a, Prop. 2.2.3], explains how to determine whether two cusps are equivalent.

Proposition 3.21

Let $(c_i,d_i)$ , $i=1,2$ , be pairs of integers with $\gcd(c_i,d_i)=1$ and possibly $d_i=0$ . There is $g\in\Gamma_0(N)$ such that $g (c_1/d_1)=c_2/d_2$ in $\P^1(\Q)$ if and only if

$s_1 d_2 \con s_2 d_1 \pmod{\gcd(d_1 d_2,N)}$
where $s_j$ satisfies $c_j s_j\con 1\pmod{d_j}$ .

In Sage the command boundary_map() computes the boundary map from $\sM_2(\Gamma_0(N))$ to $\sB_2(\Gamma_0(N))$ , and the cuspidal_submodule command computes its kernel. For example, for level $2$ the boundary map is given by the matrix $[1 \,\,\, -1]$ , and its kernel is the $0$ space:

sage: M = ModularSymbols(2, 2)
sage: M.boundary_map()
Hecke module morphism boundary map defined by the matrix
[ 1 -1]
Domain: Modular Symbols space of dimension 1 for
Gamma_0(2) of weight ...
Codomain: Space of Boundary Modular Symbols for
Congruence Subgroup Gamma0(2) ...
sage: M.cuspidal_submodule()
Modular Symbols subspace of dimension 0 of Modular
Symbols space of dimension 1 for Gamma_0(2) of weight
2 with sign 0 over Rational Field

The smallest level for which the boundary map has nontrivial kernel, i.e., for which $\sS_2(\Gamma_0(N))\neq 0$ , is $N=11$ .

sage: M = ModularSymbols(11, 2)
sage: M.boundary_map().matrix()
[ 1 -1]
[ 0  0]
[ 0  0]
sage: M.cuspidal_submodule()
Modular Symbols subspace of dimension 2 of Modular
Symbols space of dimension 3 for Gamma_0(11) of weight
2 with sign 0 over Rational Field
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular
Symbols space of dimension 3 for Gamma_0(11) of weight
2 with sign 0 over Rational Field
sage: S.basis()
((1,8), (1,9))

The following illustrates that the Hecke operators preserve $\sS_2(\Gamma_0(N))$ :

sage: S.T(2).matrix()
[-2  0]
[ 0 -2]
sage: S.T(3).matrix()
[-1  0]
[ 0 -1]
sage: S.T(5).matrix()
[1 0]
[0 1]

A nontrivial fact is that for $p$ prime the eigenvalue of each of these matrices is $p+1 - \#E(\F_p)$ , where $E$ is the elliptic curve $X_0(11)$ defined by the (affine) equation $y^2 + y = x^3 - x^2 - 10x - 20.$ For example, we have

sage: E = EllipticCurve([0,-1,1,-10,-20])
sage: 2 + 1 - E.Np(2)
-2
sage: 3 + 1 - E.Np(3)
-1
sage: 5 + 1 - E.Np(5)
1
sage: 7 + 1 - E.Np(7)
-2

The same numbers appear as the eigenvalues of Hecke operators:

sage: [S.T(p).matrix()[0,0] for p in [2,3,5,7]]
[-2, -1, 1, -2]

In fact, something similar happens for every elliptic curve over $\Q$ . The book [DS05] (especially Chapter~8) is about this striking numerical relationship between the number of points on elliptic curves modulo $p$ and coefficients of modular forms.

Computing a Basis for $S_2(\Gamma_0(N))$ ¶

This section is about a method for using modular symbols to compute a basis for $S_2(\Gamma_0(N))$ . It is not the most efficient for certain applications, but it is easy to explain and understand. See Section Computing Using Eigenvectors for a method that takes advantage of additional structure of $S_2(\Gamma_0(N))$ .

Let $\sM_2(\Gamma_0(N);\Q)$ and $\sS_2(\Gamma_0(N);\Q)$ be the spaces of modular symbols and cuspidal modular symbols over $\Q$ . Before we begin, we describe a simple but crucial fact about the relation between cusp forms and Hecke operators.

If $f = \sum b_n q^n \in \C[[q]]$ is a power series, let $a_n(f) = b_n$ be the $n$ coefficient of $f$ . Notice that $a_n$ is a $\C$ -linear map $\C[[q]] \to \C$ .

As explained in [DS05, Prop. 5.3.1] and [Lan95, Section VII.3] (recall also Proposition 2.31), the Hecke operators $T_n$ act on elements of $M_2(\Gamma_0(N))$ as follows (where $k=2$ below):

(6) $T_n\left(\sum_{m=0}^{\infty} a_m q^m\right) = \sum_{m=0}^{\infty} \left(\sum_{1\leq d\,\mid\, \gcd(n,m)} \eps(d) \cdot d^{k-1} \cdot a_{mn/d^2}\right) q^m,$

where $\eps(d)=1$ if $\gcd(d,N)=1$ and $\eps(d)=0$ if $\gcd(d,N)\neq 1$ . (Note: More generally, if $f \in M_k(\Gamma_1(N))$ is a modular form with Dirichlet character $\eps$ , then the above formula holds; above we are considering this formula in the special case when $\eps$ is the trivial character and $k=2$ .)

Lemma 3.22

Suppose $f\in \C[[q]]$ and $n$ is a positive integer. Let $T_n$ be the operator on $q$ -expansions (formal power series) defined by (6). Then

$a_1(T_n(f)) = a_n(f).$

Proof

The coefficient of $q$ in (6) is $\eps(1)\cdot 1 \cdot a_{1 \cdot n/1^2} = a_n$ .

The Hecke algebra $\T$ is the ring generated by all Hecke operators $T_n$ acting on $M_k(\Gamma_1(N))$ . Let $\T'$ denote the image of the Hecke algebra in $\End(S_2(\Gamma_0(N)))$ , and let $\T'_\C=\T'\tensor_\Z\C$ be the $\C$ -span of the Hecke operators. Let $\tT_\C$ denote the subring of $\End(\C[[q]])$ generated over $\C$ by all Hecke operators acting on formal power series via definition (6).

Proposition 3.23

There is a bilinear pairing of complex vector spaces

$\C[[q]] \cross \tT_\C \to \C$

given by

$\langle f, t \rangle = a_1(t(f)).$

If $f$ is such that $\langle f, t\rangle = 0$ for all $t\in \tT_\C$ , then $f=0$ .

Proof

The pairing is bilinear since both $t$ and $a_1$ are linear.

Suppose $f\in \C[[q]]$ is such that $\langle f, t\rangle = 0$ for all $t\in \tT_\C$ . Then $\langle f, T_n\rangle = 0$ for each positive integer $n$ . But by Lemma 3.22 we have

$a_n(f) = a_1(T_n(f)) = 0$

for all $n$ ; thus $f=0$ .

Proposition 3.24

There is a perfect bilinear pairing of complex vector spaces

$S_2(\Gamma_0(N)) \cross \T'_\C \to \C$

given by

$\langle f, t \rangle = a_1(t(f)).$

Proof

The pairing has $0$ kernel on the left by Proposition 3.23. Suppose that $t\in\T'_\C$ is such that $\langle f, t\rangle =0$ for all $f \in S_2(\Gamma_0(N))$ . Then $a_1(t(f)) = 0$ for all $f$ . For any $n$ , the image $T_n(f)$ is also a cusp form, so $a_1(t(T_n(f))) = 0$ for all $n$ and $f$ . Finally the fact that $\T'$ is commutative and Lemma 3.22 together imply that for all $n$ and $f$ ,

$0 = a_1(t(T_n(f))) = a_1(T_n(t(f))) = a_n(t(f)),$

so $t(f)=0$ for all $f$ . Thus $t$ is the $0$ operator.

Since $S_2(\Gamma_0(N))$ has finite dimension and the kernel on each side of the pairing is $0$ , it follows that the pairing is perfect, i.e., defines an isomorphism

$\T'_\C \isom \Hom_{\C}(S_2(\Gamma_0(N));\C).$

By Proposition 3.24 there is an isomorphism of vector spaces

(7) $\Psi: S_2(\Gamma_0(N)) \xrightarrow{\,\,\isom\,\,} \Hom(\T'_\C,\C)$

that sends $f \in S_2(\Gamma_0(N))$ to the homomorphism

$t \mapsto a_1(t(f)).$

For any $\C$ -linear map $\vphi:\T'_\C\to \C$ , let

$f_{\vphi} = \sum_{n=1}^{\infty} \vphi(T_n)q^n \in\C[[q]].$

Lemma 3.25

The series $f_{\vphi}$ is the $q$ -expansion of $\Psi^{-1}(\vphi) \in S_2(\Gamma_0(N))$ .

Proof

Note that it is not even a priori obvious that $f_{\vphi}$ is the $q$ -expansion of a modular form. Let $g = \Psi^{-1}(\vphi)$ , which is by definition the unique element of $S_2(\Gamma_0(N))$ such that $\langle g, T_n \rangle = \vphi(T_n)$ for all $n$ . By Lemma 3.22, we have

$\langle f_{\vphi}, T_n \rangle = a_1(T_n(f_\vphi)) = a_n(f_{\vphi}) = \vphi(T_n),$

so $\langle f_{\vphi} - g, T_n \rangle = 0$ for all $n$ . Proposition 3.23 implies that $f_\vphi - g = 0$ , so $f_{\vphi} = g = \Psi^{-1}(\vphi)$ , as claimed.

Conclusion: The cusp forms $f_{\vphi}$ , as $\vphi$ varies through a basis of $\Hom(\T'_\C,\C)$ , form a basis for $S_2(\Gamma_0(N))$ . In particular, we can compute $S_2(\Gamma_0(N))$ by computing $\Hom(\T'_{\C},\C)$ , where we compute $\T'$ in any way we want, e.g., using a space that contains an isomorphic copy of $S_2(\Gamma_0(N))$ .

Algorithm 3.26

Given positive integers $N$ and $B$ , this algorithm computes a basis for $S_2(\Gamma_0(N))$ to precision $O(q^B)$ .

Compute $\sM_2(\Gamma_0(N);\Q)$ via the presentation of Section Manin Symbols.
Compute the subspace $\sS_2(\Gamma_0(N);\Q)$ of cuspidal modular symbols as in Section Computing the Boundary Map.
Let $d=\frac{1}{2}\cdot \dim\sS_2(\Gamma_0(N);\Q)$ . By Proposition 3.8, $d$ is the dimension of $S_2(\Gamma_0(N))$ .
Let $[T_n]$ denote the matrix of $T_n$ acting on a basis of $\sS_2(\Gamma_0(N);\Q)$ . For a matrix $A$ , let $a_{ij}(A)$ denote the $ij`th entry of `A$ . For various integers $i,j$ with $0\leq i,j \leq d-1$ , compute formal $q$ -expansions

$f_{ij}(q) = \sum_{n=1}^{B-1} a_{ij}([T_n])q^n + O(q^{B}) \in \Q[[q]]$

until we find enough to span a space of dimension $d$ (or exhaust all of them). These $f_{ij}$ are a basis for $S_2(\Gamma_0(N))$ to precision $O(q^{B})$ .

Examples¶

We use Sage to demonstrate Algorithm 3.26.

Example 3.27

The smallest $N$ with $S_2(\Gamma_0(N))\neq 0$ is $N=11$ .

sage: M = ModularSymbols(11); M.basis()
((1,0), (1,8), (1,9))
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular
Symbols space of dimension 3 for Gamma_0(11) of weight
2 with sign 0 over Rational Field

We compute a few Hecke operators, and then read off a nonzero cusp form, which forms a basis for $S_2(\Gamma_0(11))$ :

sage: S.T(2).matrix()
[-2  0]
[ 0 -2]
sage: S.T(3).matrix()
[-1  0]
[ 0 -1]

Thus

$f_{0,0} = q - 2q^2 - q^3 + \cdots \in S_2(\Gamma_0(11))$

forms a basis for $S_2(\Gamma_0(11))$ .

Example 3.28

We compute a basis for $S_2(\Gamma_0(33))$ to precision $O(q^6)$ .

sage: M = ModularSymbols(33)
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 6 of Modular
Symbols space of dimension 9 for Gamma_0(33) of weight
2 with sign 0 over Rational Field

Example 3.29

Next consider $N=23$ , where we have

$d = \dim S_2(\Gamma_0(23)) = 2.$

The command q_expansion_cuspforms computes matrices $T_n$ and returns a function $f$ such that $f(i,j)$ is the $q$ -expansion of $f_{i,j}$ to some precision. (For efficiency reasons, $f(i,j)$ in Sage actually computes matrices of $T_n$ acting on a basis for the linear dual of $\sS_2(\Gamma_0(N))$ .)

sage: M = ModularSymbols(23)
sage: S = M.cuspidal_submodule()
sage: S
Modular Symbols subspace of dimension 4 of Modular
Symbols space of dimension 5 for Gamma_0(23) of weight
2 with sign 0 over Rational Field
sage: f = S.q_expansion_cuspforms(6)
sage: f(0,0)
q - 2/3*q^2 + 1/3*q^3 - 1/3*q^4 - 4/3*q^5 + O(q^6)
sage: f(0,1)
O(q^6)
sage: f(1,0)
-1/3*q^2 + 2/3*q^3 + 1/3*q^4 - 2/3*q^5 + O(q^6)

Thus a basis for $S_2(\Gamma_0(23))$ is

$f_{0,0} &= q - \frac{2}{3}q^{2} + \frac{1}{3}q^{3} - \frac{1}{3}q^{4} - \frac{4}{3}q^{5} + \cdots, \\ f_{1,0} &= -\frac{1}{3}q^{2} + \frac{2}{3}q^{3} + \frac{1}{3}q^{4} - \frac{2}{3}q^{5} + \cdots.$

Or, in echelon form,

$& q - q^{3} - q^{4} + \cdots\\ & \,\,\,\,\,q^{2} - 2q^{3} - q^{4} + 2q^{5} + \cdots$

which we computed using

sage: S.q_expansion_basis(6)
[
q - q^3 - q^4 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6)
]

Computing $S_2(\Gamma_0(N))$ Using Eigenvectors¶

In this section we describe how to use modular symbols to construct a basis of $S_2(\Gamma_0(N))$ consisting of modular forms that are eigenvectors for every element of the ring $\T^{(N)}$ generated by the Hecke operator $T_p$ , with $p\nmid N$ . Such eigenvectors are called eigenforms.

Suppose $M$ is a positive integer that divides $N$ . As explained in [Lan95, VIII.1–2], for each divisor $d$ of $N/M$ there is a natural degeneracy map $\alpha_{M,d} : S_2(\Gamma_0(M))\ra S_2(\Gamma_0(N))$ given by $\alpha_{M,d}(f(q)) = f(q^d)$ . The new subspace of $S_2(\Gamma_0(N))$ , denoted $S_2(\Gamma_0(N))_{\new}$ , is the complementary $\T$ -submodule of the $\T$ -module generated by the images of all maps $\alpha_{M,d}$ , with $M$ and $d$ as above. It is a nontrivial fact that this complement is well defined; one possible proof uses the Petersson inner product (see [Lan95, Section VII.5]).

The theory of Atkin and Lehner [AL70] (see Theorem 9.4 below) asserts that, as a $\T^{(N)}$ -module, $S_2(\Gamma_0(N))$ decomposes as follows:

$S_2(\Gamma_0(N)) \quad = \bigoplus_{M | N, \,\, d | N/M} \beta_{M,d}(S_2(\Gamma_0(M))_{\new}).$

To compute $S_2(\Gamma_0(N))$ it suffices to compute $S_2(\Gamma_0(M))_{\new}$ for each $M\mid N$ .

We now turn to the problem of computing $S_2(\Gamma_0(N))_{\new}$ . Atkin and Lehner [AL70] proved that $S_2(\Gamma_0(N))_{\new}$ is spanned by eigenforms for all $T_p$ with $p\nmid N$ and that the common eigenspaces of all the $T_p$ with $p\nmid N$ each have dimension $1$ . Moreover, if $f\in S_2(\Gamma_0(N))_{\new}$ is an eigenform then the coefficient of $q$ in the $q$ -expansion of $f$ is nonzero, so it is possible to normalize $f$ so the coefficient of $q$ is $1$ (such a normalized eigenform in the new subspace is called a newform). With $f$ so normalized, if $T_p(f) = a_p f$ , then the $p^{th}$ Fourier coefficient of $f$ is $a_p$ . If $f=\sum_{n=1}^{\infty} a_n q^n$ is a normalized eigenvector for all $T_p$ , then the $a_n$ , with $n$ composite, are determined by the $a_p$ , with $p$ prime, by the following formulas: $a_{nm}=a_n a_m$ when $n$ and $m$ are relatively prime and $a_{p^r} = a_{p^{r-1}} a_p - p a_{p^{r-2}}$ for $p\nmid N$ prime. When $p \mid N$ , $a_{p^r}=a_p^r$ . We conclude that in order to compute $S_2(\Gamma_0(N))_{\new}$ , it suffices to compute all systems of eigenvalues $\{a_2, a_3, a_5, \ldots\}$ of the prime-indexed Hecke operators $T_2, T_3, T_5, \ldots$ acting on $S_2(\Gamma_0(N))_{\new}$ . Given a system of eigenvalues, the corresponding eigenform is $f=\sum_{n=1}^{\infty} a_n q^n$ , where the $a_n$ , for $n$ composite, are determined by the recurrence given above.

In light of the pairing $\langle\, ,\, \rangle$ introduced in Section Hecke Operators, computing the above systems of eigenvalues $\{a_2,a_3, a_5, \ldots \}$ amounts to computing the systems of eigenvalues of the Hecke operators $T_p$ on the subspace $V$ of $\sS_2(\Gamma_0(N))$ that corresponds to the new subspace of $S_2(\Gamma_0(N)).$ For each proper divisor $M$ of $N$ and each divisor $d$ of $N/M$ , let $\phi_{M,d}:\sS_2(\Gamma_0(N)) \ra \sS_2(\Gamma_0(M))$ be the map sending $x$ to $\abcd{d}{0}{0}{1}x$ . Then $V$ is the intersection of the kernels of all maps $\phi_{M,d}$ .

Computing the systems of eigenvalues of a collection of commuting diagonalizable endomorphisms is a problem in linear algebra (see Chapter Linear Algebra).

Example 3.30

All forms in $S_2(\Gamma_0(39))$ are new. Up to Galois conjugacy, the eigenvalues of the Hecke operators $T_2$ , $T_3$ , $T_5$ , and $T_7$ on $\sS_2(\Gamma_0(39))$ are $\{1, -1, 2, -4 \}$ and $\{a, 1, -2a-2, 2a+2\}$ , where $a^2+2a-1=0$ . Each of these eigenvalues occur in $\sS_2(\Gamma_0(39))$ with multiplicity two; for example, the characteristic polynomial of $T_2$ on $\sS_2(\Gamma_0(39))$ is $(x - 1)^2\cdot(x^2+2x-1)^2$ . Thus $S_2(\Gamma_0(39))$ is spanned by

$f_1&=q + q^2 - q^3 - q^4 + 2q^5 - q^6 - 4q^7 + \cdots, \\ f_2&= q + aq^2 + q^3 + (-2a - 1)q^4 + (-2a - 2)q^5 + aq^6 + (2a + 2)q^7 + \cdots, \\ f_3&= q + \sigma(a)q^2 + q^3 + (-2\sigma(a) - 1)q^4 + (-2\sigma(a) - 2)q^5 + \sigma(a)q^6 + \cdots,$

where $\sigma(a)$ is the other $\Gal(\Qbar/\Q)$ -conjugate of $a$ .

Summary¶

To compute the $q$ -expansion of a basis for $S_2(\Gamma_0(N))$ , we use the degeneracy maps so that we only have to solve the problem for $S_2(\Gamma_0(M))_{\new}$ , for all integers $M\mid N$ . Using modular symbols, we compute all systems of eigenvalues $\{a_2, a_3,a_5,\ldots\}$ , and then write down the corresponding eigenforms $\sum a_n q^n$ .

Exercises¶

Exercise 3.1

Suppose that $\lambda, \lambda' \in \h$ are in the same orbit for the action of $\Gamma_0(N)$ , i.e., that there exists $g\in \Gamma_0(N)$ such that $g(\lambda) = \lambda'$ . Let $\Lambda = \Z + \Z\lambda$ and $\Lambda' = \Z + \Z\lambda'$ . Prove that the pairs $(\C/\Lambda,(\frac{1}{N}\Z+\Lambda)/\Lambda)$ and $(\C/\Lambda', (\frac{1}{N}\Z + \Lambda')/\Lambda')$ are isomorphic. (By an isomorphism $(E,C)\to (F,D)$ of pairs, we mean an isomorphism $\phi: E\to F$ of elliptic curves that sends $C$ to $D$ . You may use the fact that an isomorphism of elliptic curves over $\C$ is a $\C$ -linear map $\C\to\C$ that sends the lattice corresponding to one curve onto the lattice corresponding to the other.)

Exercise 3.2

Let $n,m$ be integers and $N$ a positive integer. Prove that the modular symbol $\{n,m\}$ is $0$ as an element of $\sM_2(\Gamma_0(N))$ . [Hint: See Example 3.6.]

Exercise 3.3

Let $p$ be a prime.

List representative elements of $\P^1(\Z/p\Z)$ .
What is the cardinality of $\P^1(\Z/p\Z)$ as a function of $p$ ?
Prove that there is a bijection between the right cosets of $\Gamma_0(p)$ in $\SL_2(\Z)$ and the elements of $\P^1(\Z/p\Z)$ that sends $\abcd{a}{b}{c}{d}$ to $(c:d)$ . (As mentioned in this chapter, the analogous statement is also true when the level is composite; see [Cre97a, Section 2.2] for complete details.) end{enumerate}

Exercise 3.4

Use the inductive proof of Proposition 3.11 to write $\{0,4/7\}$ in terms of Manin symbols for $\Gamma_0(7)$ .

Exercise 3.5

Show that the Hecke operator $T_2$ acts as multiplication by $3$ on the space $\sM_2(\Gamma_0(3))$ as follows:

Write down right coset representatives for $\Gamma_0(3)$ in $\SL_2(\Z)$ .
List all eight relations coming from Theorem 3.13.
Find a single Manin symbols $[r_i]$ so that the three other Manin symbols are a nonzero multiple of $[r_i]$ modulo the relations found in the previous step.
Use formula (5) to compute $T_2([r_i])$ . You will obtain a sum of four symbols. Using the relations above, write this sum as a multiple of $[r_i]$ . (The multiple must be $3$ or you made a mistake.)

Modular Forms of Weight 2¶

Hecke Operators¶

Homology¶

Modular Symbols¶

Computing with Modular Symbols¶

Manin Symbols¶

Hecke Operators¶

Hecke Operators on Modular Symbols¶

Hecke Operators on Manin Symbols¶

Computing the Boundary Map¶

Computing a Basis for $S_2(\Gamma_0(N))$ ¶

Examples¶

Computing $S_2(\Gamma_0(N))$ Using Eigenvectors¶

Summary¶

Exercises¶

Table Of Contents

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Modular Forms of Weight 2¶

Hecke Operators¶

Homology¶

Modular Symbols¶

Computing with Modular Symbols¶

Manin Symbols¶

Hecke Operators¶

Hecke Operators on Modular Symbols¶

Hecke Operators on Manin Symbols¶

Computing the Boundary Map¶

Computing a Basis for ¶

Examples¶

Computing Using Eigenvectors¶

Summary¶

Exercises¶

Table Of Contents

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Purchase a hardcopy of this book from the AMS

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Computing a Basis for $S_2(\Gamma_0(N))$ ¶

Computing $S_2(\Gamma_0(N))$ Using Eigenvectors¶