Classical modular forms

Abstract

In this chapter, we introduce modular forms on the classical modular group. This chapter will provide motivation as well as important examples for generalizations in this last part of the text.

In this chapter, we introduce modular forms on the classical modular group. This chapter will provide motivation as well as important examples for generalizations in this last part of the text.

1 $\triangleright$ Functions on lattices

In this section, we pursue the interpretation of the quotient $\Gamma \backslash \boldsymbol{\mathsf {H}}^2$ with $\Gamma ={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ as a moduli space of lattices, and we study functions on the quotient. There are a wealth of references for the classical modular forms, including Apostol [Apo90, Chapters 1–2], Diamond–Shurman [DS2005], Miyake [Miy2006, Chapter 4], Lang [Lang95, §1], and Serre [Ser73, Chapter VII]. For this section, see Silverman [Sil2009, Chapter VI] for the complex analytic theory of elliptic curves and the relationship to Eisenstein series.

Recall from 35.3.3 that $Y=\Gamma \backslash \boldsymbol{\mathsf {H}}^2$ parametrizes complex lattices up to homothety, i.e., there is a bijection

$$\begin{aligned} Y=\Gamma \backslash \boldsymbol{\mathsf {H}}^2&\rightarrow \{\Lambda \subset \mathbb C \text { lattice}\}/\!\sim \nonumber \\ \Gamma \tau&\mapsto [\mathbb Z + \mathbb Z \tau ]. \end{aligned}$$

(40.1.1)

In particular, the set of homothety classes has a natural structure of a Riemann surface, and we seek now to make this explicit. We show that there are natural, holomorphic functions on the set of lattices that allow us to go beyond the bijection 40.1.1 to realize the complex structure on Y explicitly.

Let $\Lambda \subset \mathbb C$ be a lattice. To write down complex moduli, we average over $\Lambda$ in a convergent way, as follows.

Definition 40.1.2

The Eisenstein series of weight $k \in \mathbb Z _{>2}$ for $\Lambda$ is

$$\begin{aligned} G_k(\Lambda ) = \sum _{\begin{array}{c} \lambda \in \Lambda \\ \lambda \ne 0 \end{array}} \frac{1}{\lambda ^k}. \end{aligned}$$

If k is odd, then $G_k(\Lambda )=0$, so let $k \in 2\mathbb Z _{\ge 2}$.

Lemma 40.1.3

The series $G_k(\Lambda )$ converges absolutely.

Proof. Up to homothety (which does not affect convergence), we may suppose $\Lambda =\mathbb Z + \mathbb Z \tau$, with $\tau \in \boldsymbol{\mathsf {H}}^2$. Then we consider the corresponding absolute sum

$$\begin{aligned} \sum _{\begin{array}{c} \lambda \in \mathbb Z + \mathbb Z \tau \\ \lambda \ne 0 \end{array}} \frac{1}{|\lambda |^k} = \sum _{\begin{array}{c} m,n \in \mathbb Z \\ (m,n) \ne (0,0) \end{array}} \frac{1}{|m+n\tau |^k}. \end{aligned}$$

(40.1.4)

The number of pairs (m, n) with $r \le |m\tau +n| < r+1$ is the number of lattice points in an annulus of area $\pi (r+1)^2-\pi r^2 = O(r)$, so there are O(r) such points; and thus the series (40.1.4) is majorized by (a constant multiple of) $\sum _{r=1}^{\infty } r^{1-k}$, which is convergent for $k>2$. $\square$

40.1.5

For ${z \in \boldsymbol{\mathsf {H}}^2}$ and $k \in 2\mathbb Z _{\ge 2}$, define

$$\begin{aligned} G_k(z)= G_k(\mathbb Z + \mathbb Z z)=\sum _{\begin{array}{c} m,n \in \mathbb Z \\ (m,n) \ne (0,0) \end{array}} \frac{1}{(m+nz)^k}. \end{aligned}$$

(40.1.6)

Lemma 40.1.7

$G_k(z)$ is holomorphic for ${z \in \boldsymbol{\mathsf {H}}^2}$, and

$$\begin{aligned} G_k(\gamma z)=(cz+d)^k G_k(z) \end{aligned}$$

for all $\gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$.

Proof. This is true for since then

$$\begin{aligned} |m+nz|^2=m^2+2mn{{\,\mathrm{Re}\,}}z + n^2|z|^2 \ge m^2-mn + n^2 = |m+n\omega |^2 \end{aligned}$$

thus $|G_k(z)| \le |G_k(\omega )|$ and so by the Weierstrass M-test, $G_k(z)$ is holomorphic for : by Morera’s theorem, uniform convergence implies holomorphicity. But now for all $\gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in \Gamma$, we claim that

$$\begin{aligned} G_k(\gamma z)=(cz+d)^k G_k(z) \end{aligned}$$

(40.1.8)

(and note this does not depend on the choice of sign): indeed,

$$\begin{aligned} \frac{1}{m+n(\gamma z)} = \frac{cz+d}{(bn+dm)+(an+cm)z} \end{aligned}$$

(40.1.9)

and the map

$$ (n,m) \mapsto (n,m)\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} = (an+cm,bn+dm) $$

is a permutation of $\mathbb Z ^2 - \{(0,0)\}$, so by absolute convergence we may rearrange the sum to get

$$\begin{aligned} \begin{aligned} G_k(\gamma z)&=(cz+d)^k\sum _{\begin{array}{c} m,n \in \mathbb Z \\ (m,n) \ne (0,0) \end{array}} \frac{1}{((bn+dm)+(an+cm)z)^k} \\&= (cz+d)^k\sum _{\begin{array}{c} m,n \in \mathbb Z \\ (m,n) \ne (0,0) \end{array}} \frac{1}{(m+nz)^k} = (cz+d)^k G_k(z). \end{aligned} \end{aligned}$$

(40.1.10)

By transport, since , we see that $G_k(z)$ is holomorphic on all of ${\boldsymbol{\mathsf {H}}^2}$. $\square$

40.1.11

In this (somewhat long) paragraph, we connect the theory of Eisenstein series above to the theory of elliptic curves.

Let $\Lambda \subset \mathbb C$ be a lattice. We define the Weierstrass $\wp$-function (relative to $\Lambda$) by

$$\begin{aligned} \wp (z)=\wp (z;\Lambda ) = \frac{1}{z^2}+\sum _{\begin{array}{c} \lambda \in \Lambda \\ \lambda \ne 0 \end{array}} \left( \frac{1}{(z-\lambda )^2}-\frac{1}{\lambda ^2}\right) . \end{aligned}$$

(40.1.12)

We have

$$\begin{aligned} \left|\frac{1}{(z-\lambda )^2}-\frac{1}{\lambda ^2}\right|\le \frac{|z|(2|\lambda |+|z|)}{|\lambda |^2(|\lambda |-|z|)^2} =O\left( \frac{1}{|\lambda |^3}\right) \end{aligned}$$

(40.1.13)

so as above we see that $\wp (z)$ is absolutely convergent for all $z \in \mathbb C \smallsetminus \Lambda$ and uniformly convergent on compact subsets, and so defines a holomorphic function on $\mathbb C \smallsetminus \Lambda$. Since

$$\begin{aligned} \frac{1}{(z-\lambda )^2}-\frac{1}{\lambda ^2} = \sum _{n=1}^{\infty } (n+1)\frac{z^n}{\lambda ^{n+2}} \end{aligned}$$

(40.1.14)

by differentiating the geometric series, we find

$$\begin{aligned} \wp (z)=\frac{1}{z^2}+\sum _{k=3}^{\infty }(k-1)G_k(\Lambda ) z^k = \frac{1}{z^2}+3G_4(\Lambda )z^4 + 5G_6(\Lambda )z^6+\ldots . \end{aligned}$$

(40.1.15)

Differentiating with respect to z and squaring, we find that

$$\begin{aligned} \left( \frac{\mathrm {d}{\wp }}{\mathrm {d}{z}}(z)\right) ^2 = \frac{4}{z^6} - \frac{24G_4(\Lambda )}{z^2} - 80G_6(\Lambda ) + \dots .\end{aligned}$$

(40.1.16)

Expanding out the first few terms, we find that

$$\begin{aligned}f(z)=\left( \frac{\mathrm {d}{\wp }}{\mathrm {d}{z}}(z)\right) ^2 - 4\wp (z)^3+60G_4(\Lambda )\wp (z) + 140G_6(\Lambda ) = O(z^2)\end{aligned}$$

is holomorphic at $z=0$ and satisfies $f(z+\lambda )=f(z)$ for all $\lambda \in \Lambda$. By periodicity, f(z) takes its maximum in a fundamental parallelogram for $\Lambda$; then by Liouville’s theorem,f is bounded on $\mathbb C$ so constant. Since $f(0)=0$, we conclude that f(z) is identically zero.

Following convention, write

$$\begin{aligned} g_4=g_4(\Lambda )=60G_4(\Lambda ) \quad \text {and} \quad g_6=g_6(\Lambda )=140G_6(\Lambda ) \end{aligned}$$

and

$$\begin{aligned} x(z)=\wp (z;\Lambda ) \quad \text {and} \quad y(z)=\frac{\mathrm {d}{\wp }}{\mathrm {d}{z}}(z;\Lambda ). \end{aligned}$$

Then the image of the map

$$\begin{aligned} \begin{aligned} \mathbb C /\Lambda&\rightarrow \mathbb P ^2(\mathbb C ) \\ z&\mapsto (x(z):y(z):1) \end{aligned} \end{aligned}$$

(40.1.17)

is cut out by the affine equation

$$\begin{aligned} y^2=4x^3-g_4x-g_6; \end{aligned}$$

the map (40.1.17) is an isomorphism of Riemann surfaces. Looking ahead to Definition 42.1.1, this map exhibits $\mathbb C /\Lambda$ as an elliptic curve over $\mathbb C$.

To produce holomorphic functions that are well-defined on the quotient ${\Gamma \backslash \boldsymbol{\mathsf {H}}^2}$, we can take ratios of Eisenstein series; soon we will exhibit a map

$$\begin{aligned} {j:\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C }\end{aligned}$$

(40.1.18)

obtained in this way that defines a bijective holomorphic map ${\Gamma \backslash \boldsymbol{\mathsf {H}}^2 \displaystyle \mathop {\rightarrow }^{\sim } \mathbb C }$ (Theorem 40.3.8).

40.1.19

Eisenstein series can also be thought of as weighted averages over the (cosets of the) group ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ as follows.

Let $\Gamma _\infty \le \Gamma ={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ be the stabilizer of $\infty$; then $\Gamma _\infty$ is the infinite cyclic group generated by $T=\begin{pmatrix} 1 &{} 1 \\ 0 &{} 1 \end{pmatrix}$. We consider the cosets $\Gamma _\infty \backslash \Gamma$: for $t \in \mathbb Z$ and $\gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix}$ we have $T^t \gamma = \begin{pmatrix} a+tc &{} b+td \\ c &{} d \end{pmatrix}$ with the same bottom row. Thus the function $(cz+d)^2$ is well-defined on the coset $\Gamma _\infty \gamma$. Thus we can form the sum

$$\begin{aligned} E_k(z)=\sum _{\Gamma _\infty \gamma \in \Gamma _\infty \backslash \Gamma } (cz+d)^{-k} = \frac{1}{2} \sum _{\begin{array}{c} c,d \in \mathbb Z \\ \gcd (c,d)=1 \end{array}} \frac{1}{(cz+d)^k}, \end{aligned}$$

(40.1.20)

the factor 2 coming from the choice of sign in ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$. Since every nonzero $(m,n) \in \mathbb Z ^2$ can be written $(m,n)=r(c,d)$ with $r=\gcd (m,n)>0$ and $\gcd (c,d)=1$, we find that

$$\begin{aligned} G_k(z)=\zeta (k)E_k(z). \end{aligned}$$

2 $\triangleright$ Eisenstein series as modular forms

In the previous section, we saw that natural sums (Eisenstein series) defined functions on ${\boldsymbol{\mathsf {H}}^2}$ that transformed with respect to ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ with a natural invariance. In this section, we pursue this more systematically.

Definition 40.2.1

Let $k \in 2\mathbb Z$ and let $\Gamma \le {{\,\mathrm{PSL}\,}}_2(\mathbb R )$ be a Fuchsian group. A map ${f :\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C \cup \{\infty \}}$ is weight k-invariant under $\Gamma$ if

$$\begin{aligned} f(\gamma z)=(cz+d)^k f(z) \quad \text {for all}\, \gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in \Gamma . \end{aligned}$$

(40.2.2)

40.2.3

If f is weight k invariant and $f'$ is weight $k'$ invariant, then $ff'$ is weight $k+k'$ invariant, and if $k'=k$ then $f+f'$ is weight k invariant. Therefore, the set of weight k-invariant functions has the structure of a $\mathbb C$-vector space.

40.2.4

Weight k invariance under $\Gamma$ can be checked on a set of generators for $\Gamma$ lifted to ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$, as follows. For $\gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {{\,\mathrm{SL}\,}}_2(\mathbb R )$, define $\jmath (\gamma ;z)=cz+d$. Then (40.2.2) can be rewritten $f(\gamma z)=\jmath (\gamma ;z)^k f(z)$.

For $\gamma ' \in \Gamma$, we compute that $\jmath$ satisfies the cocycle relation

$$\begin{aligned} \jmath (\gamma \gamma ';z)=\jmath (\gamma ;\gamma ' z)\jmath (\gamma ';z) \end{aligned}$$

(40.2.5)

because if $\gamma '=\begin{pmatrix} a' &{} b' \\ c' &{} d' \end{pmatrix}$, then

$$\begin{aligned} (a'c+c'd)z+b(b'c+dd') = \left( c\left( \frac{a'z+b'}{c'z+d'}\right) +d\right) (c'z+d'). \end{aligned}$$

(40.2.6)

Therefore, if f is a map with $f(\gamma z)=\jmath (\gamma ;z)^k f(z)$ and $f(\gamma ' z)=\jmath (\gamma ';z)^k f(z)$, then

$$\begin{aligned} \begin{aligned} f(\gamma (\gamma ' z))&=\jmath (\gamma ;\gamma ' z)^k f(\gamma 'z)=\jmath (\gamma ;\gamma ' z)^k \jmath (\gamma ';z)^k f(z) \\&= \jmath (\gamma \gamma ';z)^k f(z). \end{aligned} \end{aligned}$$

(40.2.7)

Since ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ is generated by S, T, it follows from (40.2.7) that a map f is weight k invariant for ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ if and only if both equalities

$$\begin{aligned} \begin{aligned} f(z+1)&= f(z) \\ f(-1/z)&= z^{k} f(z) \end{aligned} \end{aligned}$$

(40.2.8)

hold for all ${z \in \boldsymbol{\mathsf {H}}^2}$.

40.2.9

Since

$$\begin{aligned} \frac{\mathrm {d}{(\gamma z)}}{\mathrm {d}{z}}=\frac{1}{(cz+d)^2} \end{aligned}$$

(40.2.10)

the weight k invariance (40.2.2) of a map f can be rewritten

$$\begin{aligned} f(\gamma z)\,\mathrm {d}{(\gamma z)}^{\otimes k/2} = f(z)\,\mathrm {d}{z}^{\otimes k/2} \end{aligned}$$

(40.2.11)

so equivalently, the differential $f(z)\,\mathrm {d}{z}^{\otimes k/2}$ is (straight up) invariant under $\Gamma$.

40.2.12

Let ${f:\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C }$ be a meromorphic map that is weight k invariant under a Fuchsian group $\Gamma \ni \begin{pmatrix} 1 &{} 1 \\ 0 &{} 1 \end{pmatrix}$. Then $f(z+1)=f(z)$. If f admits a Fourier series expansion in $q=\exp (2\pi iz)$ of the form

$$\begin{aligned} f(z) = \sum _{n=-\infty }^{\infty } a_n q^n \in \mathbb C ((q)) \end{aligned}$$

(40.2.13)

with $a_n \in \mathbb C$ and $a_n=0$ for all but finitely many $n<0$, then we say that f is meromorphic at $\infty$; if further $a_n=0$ for $n<0$, we say f is holomorphic at $\infty$.

More generally, let $\Gamma \le {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ be a subgroup of finite index. For $\gamma \in {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, we define

$$\begin{aligned} f[\gamma ]_k(z) :=\jmath (\gamma ;z)^{-k} f(\gamma z). \end{aligned}$$

(40.2.14)

Then $f[\gamma ]_k(z)$ is weight k invariant under the group $\gamma ^{-1} \Gamma \gamma$. We say that f is meromorphic at the cusps if for every $\gamma \in {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, the function $f[\gamma ]_k$ is meromorphic at $\infty$, in the above sense. Since f is weight k invariant, to check if f is meromorphic at the cusps, it suffices to take representatives of the finite set of cosets $\Gamma \backslash {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$. (The name cusp comes from the geometric description at $\infty$ coming from the parabolic stabilizer group, recalling Definition 33.4.5.)

Finally, we say that f is holomorphic at the cusps if $f[\gamma ]_k(z)$ is holomorphic at $\infty$ for all $\gamma \in {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, and vanishes at the cusps if $f[\gamma ]_k(\infty )=0$ for all $\gamma$.

Definition 40.2.15

Let $k \in 2\mathbb Z$ and let $\Gamma \le {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ be a subgroup of finite index. A meromorphic modular form of weight k is a meromorphic map ${f:\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C }$ that is weight k invariant under $\Gamma$ and meromorphic at the cusps. A meromorphic modular function is a meromorphic modular form of weight 0.

A (holomorphic) modular form of weight k is a holomorphic map ${f :\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C }$ that is weight k invariant under $\Gamma$ and holomorphic at the cusps. A cusp form of weight k  is a holomorphic modular form of weight k that vanishes at the cusps.

Let $M_k(\Gamma )$ be the $\mathbb C$-vector space of modular forms of weight k for $\Gamma$, and let $S_k(\Gamma ) \subseteq M_k(\Gamma )$ be the subspace of cusp forms.

Lemma 40.2.16

The Eisenstein series $G_k(z)$ is a holomorphic modular form of weight $k \in 2\mathbb Z _{\ge 2}$ for ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ with Fourier expansion

$$\begin{aligned} G_k(z)=2\zeta (k) + 2\frac{(2\pi i)^k}{(k-1)!} \sum _{n=1}^{\infty } \sigma _{k-1}(n)q^n \end{aligned}$$

(40.2.17)

where

$$\begin{aligned} \zeta (k)=\sum _{n=1}^{\infty } \frac{1}{n^k} \end{aligned}$$

and

$$\begin{aligned} \sigma _{k-1}(n)=\sum _{\begin{array}{c} d \mid n \\ d>0 \end{array}} d^{k-1}. \end{aligned}$$

Proof. We start with the formula

$$\begin{aligned} \pi \cot (\pi z)=\sum _{m=-\infty }^{\infty } \frac{1}{z+m} = \lim _{M \rightarrow \infty } \sum _{m=-M}^{M} \frac{1}{z+m} \end{aligned}$$

(40.2.18)

(Exercise 40.2); with $q=\exp (2\pi iz)$,

$$\begin{aligned} \cot (\pi z)=\frac{\cos (\pi z)}{\sin (\pi z)}=i\frac{q+1}{q-1}=i\left( 1+\frac{2}{q-1}\right) \end{aligned}$$

and so we obtain the Fourier expansion

$$\begin{aligned} \pi \cot (\pi z)=\pi i - 2\pi i\sum _{n=0}^{\infty } q^n. \end{aligned}$$

(40.2.19)

Equating (40.2.18)–(40.2.19) and differentiating $k-1$ times, we find that

$$\begin{aligned} \sum _{m=-\infty }^{\infty } \frac{1}{(m+z)^k} = \frac{1}{(k-1)!}(2\pi i)^k \sum _{n=1}^{\infty } n^{k-1} q^n \end{aligned}$$

(40.2.20)

(since k is even). Thus

$$\begin{aligned} G_k(z)= \sum _{(m,n) \ne (0,0)} \frac{1}{(m+nz)^k} = 2\zeta (k) + 2\sum _{n=1}^{\infty } \sum _{m=-\infty }^{\infty } \frac{1}{(m+nz)^k} \end{aligned}$$

so replacing $n \leftarrow a$ and then substituting $z \leftarrow nz$ in (40.2.20), summing over n we obtain

$$\begin{aligned} \begin{aligned} G_k(z)&= 2\zeta (k)+2\frac{(2\pi i)^k}{(k-1)!} \sum _{n=1}^{\infty } \sum _{a=1}^{\infty } a^{k-1} q^{an} \\&= 2\zeta (k) + 2\frac{(2\pi i)^k}{(k-1)!} \sum _{n=1}^{\infty } \Bigl (\sum _{d \mid n} d^{k-1}\Bigr ) q^n \end{aligned} \end{aligned}$$

(40.2.21)

grouping together terms in the second step. The fact that $G_k$ is holomorphic at $\infty$ then follows by definition. $\square$

40.2.22

We accordingly define the normalized Eisenstein series by

$$\begin{aligned} E_k(z)=\frac{1}{2\zeta (k)}G_k(z) \end{aligned}$$

(see also 40.1.19). We have

$$\begin{aligned} E_k(z) = 1 - \frac{2k}{B_k}\sum _{n=1}^{\infty } \sigma _{k-1}(n)q^n \end{aligned}$$

(40.2.23)

where $B_k \in \mathbb Q ^\times$ are the Taylor coefficients of

$$\begin{aligned} \frac{x}{e^x-1}=\sum _{k=0}^{\infty } B_k \frac{x^k}{k!} = 1 - \frac{1}{2}x + \frac{1}{6}\frac{x^2}{2!}-\frac{1}{30}\frac{x^4}{4!}+\frac{1}{42}\frac{x^6}{6!} + \ldots \end{aligned}$$

(Exercise 40.3): the numbers $B_k \in \mathbb Q$ (with $B_k \ne 0$ for $k \in 2\mathbb Z _{\ge 0}$) are Bernoulli numbers. Expanding, we find

$$\begin{aligned} E_4(z)&= 1+240q+2160q^2+6720q^3+17520q^4+\ldots \\ E_6(z)&= 1-504q-16632q^2-122976q^3-532728q^4-\ldots . \end{aligned}$$

Remark 40.2.24. The notion of Eisenstein series extends in a natural way to the Bianchi groups ${{\,\mathrm{PSL}\,}}_2(\mathbb Z _F)$ where F is an imaginary quadratic field: see Elstrodt–Grunewald–Mennicke [EGM98, Chapter 3].

3 $\triangleright$ Classical modular forms

In this section, we study modular forms for ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$; to ease notation, we abbreviate $\Gamma :={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$.

40.3.1

Let f be a meromorphic modular form of weight k for $\Gamma ={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$. If $k=0$, so f is a modular function, then f descends to a meromorphic function on ${Y :=\Gamma \backslash \boldsymbol{\mathsf {H}}^2}$. Although this is not true for (nonzero) forms of weight $k \ne 0$, the order of zero or pole ${{\,\mathrm{ord}\,}}_{z}(f)$ is well defined on the orbit $\Gamma z$ by weight k invariance (40.2.2). With the Fourier expansion (40.2.13), we define

$$\begin{aligned} {{\,\mathrm{ord}\,}}_\infty (f) :={{\,\mathrm{ord}\,}}_q\Bigl (\sum _n a_n q^n\Bigr ) = \min (\{n : a_n \ne 0\}). \end{aligned}$$

The form f has only finitely many zeros or poles in Y, i.e., only finitely many $\Gamma$-orbits of zeros or poles: since f is meromorphic at $\infty$, there exists $\epsilon >0$ such that f has no zero or pole with $0<|q|<\epsilon$, so with

$$\begin{aligned} {{\,\mathrm{Im}\,}}z > M=\frac{\log (1/\epsilon )}{2\pi }; \end{aligned}$$

but the part of with ${{\,\mathrm{Im}\,}}z \le M$ is compact, and since f is meromorphic in ${\boldsymbol{\mathsf {H}}^2}$, it has only finitely many zeros or poles in this part as well.

40.3.2

In a similar way, the order of the stabilizer $e_z :=\#{{\,\mathrm{Stab}\,}}_{\Gamma }(z)$ is well defined on the orbit $\Gamma z$, since points in the same orbit have conjugate stabilizers. By 35.1.14,

$$\begin{aligned} \begin{aligned} e_z = {\left\{ \begin{array}{ll} 3, &{} \text { if}\, \Gamma z = \Gamma \omega ; \\ 2, &{} \text { if}\, \Gamma z = \Gamma i; \\ 1, &{} \text { otherwise.} \end{array}\right. } \end{aligned} \end{aligned}$$

(40.3.3)

Proposition 40.3.4

Let ${f :\boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C }$ be a meromorphic modular form of weight k for $\Gamma ={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, not identically zero. Then

$$\begin{aligned} {{{\,\mathrm{ord}\,}}_\infty (f)+ \sum _{\Gamma z \in \Gamma \backslash \boldsymbol{\mathsf {H}}^2} \frac{1}{e_z}{{\,\mathrm{ord}\,}}_z(f) = \frac{k}{12}} \end{aligned}$$

(40.3.5)

where $e_z :=\#{{\,\mathrm{Stab}\,}}_{\Gamma }(z)$.

The sum (40.3.5) has only finitely many terms, by 40.3.1, and the stabilizers are given in 40.3.2.

Proof. See Serre [Ser73, §3, Theorem 3]: the proof consists of performing a contour integration $\displaystyle {\frac{1}{2\pi i}\frac{\mathrm {d}{f}}{f}}$ on the boundary of . Alternatively, this statement can be seen as a manifestation of the Riemann–Roch theorem: see Diamond–Shurman [DS2005, §3.5]. $\square$

40.3.6

We have $E_4(STz)=(z+1)^4 E_4(z)$, so since $(ST)(\omega )=\omega$,

$$\begin{aligned} E_4(\omega )=(\omega +1)^4 E_4(\omega )=\omega ^2 G_4(\omega ) \end{aligned}$$

so $E_4(\omega )=0$. Since $E_4$ is holomorphic in ${\boldsymbol{\mathsf {H}}^2}$, we have ${{\,\mathrm{ord}\,}}_z(E_4) \in \mathbb Z _{\ge 0}$ for all ${z \in \boldsymbol{\mathsf {H}}^2}$, and thus by Proposition 40.3.4, we must have that $E_4(z)$ has no other zeros in . Similarly,

$$\begin{aligned} E_6(i)=E_6(Si)=i^6 E_6(i)=-E_6(i) \end{aligned}$$

so $E_6(i)=0$, and $E_6(z)$ has no other zeros.

For the same reason, the function

$$\begin{aligned} \Delta (z)=\frac{E_4(z)^3-E_6(z)^2}{1728} = q - 24q^2 + 252q^3 - 1472q^4 + \ldots \end{aligned}$$

(40.3.7)

is a modular form of weight 12 with no zeros in ${\boldsymbol{\mathsf {H}}^2}$ with ${{\,\mathrm{ord}\,}}_\infty (\Delta )=1$.

We give two applications of Proposition 40.3.4. First, we obtain the identification promised in (40.1.18).

Theorem 40.3.8

The function

$$\begin{aligned} j(z)=\frac{E_4(z)^3}{\Delta (z)}=\frac{1}{q}+744+196884q+21493760q^2+\ldots \end{aligned}$$

(40.3.9)

is a meromorphic modular function for ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, holomorphic in ${\boldsymbol{\mathsf {H}}^2}$, defining a bijection

$$\begin{aligned}{Y=\Gamma \backslash \boldsymbol{\mathsf {H}}^2 \rightarrow \mathbb C .}\end{aligned}$$

Proof. The function j is weight 0 invariant under $\Gamma$ as the ratio of two forms that are weight 12 invariant. Since $E_4$ is holomorphic in ${\boldsymbol{\mathsf {H}}^2}$, and $\Delta$ is holomorphic and has no zeros in ${\boldsymbol{\mathsf {H}}^2}$, the ratio is holomorphic in ${\boldsymbol{\mathsf {H}}^2}$ and j(z) has a simple pole at $z=\infty$, corresponding to a simple zero of $\Delta$ at $z=\infty$. From 40.3.6, we have $j(i)=1728$, and $j(z)-1728$ has a double zero at $z=i$, and j(z) has a triple zero at $z=\omega$.

To conclude that j is bijective, we show that $j(z)-c$ has a unique zero $\Gamma z \in Y$. If $c \ne 0,1728$, this follows immediately from Proposition 40.3.4; if $c=0,1728$, the results follow for the same reason from the multiplicity of the zero. $\square$

Remark 40.3.10. The definition of j(z) is now standard, but involves some choices. In some circumstances (including the generalization to abelian surfaces, see 43.5.7), it is more convenient to remember the values of the Eisenstein series themselves, as follows. To $z \in \mathcal H$, we associate the pair $(E_4(z),E_6(z)) \in \mathbb C ^2$; if $\gamma \in \Gamma$ and $z'=\gamma z$, then

$$\begin{aligned} (E_4(z'),E_6(z'))=(\delta ^4 E_4(z), \delta ^6 E_6(z)) \end{aligned}$$

where $\delta =\jmath (\gamma ;z) \in \mathbb C ^\times$. We therefore define the weighted projective (4, 6)-space by

$$\begin{aligned} \mathbb P (4,6)(\mathbb C ) :=(\mathbb C ^2 \smallsetminus \{(0,0)\})/\sim \end{aligned}$$

where

$$\begin{aligned} (E_4,E_6) \sim (\delta ^4 E_4, \delta ^6 E_6) \end{aligned}$$

for $\delta \in \mathbb C ^\times$. We write equivalence classes $(E_4:E_6) \in \mathbb P (4,6)(\mathbb C )$. The map

$$\begin{aligned} j:\mathbb P (4,6)(\mathbb C )&\rightarrow \mathbb P ^1(\mathbb C ) \\ (E_4:E_6)&\mapsto j(E_4:E_6)=\frac{1728E_4^3}{E_4^3-E_6^2} \end{aligned}$$

is well-defined and bijective—see Silverman [Sil2009, Proposition III.1.4(b)].

To conclude, we give a complete description of the ring of (holomorphic) modular forms. By 40.2.3, the $\mathbb C$-vector space

$$\begin{aligned} M(\Gamma )=\bigoplus _{k \in 2\mathbb Z } M_k(\Gamma ) \end{aligned}$$

under multiplication has the structure of a (graded) $\mathbb C$-algebra; we call $M(\Gamma )$ the ring of modular forms for $\Gamma$.

Theorem 40.3.11

We have $M(\Gamma )=\mathbb C [E_4,E_6]$, i.e., every modular form for $\Gamma ={{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ can be written as a polynomial in $E_4,E_6$.

In particular, $M_k(\Gamma )=\{0\}$ for $k<0$.

Proof. We have $M(\Gamma ) \supseteq \mathbb C [E_4,E_6]$, so we prove the reverse inclusion. We refer to Proposition 40.3.4, and ask for solutions $a_1,a_2,a_3 \in \mathbb Z _{\ge 0}$ to $a_1+a_2/2+a_3/3=k/12$. When $k<0$, there are no such solutions; when $k=0,2,4,6,8,10$, there is a unique solution, and we find that $M_k(\Gamma )$ is spanned by $1,0,E_4,E_6,E_4^2,E_4E_6$, respectively.

For all even $k\ge 4$, there exist $a,b \in \mathbb Z _{\ge 0}$ such that $4a+6b=k$ (if $k \ge 10$ and $k \equiv 2 \pmod {4}$, then $k-6 \ge 4$ and $4 \mid k$), so $E_4^aE_6^b \in M_k(\Gamma )$ and $(E_4^aE_6^b)(\infty )=1$ (by 40.2.22). Let $S_k(\Gamma ) \subseteq M_k(\Gamma )$ be the subspace of forms that vanish at $\infty$. Then $M_k(\Gamma )=\mathbb C E_4^aE_6^b \oplus S_k(\Gamma )$ by linear algebra, and by the previous paragraph, $S_k(\Gamma )=\{0\}$ for $k \le 10$.

We claim that multiplication by $\Delta$ furnishes an isomorphism ${M_k(\Gamma ) \displaystyle \mathop {\rightarrow }^{\sim } S_{k+12}(\Gamma )}$ of $\mathbb C$-vector spaces for all k: division by $\Delta$ defines an inverse because $\Delta$ has a simple zero at $\infty$ by 40.3.6 and no zeros in ${\boldsymbol{\mathsf {H}}^2}$. The result now follows by induction on $k \ge 0$. $\square$

More generally, one can study modular forms for congruence subgroups (section 35.4) of ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ in an explicit way, as the following example illustrates.

Example 40.3.12

At the end of section 35.4, we examined a fundamental domain for the group $\Gamma (2)$, defined by (35.4.8). As with X(1), the homeomorphism (35.4.11) can be given by a holomorphic map

$$\begin{aligned}{\lambda :X(2) \displaystyle \mathop {\rightarrow }^{\sim } \mathbb P ^1(\mathbb C )}\end{aligned}$$

obtained from Eisenstein series for $\Gamma (2)$, analogous to j(z). The map $\lambda$ satisfies $\lambda (\gamma z)=\lambda (z)$ for all $\gamma \in \Gamma (2)$ and in particular is invariant under $z \mapsto z+2$. One can compute its Fourier expansion in terms of $q^{1/2}=e^{\pi i z}$ as:

$$\begin{aligned} \lambda (z)=16q^{1/2} - 128q + 704q^{3/2} - 3072q^2 + 11488q^{5/2} - 38400q^3 + \dots . \end{aligned}$$

(40.3.13)

Since j(z) induces a degree $6=[\Gamma (2):\Gamma (1)]$ map $X(2) \rightarrow X(1)$, we find the relationship

$$\begin{aligned} j = 256\frac{(\lambda ^2-\lambda +1)^3}{\lambda ^2(\lambda -1)^2}. \end{aligned}$$

(40.3.14)

From (40.3.14) (and the first term), the complete series expansion (40.3.13) can be obtained recursively.

As a uniformizer for a congruence subgroup of ${{\,\mathrm{PSL}\,}}_2(\mathbb Z )$, the function $\lambda (z)$ has a moduli interpretation (cf. 40.1.11): there is a family of elliptic curves over X(2) equipped with extra structure. Specifically, given $\lambda \in \mathbb P ^1(\mathbb C ) \setminus \{0,1,\infty \}$, the corresponding elliptic curve with extra structure is given by the Legendre curve

$$\begin{aligned} E_{\lambda } :y^2 = x(x-1)(x-\lambda ), \end{aligned}$$

equipped with the isomorphism ${(\mathbb Z / 2 \mathbb Z )^2 \displaystyle \mathop {\rightarrow }^{\sim } E[2]}$ determined by sending the standard generators to the 2-torsion points (0, 0) and (1, 0). The map j is the map that forgets this additional torsion structure on a Legendre curve and remembers only isomorphism class.

4 Theta series

An important class of classical modular forms arise via theta series, which count the number of representations of an integer by a positive definite quadratic form. We present only a small fraction of the general theory here.

Let $Q:\mathbb Z ^m \rightarrow \mathbb Z$ be a positive definite integral quadratic form in m variables and suppose that $m=2k$ is a positive even integer. We define the theta series of Q by

$$\begin{aligned} \Theta _Q : \boldsymbol{\mathsf {H}}^2&\rightarrow \mathbb C \nonumber \\ \Theta _Q(z)&= \sum _{x \in \mathbb Z ^m} e^{2\pi i Q(x) z} = \sum _{n=0}^{\infty } r_Q(n) q^n \end{aligned}$$

(40.4.1)

where $q=e^{2\pi iz}$ and

$$\begin{aligned} r_Q(n) = \#\{x \in \mathbb Z ^m : Q(x)=n\} < \infty \end{aligned}$$

counts the number of lattice points on the sphere of radius $\sqrt{n}$.

Lemma 40.4.2

$\Theta _Q(z)$ is a holomorphic function.

Proof. Since Q is positive definite, there exists $c \in \mathbb R _{>0}$ such that

$$\begin{aligned} Q(x) \ge c(x_1^2+\dots +x_m^2). \end{aligned}$$

Thus $r_Q(n)=O(n^{k})$, and the series $\Theta _q(z)$ is majorized by (a constant multiple of) $\sum _{n=1}^{\infty } n^k q^n$, so converges to a holomorphic function. $\square$

Let [T] be the Gram matrix for the symmetric bilinear form associated to Q; then $[T] \in {{\,\mathrm{M}\,}}_m(\mathbb Z )$ is an integral symmetric matrix with even diagonal entries. Let $d=\det Q=\det [T] \in \mathbb Z$. Then $dA^{-1} \in {{\,\mathrm{M}\,}}_m(\mathbb Z )$ is the adjugate matrix: it is again symmetric.

Definition 40.4.3

The least positive integer $N \in \mathbb Z _{>0}$ such that $NA^{-1}$ is integral with even diagonal entries is called the level of Q.

We recall the definition of the congruence subgroups 35.4.5.

Theorem 40.4.4

The theta series $\Theta _Q(z)$ is a modular form of weight k for $\Gamma _1(N)$.

Proof. Unfortunately, in this generality the proof would take us too far afield. Fundamentally, the transformation formula for $\Theta _Q$ follows from Poisson summation and careful computations: see Eichler [Eic73, §I.3, Proposition 2], Miyake [Miy2006, Corollary 4.9.5], or Ogg [Ogg69, Chapter VI]. $\square$

40.4.5

We can be a bit more specific about the transformation group for $\Theta _Q(z)$ as follows. To Q, we associate the character $\chi$ defined by $\chi (n)=\biggl (\displaystyle {\frac{(-1)^k \det Q}{n}}\biggr )$. Then

$$\begin{aligned} \Theta _Q(\gamma z) = \chi (d)(cz+d)^k \Theta _Q(z) \qquad \text {for all}\, \gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in \Gamma _0(N); \end{aligned}$$

accordingly, we say $\Theta _Q$ is a modular form of level N with character $\chi$.

5 Hecke operators

Significantly, the $\mathbb C$-vector space $M_k(\Gamma _0(N))$ of modular forms of weight k for $\Gamma _0(N)$ carries with it an action of commuting semisimple operators, called Hecke operators. These operators may be interpreted as averaging modular forms over sublattices of a fixed index; for efficiency, we work with a more explicit definition. For further reference, see e.g. Diamond–Shurman [DS2005, Chapter 5] or Miyake [Miy2006, §2.7, §4.5].

Throughout, let $N \in \mathbb Z _{\ge 1}$. Let

$$\begin{aligned} \mathcal {O}=\mathcal {O}_0(N) :=\left\{ \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {{\,\mathrm{M}\,}}_2(\mathbb Z ) : N \mid c\right\} \subseteq {{\,\mathrm{M}\,}}_2(\mathbb Z ) \end{aligned}$$

be the standard Eichler order of level N in ${{\,\mathrm{M}\,}}_2(\mathbb Q )$, so that $\Gamma = \Gamma _0(N)=\mathcal {O}^1/\{\pm 1\}$.

Let $n \in \mathbb Z _{\ge 1}$ with $\gcd (n,N)=1$. We consider the set of matrices

$$\begin{aligned} \mathcal {O}_n = \{\alpha \in \mathcal {O}: \det (\alpha )=n \}. \end{aligned}$$

(40.5.1)

Visibly, there is a left (and right) action of $\mathcal {O}^1$ on $\mathcal {O}_n$ by multiplication.

Lemma 40.5.2

A system of representatives of $\mathcal {O}^1 \backslash \mathcal {O}_n$ is given by the set of matrices of the form $\begin{pmatrix} a &{} b \\ 0 &{} d \end{pmatrix}$ with $ad=n$, $a>0$, and $0 \le b < d$.

Proof. The lemma follows as in Lemma 26.4.1(b) using the theory of elementary divisors, but applying row operations (acting on the left). $\square$

Example 40.5.3

When $p \not \mid N$ is prime, the set $\mathcal {O}^1 \backslash \mathcal {O}_p$ is represented by the $p+1$ matrices

$$ \begin{pmatrix} p &{} 0 \\ 0 &{} 1 \end{pmatrix}, \begin{pmatrix} 1 &{} 0 \\ 0 &{} p \end{pmatrix}, \begin{pmatrix} 1 &{} 1 \\ 0 &{} p \end{pmatrix}, \cdots , \begin{pmatrix} 1 &{} p-1 \\ 0 &{} p \end{pmatrix}. $$

Definition 40.5.4

For $n \in \mathbb Z _{\ge 1}$ with $\gcd (n,N)=1$, we define the Hecke operator

$$\begin{aligned} \begin{aligned} T(n) : M_k(\Gamma )&\rightarrow M_k(\Gamma ) \\ (T(n) f)(z)&= n^{k/2-1} \sum _{\mathcal {O}^1 \alpha \in \mathcal {O}^1 \backslash \mathcal {O}_n}\jmath (\alpha ;z)^{-k} f(\alpha z). \end{aligned} \end{aligned}$$

(40.5.5)

By the condition of automorphy $f(\gamma z)=\jmath (\gamma ;z)^k f(z)$ and the cocycle relation (40.2.5), the Hecke operators are well-defined and preserve weight k invariance.

40.5.6

By Lemma 40.5.2, we have more explicitly

$$\begin{aligned} (T(n) f)(z) = n^{k-1} \sum _{\begin{array}{c} ad=n \\ a>0 \end{array}} \frac{1}{d^k} \sum _{b=0}^{d-1} f\left( \frac{az+b}{d}\right) . \end{aligned}$$

(40.5.7)

Accordingly, if $f(z)=\sum _{n=0}^{\infty } a_n q^n$, then $(T(n) f)(z) = \sum _{m=0}^{\infty } b_m q^m$ where

$$\begin{aligned} b_m = \sum _{\begin{array}{c} d \mid \gcd (m,n) \\ d > 0 \end{array}} d^{k-1} a_{mn/d^2} \end{aligned}$$

(40.5.8)

so in particular for $n=p$ prime we have

$$\begin{aligned} b_m = a_{pm} + {\left\{ \begin{array}{ll} p^{k-1} a_{m/p}, &{} \text { if}\, p \mid m; \\ 0, &{} \text { if}\, p \not \mid m. \end{array}\right. } \end{aligned}$$

(40.5.9)

Applying (40.5.9), we see that if $f \in M_k(\Gamma )$ has $f(\infty )=0$ (equivalently, $a_0=0$), then the same is true for T(n)f. Repeating this for the functions $f[\gamma ]_k$ with $\gamma \in {{\,\mathrm{PSL}\,}}_2(\mathbb Z )$ (as in (40.2.14)), we conclude that the operators T(n) act on the space of cusp forms $S_k(\Gamma ) \subset M_k(\Gamma )$.

Proposition 40.5.10

For $m,n \in \mathbb Z _{\ge 1}$, we have

$$\begin{aligned} T(m) T(n) = \sum _{d \mid \gcd (m,n)} d^{k-1} T(mn/d^2). \end{aligned}$$

(40.5.11)

In particular, if $\gcd (m,n)=1$ then

$$\begin{aligned} T(m) T(n) = T(n) T(m) = T(mn) \end{aligned}$$

and if p is prime and $r \ge 1$ then

$$\begin{aligned} T(p) T(p^r) = T(p^{r+1}) + p^{k-1} T(p^{r-1}). \end{aligned}$$

Proof. These statements follow directly from the expansion (40.5.8). $\square$

Theorem 40.5.12

The Hecke operators T(n) for $\gcd (n,N)=1$ on $M_k(\Gamma _0(N))$ generate a commutative, semisimple $\mathbb Z$-algebra.

Proof. See e.g. Diamond–Shurman [DS2005, Theorem 5.5.4]. Briefly, we treat Eisenstein series separately and work with cusp forms $S_k(\Gamma _0(N))$. To prove that the operators are semisimple, we would need to show that the Petersson inner product

$$\begin{aligned} {\langle f,g \rangle = \int _{\Gamma \backslash \boldsymbol{\mathsf {H}}^2} f(z)\overline{g(z)} y^k \,\mathrm {d}{\mu (z)}} \end{aligned}$$

is well-defined, positive, and nondegenerate, and then verify that the operators are normal with respect to this inner product. $\square$

By Theorem 40.5.12 and linear algebra, there exists a $\mathbb C$-basis $f_i(z)$ of $M_k(\Gamma _0(N))$ consisting of simultaneous eigenfunctions for all T(n).

Exercises

$\triangleright$ 1.:

Let $f :U \rightarrow \mathbb C$ be a function that is meromorphic in an open neighborhood $U \supseteq \mathbb C$ with $z \in U$, and let C be a contour along an arc of a circle of radius $\epsilon >0$ centered at z contained in U with total angle $\theta$. Show that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \int _C \frac{\mathrm {d}{f}}{f} = \theta i {{\,\mathrm{ord}\,}}_z(f). \end{aligned}$$

$\triangleright$ 2.:

Prove the formula

$$\begin{aligned} \pi \cot (\pi z)=\sum _{m=-\infty }^{\infty } \frac{1}{z+m} \end{aligned}$$

for $z \in \mathbb C$. [Hint: the difference h(z) of the left- and right-hand sides is bounded away from $\mathbb Z$, invariant under $z \mapsto T(z)=z+1$, and in a neighborhood of 0 is holomorphic (both sides have principal part 1/z) so bounded; thus h(z) is bounded in $\mathbb C$ and hence constant.]

$\triangleright$ 3.:

In this exercise, we give Euler’s evaluation of $\zeta (k)$ in terms of Bernoulli numbers. As in Exercise 36.12, define the series

$$\begin{aligned} \frac{x}{e^x-1}=\sum _{k=0}^{\infty } B_k \frac{x^k}{k!}=1-\frac{x}{2}+\frac{1}{6}\frac{x^2}{2!}-\frac{1}{30}\frac{x^4}{4!}+\ldots \in \mathbb Q [[x]]. \end{aligned}$$

(40.5.13)

(a):

Plug in $x=2iz$ into (40.5.13) to obtain

$$\begin{aligned} z \cot z = 1 + \sum _{k=2}^{\infty } B_{k} \frac{(2iz)^{k}}{k!}. \end{aligned}$$

(b):

Take the logarithmic derivative of

$$\begin{aligned} \sin z = z \prod _{n=1}^{\infty } \left( 1-\frac{z^2}{n^2\pi ^2}\right) \end{aligned}$$

to show

$$\begin{aligned} z \cot z = 1 - 2\sum _{\begin{array}{c} k=2 \\ k \text { even} \end{array}}^{\infty }\sum _{n=1}^{\infty } \left( \frac{z}{n\pi }\right) ^{k}. \end{aligned}$$

(c):

Conclude that

$$\begin{aligned} \zeta (k)=\sum _{n=1}^{\infty } \frac{1}{n^k} = -\frac{1}{2}\frac{(2\pi i)^k}{k!} B_k \end{aligned}$$

for $k \in 2\mathbb Z _{\ge 1}$.

4.:

We defined Eisenstein series $G_k(z)$ for $k \ge 4$, and found $G_k(z) \in {{\,\mathrm{M}\,}}_k({{\,\mathrm{SL}\,}}_2(\mathbb Z ))$ are modular forms of weight k for ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$. The case $k=2$ is also important, though we must be a bit more careful in its analysis. Let

$$\begin{aligned} G_2(z) :=\sum _{c \in \mathbb Z } \sum _{\begin{array}{c} d \in \mathbb Z \\ (c,d) \ne (0,0) \end{array}} \frac{1}{(cz+d)^2}. \end{aligned}$$

(a):

Show that $G_2(z)$ converges (conditionally) and satisfies

$$\begin{aligned} G_2(z) = 2\zeta (2) - 8\pi ^2 \sum _{n=1}^{\infty } \sigma (n)q^n \end{aligned}$$

where $q=e^{2\pi iz}$.

(b):

Show that $G_2(z+1)=G_2(z)$ and

$$\begin{aligned} G_2\left( \frac{-1}{z}\right) = G_2(z) - \frac{2\pi i}{z}. \end{aligned}$$

[Hint: use a telescoping series and rearrange terms.]

(c):

Conclude that

$$\begin{aligned} G_2(\gamma z) = G_2(z) - \frac{2\pi i c}{cz+d} \quad \text { for all}\, \gamma =\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix}. \end{aligned}$$

[Hint: use the cocycle relation.]

(d):

Define

$$\begin{aligned} G_2^*(z) = G_2(z) - \frac{\pi }{{{\,\mathrm{Im}\,}}z}. \end{aligned}$$

Show that $G_2^*(z)$ is weight 2 invariant under ${{\,\mathrm{SL}\,}}_2(\mathbb Z )$.

5.:

In this exercise, we give a proof using modular forms of the formula for the number of ways of representing an integer as the sum of four squares, due to Jacobi. Consider the function

$$\begin{aligned} \vartheta (z) :=\sum _{n=-\infty }^{\infty } q^{n^2} \end{aligned}$$

where $q :=e^{2\pi iz}$ and ${z \in \boldsymbol{\mathsf {H}}^2}$. [The function $\vartheta (z)$ is a theta series for the univariate quadratic form $x \mapsto x^2$.] Let $r_4(n)$ be the number of ways of representing $n \ge 0$ as the sum of 4 squares.

(a):

Show that

$$\begin{aligned} \Theta _Q(z) :=\vartheta (z)^4 = 1+\sum _{n=1}^{\infty } r_4(n)q^n = 1+8q+12q^2+\ldots . \end{aligned}$$

(b):

Show that $\Theta _Q(q) \in {{\,\mathrm{M}\,}}_2(\Gamma _0(4))$ is a modular form of weight 2 on $\Gamma _0(4)$.

(c):

Show that $\dim _\mathbb C M_2(\Gamma _0(4))=2$ and $\dim _\mathbb C S_2(\Gamma _0(4)) = 0$.

(d):

Let

$$\begin{aligned} G_{2,2}(z)&= G_2(z) - 2G_2(2z) \\ G_{2,4}(z)&= G_2(z) - 4G_2(4z). \end{aligned}$$

Show that $G_{2,2},G_{2,4}$ are a basis for ${{\,\mathrm{M}\,}}_2(\Gamma _0(4))$. [Hint: use Exercise 40.4(c).]

(e):

Show that

$$\begin{aligned} E_{2,2}(z) :=-\frac{3}{\pi ^2} G_{2,2}(z)&= 1+24 \sum _{n=1}^{\infty } \sigma ^{(2)}(n) q^n \\ E_{2,4}(z) :=-\frac{1}{\pi ^2} G_{2,4}(z)&= 1+8 \sum _{n=1}^{\infty } \sigma ^{(4)}(n) q^n \end{aligned}$$

where

$$\begin{aligned} \sigma ^{(m)}(n) = \sum _{m \not \mid d \mid n} d. \end{aligned}$$

(f):

Matching the first few coefficients, show that

$$\begin{aligned} \Theta _Q(z)=E_{2,4}(z). \end{aligned}$$

Conclude that $r_4(n) = 8\sigma ^{(4)}(n)$ for all $n>0$.