Lecture 13: Modular forms

$$ \newcommand{\ord}{\operatorname{ord}} \newcommand{\SL}{\mathrm{SL}} \def\mat#1#2#3#4{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array} \right)} \newcommand{\fh}{\mathfrak{h}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bZ}{\mathbf{Z}} \newcommand{\bA}{\mathbf{A}} \newcommand{\cL}{\mathcal{L}} \let\wt\widetilde \newcommand{\bP}{\mathbf{P}} \newcommand{\rH}{\mathrm{H}} \newcommand{\bF}{\mathbf{F}} $$

This lecture introduced modular forms and Hecke operators. I started by introducing modular forms of level 1, and gave several interpretations of them, e.g., as sections of line bundles on the modular curve, or as functions of lattices. I then talked about modular forms of higher level. Finally, I introduced Hecke operators and their action on modular forms, and proved that they commute.

Modular forms of level 1

Definition

In the previous lecture, we established bijections

$$ \fh/\Gamma(1) \to \{ \textrm{lattices in $\bC$} \} / \textrm{homothety} \to \{ \textrm{isom. cl. of elliptic curves/$\bC$} \} = Y(1) $$

The first map takes a point $z \in \fh$ to the lattice $\Lambda_z=\langle 1, z \rangle$, while the second map takes a lattice $\Lambda$ to the elliptic curve $\bC/\Lambda$. Recall that

$$ \Lambda_{\gamma(z)} = (cz+d)^{-1} \Lambda_z, \qquad \gamma = \mat{a}{b}{c}{d} \in \Gamma(1), $$

and $\Gamma(1)$ is just notation for $\SL_2(\bZ)$. Furthermore, this space is identified with $\bA^1$ via the $j$-invariant.

A modular function is a meromorphic function on this space which is meromorphic at infinity. The $j$-invariant is an example, as is any rational function of the $j$-invariant. Since $Y(1)=\bA^1$, every modular function is a rational function of the $j$-invariant.

A modular form is a function on the space of lattices which is homogeneous under homothety, but not necessarily of degree 0. Precisely, a modular form of weight $k$ is a function $f$ on the set of lattices in $\bC$ satisfying the following conditions:

Clearly, a modular form of weight $k$ is the same thing as a holomorphic function $f$ on $\fh^*$ which satisfies $f(\gamma z)=(cz+d)^k f(z)$ for all $\gamma \in \Gamma(1)$, where we use our usual notation. A modular form is a cusp form if it vanishes at the cusps.

Remark. The above condition applied to $\gamma=-1$ shows that $f(z)=(-1)^k f(z)$. So if $k$ is odd then $f=0$. Therefore, there are only interesting modular forms of even weight.

Modular interpretation

A modular form does not have a well-defined value on an elliptic curve -- in other words, it is not a function on $Y(1)$. Rather, it is a section of a line bundle. This line bundle has a nice modular interpretation.

Let $\cL$ be the space of lattices in $\bC$. There is a natural family of elliptic curves $\wt{\pi} \colon E \to \cL$: the fiber above $\Lambda \in \cL$ is $E_{\Lambda}=\bC/\Lambda$. Let $w$ be a parameter on $\bC$, so that $dw$ spans the space of holomorphic 1-forms on $E_{\Lambda}$ for any $\Lambda$. Note that $dw$ is not invariant under homothety: indeed, $\alpha^*(dw)=\alpha dw$. However, if $f$ is a weight $k$ modular form then $f (dw)^k$ is invariant under homothety, as a section of the $k$-fold tensor power of $\wt{\pi}_*(\Omega^1_{E/\cL})$.

Let $\pi \colon E \to Y(1)$ be the universal elliptic curve over $Y(1)$. We must treat $Y(1)$ as a stack for this to work properly, something we'll briefly discuss in the future. Let $\omega=\pi_*(\Omega^1_{E/Y(1)})$. Then $\omega$ is a line bundle on $Y(1)$ whose fiber at an elliptic curve $E$ is $\Gamma(E, \Omega^1_{E/\bC})$. The bundle $\omega$ is called the Hodge bundle. The above discussion shows that a weight $k$ modular form defines a section of $\omega^{\otimes k}$. In fact, every section that satisfies an appropriate condition at $\infty$ defines a weight $k$ modular form.

It is possible to state the above description more concretely. Let $f$ be a weight $k$ modular form, let $E$ be an elliptic curve, and let $\omega$ be a non-zero 1-form on $E$. Writing $E=E_{\Lambda}$, the above discussion shows that $f(\Lambda) (dw)^k$ is a well-defined element of $\Gamma(E, \Omega^1)^{\otimes k}$. However, $\omega^k$ is also such an element, and so we can divide to get a well-defined number. In other words, $f$ defines a function $F$ from the set of pairs $(E, \omega)$ to $\bC$. The function $F$ has two important properties:

Any $F$ satisfying these two properties, and a holomorphicity condition that we do not state now, comes from a modular form $f$ of weight $k$.

Geometric interpretation

There is another useful way to think about modular forms, in terms of the geometry of the modular curve $X(1)$. Suppose $f$ is a modular form of weight $2k$ on $\fh$. Then $f(\gamma z)=(cz+d)^k f(z)$, by definition. A simple computation shows that $\gamma^*(dz)=(cz+d)^{-2} dz$. Thus $f(z) (dz)^k$ is invariant under $\Gamma$. Let $\pi \colon \fh^* \to X(1)$ be the quotient map. Then $f(z) (dz)^k=\pi^*(\omega)$ for some meromorphic section $\omega$ of $(\Omega^1)^{\otimes k})$ over $X(1)$. The local behavior of $f$ and $\omega$ are related as follows.

Proposition. Let $x \in \fh^*$ and let $y=\pi(x) \in X(1)$. Then

$$ \ord_y(\omega) = \begin{cases} \tfrac{1}{2} (\ord_x(f)-k) & \textrm{if $x=i$} \\ \tfrac{1}{3} (\ord_x(f)-2k) & \textrm{if $x=\rho$} \\ \ord_x(f)-k & \textrm{if $x=\infty$} \\ \ord_x(f) & \textrm{otherwise} \end{cases} $$

Proof. We just explain the $x=i$ case. Let $z$ be a uniformizing parameter of $\fh$ at $x$ and let $w$ be one on $X(1)$ at $y$. Then $\pi^*(w)=z^2$ and $\pi^*(dw)=zdz$ (up to higher order terms and constants). So if $\omega = w^n (dw)^k$ then $\pi^*(\omega)=z^{2n+k} dz$. Thus $\ord_x(f)=2n+k=2\ord_y(\omega)+k$. ◾

Corollary. The space of modular forms of weight $2k$ is isomorphic to the space of sections $\omega$ of $(\Omega^1)^{\otimes k}$ over $X(1)=\bP^1$ which are holomorphic away from $\pi(i)$, $\pi(\rho)$, and $\pi(\infty)$, and satisfy $\ord_{\pi(i)}(\omega) \ge -k/2$, $\ord_{\pi(\rho)}(\omega) \ge -2k/3$, $\ord_{\pi(\infty)}(\omega) \ge -k$. A similar statement is true for cusp forms, but where the last condition is changed to $\ord_{\pi(\infty)}(\omega) \ge 1-k$.

Corollary. The space of modular forms of weight $2k$ (for $k \gt 0$) has dimension $\lfloor k/6 \rfloor + \epsilon$, where $\epsilon$ is 1 if $k \ne 1 \pmod{6}$. The space of cusp forms has dimension one less.

Proof. Let $P=\pi(i)$, $Q=\pi(\rho)$, $\infty=\pi(\infty)$. Then the above corollary says the space of modular forms of weight $2k$ is identified with the space of sections of $(\Omega^1)^{\otimes k}(nP+mQ+k\infty)$, where $n=\lfloor k/2 \rfloor$ and $m=\lfloor 2k/3 \rfloor$. This bundle has degree $-2k+n+m+k=n+m-k$, and thus $n+m-k+1$ sections. It is elementary to show that this agrees with the stated formula. A modified argument applies to the cuspidal case. ◾

Example. There are no non-zero modular forms of weight 2. There is exactly one non-zero form, up to scalars, of weight 4, 6, 8, and 10. Then there are two of weight 12, one of which is cuspidal.

Fourier expansion

Any modular form $f$ on $\fh$ is invariant under the translation $z \mapsto z+1$, and can therefore be expanded in powers of $q=e^{2\pi i z}$. The expansion

$$ f(z) = \sum_{n \in \bZ} a_n q^n $$

is called the Fourier expansion or $q$-expansion of $f$. The condition that $f$ be holomorphic at $\infty$ amounts to $a_n=0$ for $n \lt 0$; given this, cuspidality is equivalent to $a_0=0$.

Examples

Given a lattice $\Lambda \subset \bC$ and an even integer $k \ge 4$, put

$$ G_k(\Lambda) = \sum_{\lambda \in \Lambda}' \frac{1}{\lambda^k}, $$

where the prime means to omit $\lambda=0$. Clearly, $G_k(\alpha \Lambda)=\alpha^{-k} G_k(\Lambda)$, and so $G_k$ has the right homogeneity property to be a modular form. Put $G_k(z)=G_k(\Lambda_z)$. Then

$$ G_k(z) = \sum_{n,m}' \frac{1}{(nz+m)^k}, $$

which shows that $G_k$ is a holomorphic function of $z$. Furthermore, as $z \to \infty$, only the terms with $n=0$ survive, and so $G_k(\infty)=2 \zeta(k)$, where $\zeta$ is the Riemann zeta function; in particular, $G_k$ is holomorphic (but non-zero) at $\infty$. Thus $G_k$ is a modular form of weight $k$. It is called the Eisenstein series of weight $k$. The modular form $E_k=(2\zeta(k))^{-1} G_k$ is called the normalized Eisenstein series of weight $k$. Its $q$-expansion is given by:

Proposition. We have

$$ E_k(z) = 1-\frac{4k}{B_k} \sum_{n \ge 1} \sigma_{k-1}(n) q^n $$

where $B_k$ is the Bernoulli number and $\sigma_{k-1}(n)$ is the sum of the $(k-1)$st powers of the divisors of $n$.

Let $\Delta=E_4^3-E_6^2$. Then $\Delta$ is a modular form of weight 12 whose constant term vanishes. Computing with the first two terms of the $q$-series of $E_2$ and $E_4$, one finds that $\Delta(z)=q+\cdots$, and so $\Delta$ is non-zero. It is therefore the unique (up to scaling) non-zero cusp form of weight 12. Its $q$-expansion is complicated, but it admits the following nice product formula (due to Jacobi):

Proposition. $\Delta(z)=q \prod_{n \ge 1} (1-q^n)$.

The modular forms $E_4$, $E_6$, and $\Delta$ admit nice modular interpretations. Recall that every elliptic curve over $\bC$ is isomorphic to one of the form

$$ y^2 = x^3+ax+b. $$

Call this curve $E_{a,b}$. Then $E_{a,b}$ is isomorphic to $E_{u^4a,u^6b}$, but there are no other isomorphisms. Let $\omega_{a,b}$ be the holomorphic 1-form on $E_{a,b}$ given by $y^{-1} dx$. Then under the natural isomorphism $f \colon E_{a,b} \to E_{u^4a,u^6b}$, we have $f^*(\omega_{u^4a,u^6b})=u^{-1} \omega_{a,b}$. Thus a given pair $(E, \omega)$ is isomorphic to $(E_{a,b}, \omega_{a,b})$ for a unique value of $(a,b)$. Furthermore, if $(E, \omega)$ is isomorphic to $(E_{a,b}, \omega_{a,b})$ then $(E, u \omega)$ is isomorphic to $(E_{u^{-4} a, u^{-6}b}, \omega_{u^{-4}a,u^{-6}b})$.

Define $F_4(E, \omega)$ to be the unique value of $a$ such that $(E, \omega)$ is isomorphic to $(E_{a,b}, \omega_{a,b})$ then we find $F_4(E, u \omega)=u^{-4} F_4(E, \omega)$. Thus $F_4$ defines a modular form of weight 4. Similarly, if we define $F_6$ using $b$ then $F_6$ is a modular form of weight 6. The function taking $(E, \omega)$ to the discriminant of the corresponding $E_{a,b}$ is a modular form of weight 12. These coincide with $E_4$, $E_6$, and $\Delta$, up to constants.

Modular forms of higher level

The above theory can be generalized by replacing $\Gamma(1)$ with an arbitrary finite-index subgroup $\Gamma$. We sketch the general picture.

A modular form of weight $k$ for $\Gamma$ is a function $f \colon \fh \to \bC$ satisfying the following conditions:

The last condition should be explained. At the cusp infinity, it means $f(z)$ converges as $z \to i \infty$. Suppose $x$ is some other cusp, and $\gamma(\infty)=x$ for some $\gamma \in \Gamma(1)$. Then $g(z)=(cz+d)^{-k} f(\gamma z)$ is a modular form for $\gamma^{-1} \Gamma \gamma$, and $f$ is holomorphic at $x$ if and only if $g$ is holomorphic at $\infty$.

The modular interpretation carries over: weight $k$ modular forms can be identified with sections of $\omega^{\otimes k}$ over $Y_{\Gamma}$ satisfying appropriate conditions at the cusps. Concretely, this means that a modular form assigns to every triple $(E, ?, \omega)$ a number, and is homogenous in $\omega$ of the appropriate degree. Here, the ? is the extra data associated to the moduli problem: for instance, for $\Gamma=\Gamma_1(N)$ it would be a point of order $N$.

The geometric interpretation carries over as well: a weight $2k$ modular form gives a meromorphic section of $(\Omega^1)^{\otimes k}$ over $X_{\Gamma}$. As before, one can specify the local conditions on $X_{\Gamma}$ that correspond to holomorphicity and cuspidality. The most important case the following:

Proposition. The space of weight 2 cusp forms for $\Gamma$ is identified with $\rH^0(X_{\Gamma}, \Omega^1)$. In particular, the dimension of the space of weight 2 cusp forms for $\Gamma$ is the genus of $X_{\Gamma}$.

Hecke operators

On lattices

Recall that $\cL$ is the set of lattices in $\bC$. Let $\bZ[\Lambda]$ denote the free abelian group of $\Lambda$. Let $n$ be an integer. We define an endomorphism $T(n)$ of $\bZ[\Lambda]$ by

$$ T(n) [\Lambda] = \sum_{[\Lambda':\Lambda]=n} [\Lambda'] $$

and extend linearly to all of $\bZ[\Lambda]$. For a complex number $\alpha$, define an operator $H_{\alpha}$ by $H_{\alpha} [\Lambda]=[\alpha \Lambda]$.

Proposition. (a) If $n$ and $m$ are coprime then $T(nm)=T(n) T(m)$. (b) We have $T(p^{n+1})=T(p^n) T(p)-pT(p^{n-1}) H_p$ for $p$ prime. (c) The operators $T(n)$ and $T(m)$ commute for all $n$ and $m$.

Proof. (a) Let $\Lambda''$ be a lattice of index $nm$ in $\Lambda$. Since $n$ and $m$ are coprime, there is a unique intermediate lattice $\Lambda'$ in $\Lambda$ of index $m$. Thus

$$ T(nm) \Lambda = \sum_{[\Lambda'':\Lambda]=nm} [\Lambda''] = \sum_{[\Lambda':\Lambda]=m} \sum_{[\Lambda'':\Lambda']=n} [\Lambda''] = T(n) T(m) \Lambda. $$

(b) We prove the $n=2$ case, for simplicity. We have

$$ T(p) T(p) \Lambda = \sum_{\Lambda'' \subset \Lambda' \subset \Lambda} [\Lambda''], $$

where each inclusion has index $p$. We thus find

$$ T(p)^2 [\Lambda] = \sum_{[\Lambda'':\Lambda]=p^2} n_{\Lambda''} [\Lambda''], $$

where the coefficient is the number of subgroups of $\Lambda/\Lambda''$ of order $p$. If this quotient is cyclic of order $p^2$, the coefficient is 1. Otherwise, it is the cardinality of $\bP^1(\bF_p)$, which is $p+1$. In $T(p^2) \Lambda$, we get the same sum, but with all coefficients equal to 1. Thus the only difference is that in $T(p)^2 [\Lambda]$ the coefficient of $[p\Lambda]$ is $p+1$, while in $T(p^2) [\Lambda]$ it has coefficient 1. Thus $T(p)^2 [\Lambda]-T(p^2) [\Lambda] = p H_p [\Lambda]$.

(c) By (b), $T(p^n)$ is a polynomial in $T(p)$. Thus the $T(p^n)$ commute with each other. The result now follows from (a). <h3>On modular forms of level 1</h3> A modular form $f$ of weight $k$ for $\Gamma(1)$ is a function $f \colon \cL \to \bC$ satisfying $H_{\alpha} f=\alpha^{-k} f$, together with some holomorphicity conditions. For a modular form $f$ of weight $2k$ and an integer $n$, we put

$$ (T(n) f)(\Lambda)=n^{2k-1} \sum_{[\Lambda':\Lambda]=n} f(\Lambda'). $$

Since $T(n)$ and $H_{\alpha}$ commute, this still has the appropriate homogeneity properties to be a modular form of weight $2k$. The following proposition shows that it has the appropriate holomorphicity properties. ◾

Proposition. Suppose $f(z)=\sum_{n \ge 0} a_n q^n$. Let $p$ be a prime. Then $(T(p) f)(z)=\sum_{n \ge 0} (a_{pn}+p^{2k-1}a_{n/p}) q^n$, where $a_{n/p}=0$ if $p$ does not divide $n$.

Proof. We need to compute $(T(p) f)(\Lambda_z)$. The index $p$ sublattices of $\Lambda_z$ are $\langle p, z+i \rangle$ for $0 \le i \le p-1$ and $\langle 1, pz \rangle=\Lambda_{pz}$. We have

$$ p^{2k-1} \sum_{i=0}^{p-1} f(\langle p, z+i \rangle) = p^{-1} \sum_{i=0}^{p-1} f(\Lambda_{(z+i)/p}) = p^{-1} \sum_{n \ge 0} \sum_{i=0}^{p-1} a_n e^{2\pi i n (z+i)/p}. $$

The sum over $i$ is equal to $p$ when $p \mid n$, and 0 otherwise. We thus obtain

$$ \sum_{p \mid n} a_n e^{2\pi i n z/p} = \sum_{n \ge 0} a_{np} q^n. $$

On the other hand,

$$ f(\Lambda_{pz}) = \sum_{n \ge 0} a_n q^{np}=\sum_{n \ge 0} a_{n/p} q^n. $$

Combining, we obtain the stated result. ◾

Corollary. $T(n) f$ is a modular form of weight $2k$ for any $n$. If $f$ is cuspidal, so is $T(n) f$.

Proof. The above calculations establish holomorphicity/cuspidality for $T(p) f$. The general case follows from this, since the $T(p)$ generate the $T(n)$. ◾

Remark. Suppose $f$ is a cusp form with $a_1=1$ (normalized) which is an eigenvector for $T(p)$. Then its eigenvalue is equal to $a_p$, as the linear coefficient of $T(p) f$ is $a_p$. In fact, this holds for composite $p$ as well.

Moduli description

If $\Lambda'$ is an index $n$ sublattice of $\Lambda$, then there is a degree $n$ isogeny $\varphi \colon E_{\Lambda'} \to E_{\Lambda}$ whose kernel has cardinality $n$. Furthermore, $\varphi^*(dw)=dw$ for this isogeny. It follows that we can expression the Hecke operators moduli-theoretically as follows:

$$ f(E, \omega) = \sum_{\varphi \colon E' \to E} f(E', \varphi^*(\omega)) $$

where the sum is over all isomorphism classes of isogenies $\varphi \colon E' \to E$ whose kernel has cardinality $n$.

In higher level

Suppose $\Gamma$ is a finite index subgroup of $\Gamma(1)$ of level $N$, meaning it contains $\Gamma(N)$. Then $Y_{\Gamma}$ can be described as elliptic curves together with some $N$-torsion data. The Hecke operators $T(n)$ act on modular forms for $\Gamma$ so long as $n$ is prime to $N$: indeed, an isogeny $\varphi \colon E' \to E$ induces an isomorphism on $N$-torsion, and so any $N$-torsion data can be transported along $\varphi$. These operators commute, as before.