Lecture 13: Modular forms
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:

Homogeneity: $f(\alpha \Lambda)=\alpha^{k} f(\Lambda)$ for $\alpha \in \bC^{\times}$.

Holomorphicity: $f$ is a holomorphic function of $\Lambda$, in the sense that $z \mapsto f(\Lambda_z)$ is holomorphic on $\fh$.

Holomorphicity at $\infty$: $f$ is holomorphic at $\infty$, in the sense that $z \mapsto f(\Lambda_z)$ converges as $z \to \infty$. We denote this value by $f(\infty)$.
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 welldefined 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 1forms 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 nonzero 1form on $E$. Writing $E=E_{\Lambda}$, the above discussion shows that $f(\Lambda) (dw)^k$ is a welldefined element of $\Gamma(E, \Omega^1)^{\otimes k}$. However, $\omega^k$ is also such an element, and so we can divide to get a welldefined 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:

Homogeneity: $F(E, \alpha \omega)=\alpha^{k} F(E, \omega)$.

Invariance: if $(E, \omega)$ is isomorphic to $(E', \omega')$ (in the obvious sense), then $F(E, \omega)=F(E', \omega')$.
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 $\ord_{\pi(\infty)}(\omega) \ge 1k$.
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 $2k+n+m+k=n+mk$, and thus $n+mk+1$ 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 nonzero modular forms of weight 2. There is exactly one nonzero 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 nonzero) 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
$$ E_k(z) = 1\frac{4k}{B_k} \sum_{n \ge 1} \sigma_{k1}(n) q^n $$where $B_k$ is the Bernoulli number and $\sigma_{k1}(n)$ is the sum of the $(k1)$st powers of the divisors of $n$.
Let $\Delta=E_4^3E_6^2$. 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 nonzero. It is therefore the unique (up to scaling) nonzero cusp form of weight 12. Its $q$expansion is complicated, but it admits the following nice product formula (due to Jacobi):
Proposition. $\Delta(z)=q \prod_{n \ge 1} (1q^n)$.
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 1form 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 finiteindex 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:

$f(\gamma z)=(cz+d)^k f(z)$ for all $\gamma \in \Gamma$.

$f$ is holomorphic on $\fh$.

$f$ is holomorphic at the cusps.
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 $T(p^{n+1})=T(p^n) T(p)pT(p^{n1}) H_p$ 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
$$ (T(n) f)(\Lambda)=n^{2k1} \sum_{[\Lambda':\Lambda]=n} f(\Lambda'). $$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 $(T(p) f)(z)=\sum_{n \ge 0} (a_{pn}+p^{2k1}a_{n/p}) q^n$, 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 $0 \le i \le p1$ and $\langle 1, pz \rangle=\Lambda_{pz}$. We have
$$ p^{2k1} \sum_{i=0}^{p1} f(\langle p, z+i \rangle) = p^{1} \sum_{i=0}^{p1} f(\Lambda_{(z+i)/p}) = p^{1} \sum_{n \ge 0} \sum_{i=0}^{p1} a_n e^{2\pi i n (z+i)/p}. $$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 modulitheoretically 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.