The importance of modular forms

Question

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called $E(\Lambda)=\mathbb C(\wp,\wp')$ and they "represent" all meromorphic functions on the torus $T=\mathbb C/\Lambda$. Is demonstrated that two tori $\mathbb C/\Lambda$ and $\mathbb C/\Lambda'$ are conformally equivalent iff $\Lambda'=a\Lambda$ for some $a\in\mathbb C^\ast$; moreover the surface of all equivalence classes of conformally equivalent tori is in bijection with the $G$-space $\mathcal H/ SL_2(\mathbb Z)$ where $\mathcal H$ is the upper half plane. But $\mathcal H/ SL_2(\mathbb Z)$ is homeomorphic to $\mathbb C$ through the function (absolute invariant) $j$, so the space of moduli of genus $g=1$ can be identified with $\mathbb C$. This is for me quite clear, but the my professor then introduced modular forms and modular functions, that are so important because they are related to many theory of number's problems, such as the solution of $Q(x_1,\ldots, x_k)=n$ where $Q(\cdots)$ is a quadratic form in $\mathbb Z$.

Despite this I don't understand in which way modular forms and modular functions are related to complex tori, Riemann Surfaces and elliptic functions. Why it was necessary to start the course with the concept of elliptic function?

Answer 1

Consider $\Gamma = SL_2(\mathbb{Z})$. Remember that $\gamma = \begin{pmatrix} a & b \cr c& d \end{pmatrix} \in \Gamma$ operates on the upper half plane by $T_\gamma : z \mapsto \dfrac{az+b}{cz+d}$. Write $\pi: H \to H/\Gamma$ for the quotient map.

What is a (meromorphic) differential form $\omega$ on $H/\Gamma$? It should be nothing but a (meromorphic) differential form $\tilde{\omega} := \pi^*\omega$ on $H$ which is invariant under $\Gamma$. Writing $\tilde{\omega} = f(z)dz$ this reads $$ T_\gamma^*\tilde{\omega} = T_\gamma^*(f(z)dz) = f(T_\gamma(z))dT_\gamma(z) = f\left(\dfrac{az+b}{cz+d} \right) d\left(\dfrac{az+b}{cz+d} \right) = f(\dfrac{az+b}{cz+d}) \frac{ad-bc}{(cz+d)^2} dz $$ So the invariance property $T_\gamma^*\tilde{\omega} = \tilde{\omega}$ translates to $f\left(\dfrac{az+b}{cz+d} \right) = (cz+d)^2f(z)$ which should be familiar.

So basically modular forms of weight 2 correspond to differential forms on the space $H/\Gamma$ parametrizing complex elliptic curves.

Obviously I omitted some important details like behaviour at infinity, so called cusps, or how to interpret higher weights modular forms (they correspond to sections $f(z)(dz)^n \in H^0(H/\Gamma,(\Omega^1_{H/\Gamma})^{\otimes n})$ of the tensor product sheaf). But I hope the principle is clear.