I encountered this proof of the four squares theorem using modular forms:
1) Define the function $f: \mathbb H^* \to \mathbb C,~ \tau \mapsto \sum_{n=0}^\infty r(n,k) q^n$, where $r(n,k)$ the number to write $n$ as a sum of $k$ squares.
2) Check that $f(\cdot, 4) \in M_2(\Gamma_0(4))$ i.e. it is a modular form of level $4$.
3) Note that $M_2(\Gamma_0(4))$ has the basis $\{G_{2,2}, G_{2,4}\}$ with $G_{2,N}(\tau) = G_2(\tau)-N G_2(N\tau)$, $G_2$ the second Eisenstein series.
4) From the first fourier coefficients of $f(\cdot, 4)$ conclude that $f(\cdot, 4) = - \frac 1 {\pi^2} G_{2,4}$ which has positive Fourier coefficients.
I understand the idea behind each but I am not sure how to think about what is going on "behind the scenes".
Since modular forms can be introduced and motivated by studying equivlance classes of complex tori, I was wondering how exactly the geometry of complex tori comes in to play, as of now the proof reads just as a technical calculation for me. For example, if we already knew the four squares theorem, what would that tell us about complex tori; what geometrical meaning does the function $f$ a priori have?
So far I know that the moduar curve $\Gamma_0(N) \setminus H$ consists of the equivalence classes of pairs ($E,C)$ of an elliptic curve $E$ and a cyclic subgroup $C$ of $E$, with $(E,C) \sim (E',C')$ if there exists an isomorphism $\phi:E \to E'$ mapping $C$ to $C'$.
Further, I know that modular forms correspond to sections of some holomorphic line bundle over the compactified modular curve.
Something just doesn't click for me. I would appreciate some help.