I am learning about modular forms for the first time this term and am just starting to wrap my head around what **might** be the big picture of things.

I was wondering if the following interpretation of why modular forms are important is correct **a)** technically and **b)** in terms of getting the right picture.

Possible Intuition for Importance of Modular Forms:We want arithmetic data about elliptic curves. Given a congruence subgroup $$\Gamma\in\{\Gamma_1(N),\Gamma_0(N),\Gamma(N)\}$$ we let $X(\Gamma)$ denote $\Gamma/\mathfrak{h}^\ast$ (where $\mathfrak{h}^\ast$ is the upper half-plane $\mathfrak{h}$ union the cusps $\mathbb{Q}\cup\{\infty\}$). We know then that $X(\Gamma)$ is the compactification of a moduli space whose points classify elliptic curves with some torsion data. Because of this, we want to understand the geometry of $X(\Gamma)$ because hopefully this will translate, via the moduli space concept, back to arithmetic data about elliptic curves.

But, given a compact Riemann surface, one ideologically gets a lot of the information concerning the surface by studying meromorphic sections of certain holomorphic line bundles over $X$. A very natural line bundle attached to $X(\Gamma)$ is $(T_{X(\Gamma)}^{\ast1,0})^{\otimes n}$ (which is the $n^{\text{th}}$-tensor power of its holomorphic cotangent bundle). Thus, a natural place to look for geometric information about $X(\Gamma)$ is in the meromorphic sections of this bundle, denoted $\Omega^{\otimes n}(X(\Gamma))$. More specifically one may focus on the holomorphic sections $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast1,0})^{\otimes n})$ of this line bundle.

That said, the natural equivalence map $\pi:\mathfrak{h}\to X(\Gamma)$ gives rise to the pullback map $\pi^\ast:\Omega^{\otimes n}(X(\Gamma))\to\Omega^{\otimes n}(\mathfrak{h})$. But, since $\mathfrak{h}$ has only one chart, we can naturally identify $\Omega^{\otimes n}(\mathfrak{h})$ with $\text{Mer}(\mathfrak{h})$. Thus, we have a linear embedding $\pi^\ast:\Omega^{\otimes n}(X(\Gamma))\to \text{Mer}(\mathfrak{h})$. Since $\text{Mer}(\mathfrak{h})$ is easier to deal with (at least its easier to "see") we would like to identify the objects of interest, $\Omega^{\otimes n}(X(\Gamma))$ and its subspace $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$, with their image under $\pi^\ast$.

That said, one can prove that the image under $\pi^\ast$ of $\Omega^{\otimes n}(X(\Gamma))$ is $\mathcal{A}_{2n}(\Gamma)$ (the automorphic forms of weight $2n$ with respect to $\Gamma$) and the image of $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$ is

containedin $\mathcal{M}_{2n}(\Gamma)$ (the modular forms of weight $2n$ with respect to $\Gamma$).

Ok, assuming the above is correct, there are three questions that begged to be asked:

- Why do we care about all of $\mathcal{M}_{2n}(\Gamma)$? Why don't we care more specifically about the image of $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$ under $\pi^\ast$? Can we describe this image (e.g. it's the cups forms for $n=1$)?
- What do odd weight modular or automorphic forms tell us? If $-I\in\Gamma$ then there are no non-zero such objects, but in the cases when $-I\notin\Gamma$, do we gain anything by looking at odd weights?
- What does the geometry of $X(\Gamma)$ tell us about elliptic curves? For example, the genus of $X(\Gamma)$ tells us things about the objects we parameterize. That said, we don't need to study sections of line bundles to get this geometric data. Indeed, the genus for $X_0(1)$ can be deduced from the fact that the $j$-invariant has one simple pole, and thus must be induced a biholomorphism $j:X_0(1)\to\mathbb{P}^1$. From there, we can find the genus of $X(\Gamma)$ by using the natural projection $X(\Gamma)\to X_0(1)$ and the Riemann-Hurwitz formula. So, what geometric information about $X(\Gamma)$ is important that one needs automorphic/modular forms to get?

Thank you so, so much friends! I have been grappling with the "big picture" of modular forms of late, and this is the best I cam up with. I am very excited to hear your responses!