For the past year and a half, I have been working my way through Diamond & Shurman's "A First Course in Modular Forms", and I have just finished it. I Have Some Questions.

What is so special about two dimensions? One can think about lattices/tori in N dimensions and their moduli space $$SL_n(\mathbb{Z})\backslash GL_n(\mathbb{R})/(SO_n(\mathbb{R})x\mathbb{R})$$ (I use $GL_n/\mathbb{R}$ to represent the fact that we are scaling one element of the lattice basis to 1 instead of scaling the volume of the cell). One naturally obtains a metric on the symmetric space and one can look for forms on that space which transform appropriately under the action of $SL_n(\mathbb{Z})$ or a subgroup thereof. This gives you the whole machinery of the torsion groups, the double coset (Hecke) operators etc. What exactly do you wind up losing? Does it matter if $n$ is even or odd (specifically can we define a mapping of the torus to a complex algebraic variety if $n$ is even)?

It's stated (as several versions of the Modularity Theorem) that given an elliptic curve $E$ there is some modular curve $X_0(N)$ (alt. its Jacobian $J_0(N)$) that maps onto $E$, with the minimum value of $N$ given by the conductor. For the Jacobian this is a map from a $2g$-dimensional torus to a 2-dimensional torus. Do these maps induce a unique decomposition of $J_0(N)$? That is, can we represent $J$ as the direct sum of a bunch of inverse images of elliptic curves plus possibly some other $2d$-dimensional torus?

- I don't know how to work this example. Pick a random small conductor N. Explicitly find all elliptic curves E with mappings from $X_0(N)/J_0(N)$.