Lots of questions about modular forms

Question

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.

1. 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)?

2. 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?

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

Answer 1

(1.) This is a very good question and shows you are thinking in the right directions, but it also is asking for a summary of multiple entire fields of mathematics. Some keywords are "automorphic forms", "locally symmetric spaces", "Shimura varieties".

In brief, you can exactly do that. You have to think about what you mean by "forms" because, as Noam Elkies says, you lose the complex structure, so you can't consider holomorphic differential forms. The simplest thing to do instead is to consider real-valued functions that are eigenfunctions of the Laplace operator and the other Casimir operators. Many of the analytic results about modular forms have analogues in this setting.

These don't parameterize mappings of the torus to complex algebraic varieties, by which I believe you mean abelian varieties. To do that, you just need to replace the general linear group with the symplectic group in your construction. You can go further, and consider arbitrary algebraic groups. All of these objects have nice classes of functions on them and theories of Hecke operators, and all of them play a role in the Langlands program.

The significance of rank 2 is that, thanks roughly to exceptional isomorphisms, it's a special case of many different nice classes simultaneously (both a general linear group and thus a moduli space of lattices with a good theory of Fourier coefficients and many other useful theorems, and a symplectic group and thus a moduli space of abelian varieties, and a one-dimensional algebraic variety and thus admitting a Jacobian, and...) Any one given nice property of the modular curve and modular forms will have higher analogues, but not all of them simultaneously.

(2.) Roughly, yes. The Jacobian splits up to isogeny into a product of abelian varieties associated to eigenforms, with the dimension of the variety depending on the degree of the field of definition, so eigenforms with integral coefficients give elliptic curves. This is much, much easier to prove than the modularity theorem, and goes back to Eichler and Shimura.

Actually, the way you've stated your question, it may just be answered by the general point that the category of abelian varieties under isogeny is semsimple so they all split uniquely into a product (up to isogeny) of simple factors.

(3.) Using modular symbols, one can find all eigenforms of level $N$, extract the ones with integral coefficients, and then integrate those over loops in the modular curve, calculating their period lattice, and then use that to find equations for the modular curve. See Cremona's book on algorithms for modular elliptic curves, Chapter 2.

(4.) I'm not sure exactly what you mean, but I think most questions about other bundles over $X_0(N)$ will either reduce to questions about modular forms, or else be very difficult.