# Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.

For elliptic curves, one can give a proof using Baker's work on linear forms in logarithms. Another proof comes from the modularity of elliptic curves, owing to the fact that spaces of weight 2 cusp forms for congruence subgroups of $SL(2, \mathbb{Z})$ are finite dimensional.

For higher dimensional abelian varieties, one no longer has Baker's work on linear forms in logarithms, but one can hope to show that those of a given conductor and dimension correspond to automorphic forms which inhabit a finite dimensional vector space, and determine the dimension of this vector space to obtain a count for the number of abelian varieties. (One can even hope to write down the first few Euler factors of their L-functions.)

Is there a conjecture as to precisely which automorphic forms abelian varieties (over $\mathbb{Q}$) of a given conductor correspond to? If so, are these spaces known to be finite dimensional. If so, are the relevant automorphic forms computable?

Here I mean "automorphic form corresponding to Galois representations coming from $H^1$ of the abelian variety." Faltings proved that if two abelian varieties have the same $\ell$-adic Tate-modules then they're isogenous, and Raynaud and Masser-Wustholz proved an effective version of the finiteness of isogeny classes. So affirmative answers to the questions above together with their proofs would provide a way to count abelian varieties of a given conductor.

-
The $d$-dimensional abelian varieties over $\mathbf{Q}$ should correspond to automorphic representations $\pi$ of $\mathrm{GSpin}_{2d+1}/\mathbf{Q}$ such that the Langlands parameter $\phi(\pi_{\infty})$ of $\pi_{\infty}$ satisfies $\phi|_{\mathbf{C}^\times}(z)=\mathrm{diag}(z,\dots,z,\overline{z},\dots,\overl‌​ine{z})$ with $d$ $z$'s and $d$ $\overline{z}$'s and the characteristic polynomial of the conjugacy class corresponding to $\pi_v$ has integer coefficients for all primes $v$ away from the conductor... –  David Hansen Oct 4 '12 at 20:44
Dear Jonah, What you ask about is a special case of the general question of relating motives over $\mathbb Q$ to automorphic forms over $\mathbb Q$. The reciprocity laws are precise: they relate the Hodge numbers of the motive to the infinitesimal character of the automorphic representation; they relate conductor of the motive to the conductor (level) of the automorphic representation; they relate the char. polys of Frob. at unramified primes on the motive to the Satake parameters of the automorphic rep. at unramified primes. Once you fix the inf. char. and the level, you pin down a ... –  Emerton Oct 6 '12 at 21:26
... finite number of automorphic representations (or, more or less equivalently, a finite number of automorphic Hecke eigenforms). So the theoretical aspects of your question are well understood in the affirmative. But there are not many tables beyond the case of classical modular forms, especially for unitary groups or orthogonal groups (which are what I think would be most relevant for abelian varieties, whose $H^1$ is symplectic, thus self-dual). Also, the particular automorphic forms that will be attached to abelian vars. have non-regular inf. char. ("low weight", heuristically, ... –  Emerton Oct 6 '12 at 21:29
... akin to weight 1 for classical modular forms), and these are typically harder to enumerate than regular inf. char. forms (which can be computed in terms of cohomology). Regards, Matthew –  Emerton Oct 6 '12 at 21:30
@ Emerton: Thanks for the informative response! Can you give a reference for the theoretical aspects of my question? Is the situation that they're treated in a more general context such that it's hard for a beginner to see that they include a theoretical answer to my question as a special case? –  Jonah Sinick Oct 6 '12 at 22:59

There has been recent work in some concrete cases. There's a paper by Poor and Yuen that gives computational evidence for a special case of the so-called Paramodular Conjecture. This Conjecture is described as "a precise and testable modularity conjecture for rational abelian surfaces $\mathcal{A}$ with trivial endomorphisms, $End_\mathbb{Q} \mathcal{A} = \mathbb{Z}$ in the abstract of a paper by Brumer and Kramer. To the best of my knowledge, this is the most precise version of a general prediction made by Yoshida as described in