In [another question](https://mathoverflow.net/questions/107500/counting-higher-dimensional-abelian-varieties-of-a-given-conductor) 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](https://mathoverflow.net/questions/107287/finiteness-of-elliptic-curves-of-a-given-conductor). 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. 



Modularity of higher dimensional abelian varieties
There has been recent work in some concrete cases. There's a paper by [Poor and Yuen](http://arxiv.org/abs/0912.0049) 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](http://arxiv.org/abs/1004.4699). To the best of my knowledge, this is the most precise version of a general prediction made by Yoshida as described in


H. Yoshida, On generalization of the Shimura-Taniyama conjecture I and II, Siegel
Modular Forms and Abelian Varieties, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics, 2007, pp. 1-26.


