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. 
 A: 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
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.
A: K. Ribet in his article "Abelian Varieties over $\mathbb{Q}$ and modular forms" defines GL_2 type abelian varieties and conjectures that they should be motives for cusp forms of weight two. The conductor of the abelian variety of course corresponds to the level of the modular form and the dimension of the abelian variety equals the degree of the field generated by the coeffecients of the modular form.  
