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.

`$\phi(\pi_{\infty})$`

of`$\pi_{\infty}$`

satisfies`$\phi|_{\mathbf{C}^\times}(z)=\mathrm{diag}(z,\dots,z,\overline{z},\dots,\overline{z})$`

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... $\endgroup$