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There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3].

What is known about modularity of abelian varieties over $\mathbb{Q}$? I guess the arbitrary field case is at least as problematic as it is for elliptic curves.

Can be Serre's conjecture (now a theorem) generalized to more general representations $\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_n(\bar{\mathbb{F_p}})$? Is there a Ribet-type result that that would imply full modularity?

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  • $\begingroup$ I'm confused about the meaning of "modular" in the statement of the question. For example, if you have a polarized Abelian surface, then you expect that there is a Siegel modular form (not necessarily cuspidal) whose L-functions are related to the Hasse-Weil zeta function of the Abelian surface. If the geometric endomorphism ring is Z, then you expect to see a stable cuspidal Siegel modular form; whereas, if the geometric endomorphism ring is non-commutative, then you can prove that the Siegel modular form is going to be an Einsenstein series coming from the Klingen parabolic subgroup. $\endgroup$ – Ramin May 25 '14 at 3:28
  • $\begingroup$ @Ramin. An abelian variety is modular if there is a $J_1(n) \to A $ surjective for some positive integer n. $\endgroup$ – Myshkin May 25 '14 at 3:44
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Abelian varieties over the rationals are modular if and only if they are of "$GL_2$"-type, which is a notion introduced by Ribet who proved that this statement is a consequence of Serre's conjecture which, as you know, has since been proved. Here is a link to Ribet's paper:

http://math.berkeley.edu/~ribet/Articles/korea.pdf

Generalizing the statement of Serre's conjecture to higher dimension is non-trivial and the subject of ongoing research (which I am not an expert of). There are some special cases stated and proved. They are not needed to answer your first question.

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  • $\begingroup$ Thank you, that was the bit of information I was missing. I don't quite get the "only if GL2" part of Ribet's paper, but now I know where to look. Thanks again! $\endgroup$ – Myshkin May 25 '14 at 3:48
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Here is an easy situation, if you take $K$ to be any real quadratic field, then all elliptic curves over $K$ are modular (see http://arxiv.org/abs/1310.7088). If you take the restriction of scalars of such curves to $Q$, then you get an abelian surface, which is modular (because the elliptic curve is), but this is somehow a tricky situation. I think there is no proof of a general result (besides this base change cases). Nevertheless, for abelian surfaces whose endomorphism ring is just $Z$, there is a concrete conjecture regarding modularity of such surfaces and paramodular forms (which are Siegel modular forms for a specific subgroup). See arXiv:1004.4699v2 for the right statement.

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    $\begingroup$ The precise meaning of "modular" in the papers you mention is different than what the OP had in mind. $\endgroup$ – Jeremy Rouse Jun 19 '14 at 1:48

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