Modularity theorem for abelian varieties 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?
 A: 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.
A: 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.
