There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([[1]](https://mathoverflow.net/questions/96289/extensions-of-the-modularity-theorem), [[2]](https://mathoverflow.net/questions/12416/the-difficulties-in-proving-modularity-lifting-theorems-over-non-totally-real-fi)), and another one about general algebraic varieties [[3]](https://mathoverflow.net/questions/24137/modularity-of-algebraic-varieties).


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?



Modularity theorem for abelian varieties
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.


