Supersingular Galois representations and a generalization of at conjecture of Serre

Serre's conjecture relates two-dimensional odd irreducible Galois representations over (F) over barp, to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. in particular, for eta-dimensional Galois representations that are irreducible when restricted to the decomposition group at p, we strengthen a conjecture of Ash, Doud, and Pollack. We then give computational evidence for this conjecture in the case of three-dimensional representations.