Supersingular Galois Representations and a Generalization of a Conjecture of Serre

Serre’s conjecture relates two-dimensional odd irreducible Galois representations over F̄p to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. In particular, for n-dimensional Galois representations which 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.