Supersingular Galois Representations and a Generalization of Conjecture of Serre

Abstract

Serre's conjecture relates two-dimensional odd irreducible Galois representations over $\bar\F_p$ to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. In particular, for $n$-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.