Supersingular Galois Representations and a Generalization of a Conjecture of Serre

Abstract

Serre's conjecture relates two-dimensional odd irreducible Galois representations over 𝔽̄ 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.