Wildly Ramified Galois Representations and a Generalization of a Conjecture of Serre

Abstract

Serre's conjecture relates two-dimensional odd irreducible characteristic $p$ representations to modular forms. We discuss a generalization of this conjecture (due to Ash and Sinnott) to higher-dimensional Galois representations. In particular, we give a refinement of the conjecture in the case of wildly ramified Galois representations and we provide computational evidence for this refinement.