Serre weights and wild ramification in two-dimensional Galois representations

TL;DR

This paper proposes a new conjectural description of Serre weights for two-dimensional Galois representations over totally real fields, using local class field theory and Artin-Hasse exponential, to understand wild ramification effects.

Contribution

It introduces a conjectural alternative to p-adic Hodge theory for describing Serre weights, focusing on wild ramification in crystalline Galois representations.

Findings

Provides explicit conjectural formulas for Serre weights

Uses local class field theory and Artin-Hasse exponential in the approach

Includes numerical examples illustrating the conjecture

Abstract

A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such forms in terms of the local behavior of the Galois representation at primes over $p$. This characterization of the weights, which is formulated using $p$-adic Hodge theory, is known under mild technical hypotheses if $p > 2$. In this paper we give, under the assumption that $p$ is unramified in $F$, a conjectural alternative description for the set of weights. Our approach is to use the Artin-Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using $p$-adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.