Explicit Serre weights for two-dimensional Galois representations over a ramified base

Transactions of the American Mathematical Society, vol. 375, pp. 8739–8767

Abstract

Given a totally real number field F F and a mod p p Galois representation ρ : G F → G L 2 ( F ¯ p ) \rho \colon G_F\to \mathrm {GL}_2(\bar {\mathbf {F}}_p) , we propose an explicit definition of the set of Serre weights W ( ρ ) W(\rho ) attached to ρ \rho . We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre’s modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé–Diamond–Roberts and Calegari–Emerton–Gee–Mavrides which together give explicit and equivalent sets of weights when p p is unramified in F F .

Authors 1

1. Misja F. A. Steinmetz corresponding Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA, Leiden, the Netherlands