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

Abstract

Given a totally real number field $F$ and a mod $p$ Galois representation $\rho\colon G_F\to \mathrm{GL}_2(\bar{\mathbf{F}}_p)$, we propose an explicit definition of the set of Serre weights $W(\rho)$ attached to $\rho$. 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 Demb\'{e}l\'{e}--Diamond--Roberts and Calegari--Emerton--Gee--Mavrides which together give explicit and equivalent sets of weights when $p$ is unramified in $F$.