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 Dembélé–Diamond–Roberts and Calegari–Emerton–Gee–Mavrides which together give explicit and equivalent sets of weights when $p$ is unramified in $F$.