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 $\rho :G_F\to {\mathrm{GL}}_2({\bar{\mathbf{F}}}_p)$, 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 $F$.

Final version. To appear in Transactions of the American Mathematical Society