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

Given a totally real number field $F$ and a mod $p$ Galois representation $\rho \text{\hspace{0.17em}⁣}:G_F\to {\mathrm{G}\mathrm{L}}_2({\bar{\mathbf{F}}}_p)$, 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 $F$.