Joint IAS/Princeton University Number Theory Seminar

Serre conjectured that all continuous, irreducible, odd $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ arise from modular forms. If $\rho$ is modular, then proven refinements provide recipes for the possible weights and levels of the forms giving rise to it in terms of the local behavior of $\rho$. A natural generalization to the context of a totally real field $F$ predicts that all continuous, irreducible, totally odd $\rho:G_F \to {\mathrm{GL}}_2(\overline{\mathbf{F}}_p)$ arise from Hilbert modular forms. The recipe for the possible levels is similar to the case of $F= \mathbf{Q}$, but the (conjectural) recipe for the weights reveals features not so apparent for $F= \mathbf{Q}$. In particular, if $\rho$ is locally reducible at a prime $\mathcal P$ over $p$, then the possible weights depend strongly on the corresponding extension of local characters. If $\rho$ is locally semisimple at $\mathcal P$, then the recipe can be described in terms of the reduction of a corresponding irreducible characteristic zero representation of $\mathrm{GL}_2(\mathcal{O}_F/\mathcal{P})$.