5 problems

Let $G(k)$ be the finite reductive group in the local setup, let $\overline{\rho}_0$ be the local residual representation, and let $\overline{r}$ be its suitable globalization. Def…

Let $\overline{\rho}=\overline{r}_v^{\vee}$ be generic in the sense of Breuil–Paškūnas, and let $\pi_v^D(\overline{r})$ denote the associated mod $p$ Langlands representation. Writ…

Let $F$ be a CM field in which $p$ is unramified, let $w$ be a place of $F$ above $p$, and let…

Let $F$ be a totally real field, let $D$ be an indefinite quaternion algebra over $F$, and let $S^D(\mathbb F)$ be the direct limit of the mod $p$ étale cohomology of the associate…

Let $F$ be a totally real field, let $D$ be a quaternion algebra over $F$, and let $\overline{r}:G_F\to\mathrm{GL}_2(\mathbb F)$ be a two-dimensional continuous totally odd absolut…