10 problems

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 $G(k)$ 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…

Pure-locality conjecture. The representation $\pi(\overline{r})$ should depend only on the local representation $\overline{\rho}_0$.

Categorical mod-$p$ Langlands conjecture. The Hecke $\operatorname{Ext}$-algebra is isomorphic to the $\operatorname{Ext}$-algebra of a certain line bundle on $C_\sigma$.

Let $p$ be a prime, let $\mathbb{F}/\mathbb{F}_p$ be a sufficiently large finite extension, and let $F/F^+$ be a CM extension in which $p$ is inert in $F^+$ and splits in $F$. Let…

Let $k\geq 2$ and let $a_p\in\overline{\mathbb{Q}}_p$ have positive valuation. Let $V_{k,a_p}$ be the associated irreducible two-dimensional crystalline representation of…

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…

Let $F$ be a $p$-adic field, let $G=\mathrm{GL}_n(F)$, and let $\mathcal H$ be the pro-$p$-Iwahori Hecke algebra of $G$. Consider $n$-dimensional modulo $p$ Galois representations…