8 problems

Let $S\subseteq\Sigma_L$, let $I^c_S(\alpha,\mathbf h)=C(1,s_J)$ be the corresponding locally analytic representation, and let…

Let $x=(\rho,\underline{\delta})\in Y(U^p,\overline{\rho})$ be a crystalline generic strictly dominant point. For each $v\mid p$, let…

Let $X_{\mathfrak p}(\overline\rho)$ be the relevant trianguline eigenvariety, let $y=(r_y,\delta)$ be the point under consideration, and let $\chi_J^c$ be the characters defined f…

Let $y$ be a point of the relevant patched, global, or local Bernstein variety, and let $\varrho$ be its associated generic potentially crystalline Galois representation with disti…

Let $\rho$, $\Sigma_p$, $C(\rho)$, $\Sigma(\rho)$, $\mathcal E(U^p)_{\overline\rho}$, and $\delta_J^c$ be as in the preceding setup. Partial classicality conjecture. For every subs…

Breuil's conjecture. For a continuous character $\chi$ of $T_p$ over $E$,

Let $\rho$ and the associated sets $\mathscr{P}\operatorname{ar}(\rho)$ and $\mathscr{T}[\rho]$ be as above. For a parameter $\delta$, let $W(\delta)$ be the subgroup and…

Let $F$, $\rho$, $N(\rho)$, and $\mathscr{X}[\rho]$ be as above. Let $\mathscr{T}$ be the rigid space of continuous characters of the diagonal torus, and let…