6 problems

The local companion points conjecture. The following conditions are equivalent:

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 $y$ 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…

Let $T_p$ be the relevant diagonal torus, let $\mathcal E(U^p)_{\overline\rho}$ be the eigenvariety attached to the definite unitary group, and retain the notation for…

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