8 problems

Let $p$ be an odd prime, let $V_{k,a_p}$ be a two-dimensional crystalline representation of $\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and let $v_p(a_p)$ be a half…

Let $p$ be a prime, let $n$ be a positive integer, let $K$ be a finite extension of $\mathbf{Q}_p$, and let $\overline\rho:G_K\to\operatorname{GL}_n(k_L)$ be a continuous represent…

Let $p$ be a prime, let $n$ be a positive integer, let $K$ be a finite extension of $cQ_p$, and let $\overline\rho:G_K\to\operatorname{GL}_n(k_L)$ be a continuous representation. L…

Local constancy conjecture. There is a crystalline lift of $\bar r$ with Hodge–Tate weights $0,k-1$ and slope $n$ if and only if there is a crystalline lift of $\bar r$ with Hodge–…

Boundary irreducibility conjecture. If $v_\chi(\alpha)\notin{\mathbb{Z}}$, then $\overline{V}_{k,\alpha,\chi}$ is irreducible.

Irreducibility conjecture. If $v(a_p)\notin{\mathbb{Z}}$, then $\overline{V}_{k,a_p}$ is irreducible.

Perrin-Riou's conjecture. The element $\frak{z}_0^{\textup{BK}}$ is non-trivial if and only if

Extension conjecture. The claim of the theorem and the corollary also holds true if one replaces $x_0$ by a (locally) closed subscheme of the special fiber over which there exists…