9 problems

Let $r=k-2$, let $r_0=k_0-2$, and use the parameters $\tau=v_p(c)$ and $t=v_p(k-k_0)$ from the zig-zag conjecture. When $\tau=t+i$, let $\lambda_i$ denote the unramified constants…

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.

Let an eigenvariety for a unitary group map to a finite slope space, as constructed in the paper. Finite slope space eigenvariety conjecture. This map induces an isomorphism from t…

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…

Jannsen's conjecture. For $x$ geometric and outside the hypersurface of points where the motivic weight is $-1$, the dimension of $H^1_T(G_{K,\Sigma},V_x)$ is constant.