4 problems

Let $K$ be a nonarchimedean local field, let $d$ be a positive integer, and let $]_{ll]}$ be the coefficient ring. Let $L]_d(K)$-representations mean representations on $]_{ll]}$-m…

Local-global compatibility conjecture. The action of $R_p^{\operatorname{loc}}$ on

Local Langlands conjecture for $B(\chi,[1,b])$. Up to a twist by a character, $B(\chi,[1,b])$ is isomorphic to $\Pi(V_{\chi,[1,b]}^\vee)$ as a Banach space representation of…

Let $V$ be an irreducible crystabelian representation of $G_{{\mathbb {Q}}_p}$ with Hodge–Tate weights $(0,k-1)$ for an integer $k\geq 2$. Let $(\alpha,\beta)$ be the associated pa…