30 problems

Let $F$ be a finite extension of ${\mathbb Q}_p$, and let $G$ be a quasi-split group over $F$. Infinitesimal-character conjecture for quasi-split groups. Does the infinitesimal-cha…

Let $G$ be a connected reductive group over a local field with Lie algebra $\frak{g}$, let $L$ be a sufficiently large extension of ${\mathbb Q}_p$ over which $G$ is split, and let…

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 $V$ 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…

Locally analytic representation conjecture. If $\alpha\ne p\beta$, the continuous equivariant injection

Let … where the locally convex inductive limit is over open neighborhoods $U\subset N\backslash P^{\operatorname{an}}_{\operatorname{dR}}$ containing the closure of the image of…

The locally analytic $\operatorname{Ext}$ conjecture. Under the splitting in Proposition referred to in the source,…

Let $G=\operatorname{GL}_n(K)$, let $\operatorname{St}_{G}^{\operatorname{an}}$ be the locally analytic Steinberg representation, and let $\rho$ range over non-critical semistable…

Let $D$ be the filtered $\varphi$-module under consideration, let $\sigma$ range over the relevant embeddings, and let $I$ range over subsets that are very critical for $\sigma$. L…

Let $D$ be the regular filtered $\boldsymbol{\varphi}$-module under consideration, let $\pi(D)$ and $\pi_{\mathrm{alg}}(D)$ be the associated locally analytic and algebraic represe…

Let $\widetilde{\Lambda}$ satisfy the Tate–Sen axioms (TS1)–(TS4), and let $G$ be a compact $p$-adic Lie group. For a $G$-module $M$ in the setting of the paper, write…

Let $G$ be a compact $p$-adic Lie group, and let $M$ and the associated period rings and locally analytic-vector functor be as in Theorem 1. The Lie algebra $\mathrm{Lie}(G)$ is no…

Breuil's conjecture. There is a natural $E$-linear bijection

Emerton–Gee–Hellmann's locally analytic categorical Langlands conjecture. The functor $\mathfrak{LL}_p^{la}$ is fully faithful when restricted to a suitable category of “tempered”…

Let $v\in S_p$. For each $v$, let $\widetilde v$ be the chosen place above $v$, let $\rho_{\widetilde v}$ be the associated generic potentially crystalline Galois representation, l…

Let $V$ be irreducible, crystalline, Frobenius semisimple with distinct Hodge–Tate weights $0$ and $k-1$. Let $\breve{\alpha}$ and $\breve{\beta}$ be the relevant Frobenius eigenva…

Let $\rho_p:{\rm Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\to{\rm GL}_3(E)$ be semistable as in the preceding setup, with $\dim_E W=3$ and $N^2\ne0$. The $n=3$ local identificatio…

Let $\alpha$ be the unique simple root of ${\rm GL}_2$, and let $D$, $W$, $h_1$, $h_2$, and $\lambda$ be as in the preceding setup. The $n=2$ local conjecture. There exists a finit…

Let $G$ be the reductive group in the preceding setup, let $C_{\lambda,\alpha}$ be the specified full subcategory of finite-length admissible locally analytic representations, and…

Let $\pi^\alpha(\rho_p)$ be the nonsplit extension attached to $\rho_p$ in the cited work, and let $\pi^\alpha(\operatorname{Fil}^{\rm max}_\alpha(\rho_p))$ be the extension constr…

Let $n$, $E$, $\rho_p$, $(h_i)_{i\in\{1,\dots,n\}}$, $W$, $\alpha=e_j-e_{j+1}$, $\pi^\infty$, $\chi$, $\lambda$, $F_\alpha$, and $D_\alpha(\pi)$ be as in the preceding discussion,…

Let $H$ be a compact locally $\mathbf{Q}_p$-analytic group and let $A$ be the coefficient algebra occurring in the construction of $\mathrm{LA}(H,A)$. Completeness conjecture. The…

Breuil's conjecture. For a character $\chi:T(L)\rightarrow E^{\times}$,

Let $\rho$ be an $n$-dimensional continuous representation of $\operatorname{Gal}(\overline{F}/F)$ over $E$, with $\rho_{\tilde v}$ crystalline and very regular for every…

Suppose $\rho_p$ is crystalline and very regular, with Hodge–Tate weights $\underline{h}=(h_1,h_2,h_3)$ and a refinement $w(\underline{\phi})$ indexed by $w\in S_3$. Let…