140 problems

A $p$-adic representation that is de Rham should become log-crystalline after a finite base change. Fontaine's conjecture. De Rham representations are potentially log-crystalline.…

Let $G$ be a split classical group, and let $\pi$ be an isolated representation in the unitary dual of $G$. Automorphy conjecture. The representation $\pi$ is automorphic. The conj…

Weak Arthur packet decomposition conjecture. There is a set $\mathcal T$ and an appropriate notion of Arthur packet such that

Fukaya–Kato's local epsilon-isomorphism conjecture. For every such de Rham representation $V_\chi$, one has

Gross's conjecture. The following assertions hold: $\pi^{K_{n,m}}=0$ for $0\leq m<c_\pi$ and $\dim_\mathbb{C}\pi^{K_{n,c_\pi}}=1$; the action of $J_{n,c_\pi}/K_{n,c_\pi}$ on…

Assume that $\rho$ has Loewy length $3$. Let $\rho$ be a representation for which $\rho$-compatible representations $\Pi(\rho)$ and $\Pi(\rho)^{\mathrm{ord}}$ are defined, with…

For each $n\geq 1$, let $G_n$ be the relevant general linear group and let $\mathcal{R}_n$ be the Grothendieck group of finite-length smooth representations of $G_n$. Let…

Let $\pi$ be a generic standard representation of a reductive $p$-adic group, and let $L$ denote its Langlands quotient. Standard module conjecture. The Langlands quotient $L$ is g…

Let $p$ be an odd prime, let $k\geqslant 2$ be an integer, and let $a\in\overline{\mathbb{Z}}_p$ satisfy $v_p(a)>0$. Let $V_{k,a}$ be the associated two-dimensional crystalline $p$…

Let $\overline{\rho}=\overline{r}_v^{\vee}$ be generic in the sense of Breuil–Paškūnas, and let $\pi_v^D(\overline{r})$ denote the associated mod $p$ Langlands representation. Writ…

Let ${\bf G}$ be an $F$-split connected reductive algebraic group defined over $F$, let ${\bf T}$ be a maximal $F$-split torus, and let $\xi$ and $\zeta$ be the parameters used to…

Let $L'/L$ be a finite Galois extension of $p$-adic fields and let $K$ be a coefficient field. Fix an integer $d\geq 1$, an object $(r,N,V)$ of $\mathrm{WD}_{L'/L}$ of dimension…

Let $G$ be the $p$-adic classical group considered in the paper, and let $\pi$ be an irreducible supercuspidal representation of $G$. A representation is compactly induced if it is…

Let $\mathbf{G}$ be a reductive group over the nonarchimedean field $k$, let $\mathbf{X}$ be a spherical $\mathbf{G}$-variety, let $K$ be a maximal compact subgroup of…

Let $E$ be the number field in the deformation problem, let $V$ be an absolutely irreducible crystalline $p$-adic representation with the stated self-duality and distinct local dat…

Let $F$ be a number field, let $S$ contain the places above $p$, the places where the representation is ramified, and the infinite places, and let $F^S/F$ be the maximal extension…

Non-degeneracy conjecture. The pairing $\langle \langle\,,\,\rangle \rangle$ is always non-degenerate modulo torsion. This would characterize ${\mathfrak G}_F(T)$ in terms of the…

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

Let $K$ be a non-archimedean local field, let $G_d$ be the relevant general linear group, let $D_d^\times$ be the multiplicative group of the central division algebra of invariant…

Let $G$ be the group under consideration, let $Ell$ be the set of regular semisimple elliptic conjugacy classes in $G$ equipped with its canonical measure $d\mu$, and let $\rho_1$…

A $p$-adic differential equation is a finite-rank free module $M$ over the Robba ring $\mathcal{R}_K$, equipped with a connection $\partial_M:M\rightarrow M$. It has a Frobenius st…

Let $X$ be the building of $G$, and let ${\mathscr{S}}$ be the Schneider–Stuhler $G$-equivariant sheaf associated with an irreducible supercuspidal representation. A basic elementa…

Let $(\pi,V)$ be an irreducible supercuspidal representation of $G$, and let ${\mathscr{S}}$ be the $G$-equivariant Schneider–Stuhler sheaf on the building $X$ associated with…

Let $F$ be a nonarchimedean local field, let $\boldsymbol{\mathrm{G}}$ be a connected reductive group defined over $F$, and let $G=\boldsymbol{\mathrm{G}}(F)$ with center $Z$. Let…

Folklore irreducibility conjecture. The reduction $\bar{V}_{k,a_p}$ is irreducible for fractional slopes if $k$ is even.