27 problems

Let $R_{\rho,\mathcal F}$ be the ring pro-representing the refined deformation functor, let $\mathbb T$ be the completed local Hecke algebra at the corresponding eigenvariety point…

Let $G$ be a reductive group, let $K^p\subset G(\mathbb{A}_f^p)$ be a tame level, let $\operatorname{Iw}\subset G(\mathbb{Q}_p)$ be an Iwahori subgroup, and set…

Let $U$ be the affinoid neighbourhood of a finite-slope classical point $x_0$ on the Rankin–Selberg eigenvariety, with $U$ smooth, the weight map finite étale, and classical cuspid…

Let $\mathbf{V}$ be a datum with $\mathfrak{m}=\emptyset$, let $\xi=(\xi^n,\xi^{n+1})$ be a pair of $\Sigma^+_p\setminus\lozenge^+$-trivial dominant hermitian weights, let $\gamma$…

Let ${\mathcal E}(\mathcal N^\bullet)$ be the eigenvariety constructed from the cohomological complex $\mathcal N^\bullet$ for $\operatorname{GL}_n/\mathbb Q$, and let a cuspidal c…

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_4$, and let $P_{\tilde\pi}$ be its associated parabolic. A classical family is a family through $\tilde\pi$ consisting of cl…

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_{2n}$, and let $P_{\tilde\pi}$ be the associated parabolic. A symplectic family is a family through $\tilde\pi$ in the eigen…

Let $G$ be the reductive group considered in the paper, let $K=K^p\mathrm{Iw}$ be Iwahori level at $p$, and let $\sE_K^G$ be its eigenvariety. Let $x\in\sE_K^G$ be a non-critical c…

Let $\bar{\rho}$ be the residual Galois representation attached to a Klingen-ordinary automorphic representation, let $\mathbf{T}^{\mathrm{wKl}}_{\bar{\rho}}$ be the corresponding…

Let $L$ be the relevant local field, let $x=(\rho,\underline{x},\chi)$ be a point of $X_{\Omega,\mathbf h}(\overline\rho)$ with $\rho$ de Rham and $(\underline{x},\chi)$ generic, a…

Let $y$ be a point of the relevant patched, global, or local Bernstein variety, and let $\varrho$ be its associated generic potentially crystalline Galois representation with disti…

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 $\tilde\pi$ be a non-critical cuspidal representation, let $x_{\tilde\pi}\in\cE_{\cU,h}^\bullet$ be the associated point, and let $\cW$ be the weight space. Write $\ell$ for th…

Let $I$ index the relevant embeddings, let $\rX_D^{>r}$ be the boundary region of the eigenvariety, let $\rW^{>r}$ be the corresponding weight-space region, let $w$ be the weight m…

Let $p$ be an odd prime, let $\mathscr{W}$ be the rigid analytic space parametrizing continuous characters of $\mathbb{Z}_p^\times$, and write $T(w)=w(\exp(p))-1$ for the $T$-coord…

Nearly-overconvergent universal eigenform conjecture. There is an affinoid neighbourhood

Let $\mathcal E_{\mathrm{par}}$ be the parallel-weight eigenvariety, let $\mathcal C_M$ denote the relevant Coleman–Mazur eigencurve at level parameter $M$, and let $[\varphi]$ den…

Let $\theta$ be a finite-slope cuspidal Hecke eigensystem of $\operatorname{GL}_3$. An eigensystem is arithmetically rigid if it does not lie in any irreducible component of an eig…

Let $Y(U^p,\overline{\rho})$ be the eigenvariety locus considered in the paper, with points $x=(\rho,\underline{\delta})$ consisting of a Galois representation and a locally…

Let $\mathbb{E}_{\mathcal{H}}$ be the eigenvariety constructed from a Hecke module, let $x$ be a point belonging to exactly one irreducible component of $\mathbb{E}_{\mathcal{H}}$,…

For each $a$, let $\mathcal{E}_{a,q}^{q}$, for $q=0,\ldots,a$, be the eigenvarieties constructed over the corresponding weight spaces, and let $\mathcal{W}_a$ be the weight space.…

Let $\rho$, $\Sigma_p$, $C(\rho)$, $\Sigma(\rho)$, $\mathcal E(U^p)_{\overline\rho}$, and $\delta_J^c$ be as in the preceding setup. Partial classicality conjecture. For every subs…

Let $T_p$ be the relevant diagonal torus, let $\mathcal E(U^p)_{\overline\rho}$ be the eigenvariety attached to the definite unitary group, and retain the notation for…

Breuil's conjecture. For a continuous character $\chi$ of $T_p$ over $E$,

Let $x\in X_{\rm tri}^\square(\overline{r})$ be a crystalline strictly dominant point whose Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ are pairwise distinct. Let $w_x$ be…