55 problems

Let ${\mathcal W}$ be the weight space, viewed as a union of open unit discs, and let ${\mathcal E}$ be the Coleman–Mazur eigencurve over ${\mathcal W}$. Coleman's spectral halo co…

A Bianchi modular form varies in a one-dimensional $p$-adic family over a two-dimensional weight space; a classical family is one containing infinitely many classical forms. Calega…

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…

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 $\mathscr{X}_{\mathbf{G},K^{p}}$ be the eigenvariety for $\mathbf{G}$ at tame level $K^{p}$, let $\mathscr{W}_{K^{p}}$ be its weight space, and let $l(\mathbf{G})$ be the defec…

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…

Patched eigenvariety product conjecture. It is conjectured that

Let $\mathbf{V}$ be as in the source, let $\mathrm{J}<\mathrm{I}^0$, and let $\mathscr{E}_\mathrm{J}(\xi,\mathbf{V})_\gamma$ be the corresponding eigenvariety algebra over the Iwas…

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 $\mathcal{W}^{rig}$ be the rigid-analytic weight space for the Coleman–Mazur eigencurve, and let $\mathcal{W}$ be its adic extension obtained by adjoining the characteristic-…

Let $\tilde\pi$ be an Iwahori $p$-refinement of a RASCAR of $\mathrm{GL}_4$. The refinement may be optimally $B$-spin, $Q$-spin, or $G$-spin, where $B$, $Q$, and $G$ are respective…

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 $G$ be either $\mathrm{GL}_3$ or $\operatorname{Res}_{F/\mathbb{Q}}\mathrm{GL}_2$, where $F$ is an imaginary quadratic field, and let $\alpha$ be a refinement. A refinement var…

For a spherical cuspidal representation of $\mathrm{GL}_4$ with its $p$-refinements, a symplectic family is understood modulo one-dimensional trivial variation. Symplectic-family d…

Breuil–Hellmann–Schraen's conjecture. For any irreducible component $\Xfrak^p$ of $\Xfraknotp$, the rigid analytic subvariety $\Xaut\subseteq\Xtrip$ is equal to $\Xtriptilde$. In p…

Let $F$ be a number field, let $E$ be a coefficient field, and let $\Gamma_F$ denote the absolute Galois group of $F$. Consider a semisimple representation … which is unramified ou…

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 $A/\textbf{Q}$ be a modular abelian surface, let $\pi$ be the associated automorphic representation of $\operatorname{GSp}_4/\textbf{Q}$, and let $p$ be a prime. Say that $\pi$…

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…