13 problems

Fix a positive integer $N$ and a prime $p mid N$. For each integer $k$, let $M{k,N}=Mk(Gamma1(N)capGamma0(p))$, let $Up$ be the Hecke operator on this space, and let $P{k,N}(X)$ be…

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…

Hida's codimension conjecture. We have

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 a representation be non-essentially-self-dual if it is not essentially self-dual, and let a classical $p$-adic family mean a $p$-adic family whose members include classical aut…

Let $\mathcal X$ be sufficiently small, let $W(f)$ be the global root number of the family, and let ${}^{\rm c}V_\mathcal X$ and ${}^{\rm c}V'_\mathcal X$ be the relevant critical…

Let $\mathcal X$ be a sufficiently small positive-slope family of classical modular forms, let $W(f)$ be the global root number of the family, and let $f_x$ denote the specializati…

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…

Ash–Pollack–Stevens conjecture. Outside of the essentially self-dual case, arithmetic rigidity holds.

Let $X$ be a parameter space with a Zariski-dense subset $\Sigma$ of closed points. Let $\pi$ and $\widetilde{\pi}$ be finitely generated smooth admissible torsion-free…

Let a Coleman–Mazur eigencurve be the rigid-analytic parameter space of finite-slope $p$-adic eigenforms, equipped with its family of Galois representations. Hansen's conjecture. T…

Let $F$ be an imaginary quadratic field and let $p$ be a split prime. Suppose that $\rho:\operatorname{Gal}(\overline{F}/F)\to\operatorname{GL}_2(E)$ is continuous, irreducible, ne…

An automorphic form for $\operatorname{GL}_n$ is non-essentially self-dual if it is not essentially self-dual, and it is rigid if it cannot be placed in any nontrivial positive-dim…