16 problems

Let $E$ be the quadratic field under consideration, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, and write $\mathbf{T}:=h_\mathfrak{m}$. Let … be the Galois repre…

Assume Conjecture 5.4. Let $\mathfrak{m}\subset \mathbf{T}^{T,\lambda,\tau}(K^p)$ be a non-Eisenstein maximal ideal. For $v\in S_p(F)$, set…

Let $K=F_v$ be unramified, let $\pi$ be an automorphic admissible smooth representation of ${\rm GL}_2(K)$ over $\mathbb{F}$, and let $\overline{r}_v$ be its corresponding local Ga…

Let $F$, $S$, $T$, $Q$, $\Sigma$, the Hecke algebras $\mathbb{T}_{\Sigma,Q}$ and $\widetilde{\mathbb{T}}^S(K_{\Sigma,Q})$, and the universal deformation rings $R_{\Sigma,Q}$ be as…

Let $F$ be a number field, let $\pi=\otimes'_v\pi_v$ be an algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$, let $l$ be a rational prime, and fi…

Let $\overline{r}:{\operatorname{Gal}}(\overline F/F)\rightarrow {\operatorname{GL}}_n(\mathbb{F})$ be a continuous homomorphism satisfying conditions (i)–(iii) in the paper, and l…

Let $\Pi$ be a cuspidal automorphic form on ${\mathrm{GL}}_{2n}({\mathbb A}_{\mathbb Q})$ with infinitesimal character $\chi_H$, where $H$ is a multiset of distinct integers. A str…

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 $E\subset\overline{\mathbb{Q}_p}$ be a finite extension of $\mathbb{Q}_p$, and let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}_2(E)$ be an odd continuous absolutely irreducible re…

Local-global compatibility conjecture. The restriction morphism is bijective:

Let $F$ be a CM field in which $p$ is unramified, let $w$ be a place of $F$ above $p$, and let…

Local-global compatibility conjecture. The action of $R_p^{\operatorname{loc}}$ on

Fix a prime $u mid p$, a nonempty proper subset $J e u$ of the set $u$-adic embeddings, an integer $w$, and integers $k{\tau}2$ with $k{\tau} wpmod{2}$ for every $\tau otin J$. Let…

Let $B(\rho_x)^{\circ}$ and $\mathcal{B}_x^{\circ}$ be the lattices in the $p$-adic constructions, let $M^{\circ}$ be the specified lattice in…

Let $F$ be the relevant number field, let $K^p$ be a tame level with split ramification, and let $x\in\mathbb{X}(L)$ be a regular classical point such that $\rho_x$ is residually i…

Eigencurve intersection conjecture. If $x\in\mathcal{E}$ is such that $\pi(\rho_{x,l},N_x)$ is one dimensional, then there exist irreducible components…