30 problems

Let $F$) be a totally real field in which $p$ is inert, let $E/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let…

Let $F$ be a totally real field, let $v$ be a finite place of $F$ dividing $ell$, and let $\pi$ be a cuspidal representation of $\mathrm{GL}_2(\mathbb A_F)$ of infinity type…

Assume that $\rho_{\pi}$ is absolutely irreducible and that $\rho_{\pi,\widetilde{\wp}}$ is crystalline, with regular filtered $\varphi$-module…

Let $k\geq 2$ and let $\pi=\pi_p\otimes\pi^{(p)}\subseteq S_{k,\mathbb{C}_p}$ be an irreducible representation of $\mathrm{GL}_2(\mathbb{A}_f)$. Let … Write $\rho_p$ for the restri…

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…

Let $F$ be an imaginary CM field with maximal totally real subfield $F^+$, let $G_{/F^+}$ be the reductive group in the stated global setup, and let…

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…

Integral local-global compatibility conjecture. The natural map from the unrestricted local deformation ring to the completed Hecke algebra factors through

Assume the hypotheses imposed in the definitions of the representations in,, and, and let $\pi$ be one of these representations with $\pi\ne0$. Let $\mathcal C$ be the stated categ…

Let $K$ be an unramified local field, let $A$ be the coefficient ring, and let $\pi$ be an automorphic smooth representation of ${\rm GL}_2(K)$ over $\mathbb{F}$. Let $D_A(\pi)$ de…

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 $\lambda$, $W$, $\alpha=e_j-e_{j+1}$, $D_\alpha(\lambda,W)$, $F_\alpha$, $\pi^\infty(W)$, $L(-\lambda)$, and the global data $U^\wp$, $W^\wp$, and $\rho$ be as in the preceding…

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 $n$, $E$, $\rho_p$, $(h_i)_{i\in\{1,\dots,n\}}$, $W$, $\alpha=e_j-e_{j+1}$, $\pi^\infty$, $\chi$, $\lambda$, $F_\alpha$, and $D_\alpha(\pi)$ be as in the preceding discussion,…

Let the Galois representations constructed by Scholze be those associated with the torsion in the cohomology of locally symmetric spaces as in the cited work. Scholze's strong loca…

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

Let $F$ be the paramodular Hecke eigenform under consideration, let $p$ be the relevant prime, and let $\rho_{F,p}$ be its local Galois representation. For each auxiliary place…

Local-global compatibility conjecture. The perverse sheaves $R\pi_{HT*}\overline{\mathbb Q}_\ell$ on $\mathscr{F}\!\ell_{G,\mu}$ should be related to the conjectural perverse sheav…