51 problems

Let $\Pi$ be an algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A)$, and let $\mathbb Q(\Pi)$ be its rationality field. Clozel's conjecture. There exi…

Let $G$ be a reductive algebraic group over a local field $F$ that is a pure innerform of a quasi-split group $G_0$ over $F$, equipped with a fixed triple $(B_0,N_0,\psi_0)$ used t…

Fargues–Scholze categorical conjecture. This setting enables a version of the Fargues–Scholze categorical conjecture that includes $\textnormal{Kal}_v$-inner forms.

Fix an isomorphism $\iota:\overline{\mathbb{Q}}_\ell\simeq\mathbb{C}$. Let $F$ be a number field. Consider L-algebraic isobaric automorphic representations $\Pi$ of…

Let $X$ be a smooth projective curve over $\mathbb{F}_q$, let $n$ be a positive integer, and let $E$ be a smooth geometrically irreducible $\mathbb{\bar Q}_\ell$-sheaf of rank $n$…

Carayol's conjecture. In each setting, the construction realizes simultaneously the Jacquet–Langlands and Langlands correspondences: a tensor product…

Langlands correspondence for $\mathrm{GL}_2/K$. The representation $\rho_{\lambda}$ exists as above, is potentially semistable at every place $v$, and its associated representation…

Automorphic-to-Galois compatibility conjecture. There is an irreducible geometric strongly compatible system $\mathcal{R}$ of l-adic representations such that

Let $X$ be a smooth projective absolutely irreducible curve over $\mathbb{F}_q$, let $G$ be one of the groups considered in the standard Langlands-dual setting, and let $E$ be a…

Global sections equivalence conjecture. The functor

Let $F$ be a number field, let $n\geq 1$, and let $G_F=\operatorname{Gal}(\overline{F}/F)$. Let $\pi$ be a cuspidal, algebraic automorphic representation of…

Temperedness conjecture. The set $\Pi_\phi$ always consists of irreducible, admissible, and tempered representations of $G(\mathbb A_F)$.

Distinguished-representation moment map conjecture. If $Z(\pi)^\vee$ is $X$-distinguished, then

Let $G$ be a reductive group over the finite field $k$, let $G^{\vee}$ be its dual group, and let $\ell$ be a prime distinct from the characteristic of $k$. Write…

Let $L/\mathbf Q$ be a finite extension and let … be a two-dimensional representation. Call $\rho$ odd when the determinant of $\rho(c)$ is $-1$ for every complex conjugation $c$.…

Reeder–Yu's prediction. The Swan conductor of the adjoint Galois representation associated with the Langlands parameter of $\pi_{\infty}$ should satisfy the displayed formula. This…

Let $X$ be as in the paper, let $n\geq 2$, let $G(X)$ be a reductive group, and let $\Pi_X[\log(1+T)^{1/n}]$ be the generalized Robba ring in the imperfect setting. Let…

Let $X$ be a finite extension of a local nonarchimedean field, and consider the rigid analytic stack constructed in the cited work of EGH. Generalized rigid analytic stack conjectu…

Let $n\geq 2$, let $X$ be a finite extension of a local nonarchimedean field, let $G(X)$ be a reductive group, and let $\mathrm{Gal}_{X,n}$ and $\mathrm{Weil}_{X,n}$ denote the cor…

Let $X$ be a finite extension of a local nonarchimedean field, let $G(X)$ be a $z$-adic reductive group, and let $\mathrm{Gal}_{X,2}$ denote the two-fold covering of the Galois gro…

Let $K$ be a finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_q((z))$ with $\ell\neq p$ and $p>2$. The source considers motivic Schur-irreducible sheaves in the derived categories…

Mixed-parity Hecke conjecture. In the mixed-parity setting this Hecke operator can be defined, its image lies in

Let $K$ be a finite extension of $mathbb{Q}p$ or $mathbb{F}q((z))$ with $ellneq p$ and $p2$, let $G/K$ be reductive, and let $W{K,2}$ denote the indicated two-fold covering grou…

Gross–Prasad–Rallis conjecture. A Langlands parameter is generic if and only if its adjoint $L$-function is holomorphic at $s=1$.

Let $G$ be a connected reductive group over a global function field $K$, let $U\subset G(\mathbf A_K)$ be open compact, let $E$ be a noetherian $\mathbb Z[\frac1p]$-algebra embedde…