27 problems

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 $F$ 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 $X$ 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 $n\geq 2$, 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…

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…

Buzzard–Gee conjecture. The following are true. There exists a map

Let $K$ be a number field, let $(Y,D_Y)$ be a log smooth pair over $K$, and let $(E,\theta)$ be a Higgs bundle over $(Y,D_Y)/K$. The bundle $(E,\theta)$ is arithmetic if there is a…

Let $E/F$ be a quadratic extension with nontrivial Galois element $\sigma$. Let $G$ be a reductive group over $F$, let $G(E)$ be its base change, and let $G^{\mathrm{op}}$ be the qu…

Let $Z_e^{red}$ be the quotient of $Z_e$ by its unipotent radical. Let ${\bf Z}_e\subset (Z_e^{red})^2$ be the algebraic variety of pairs of commuting elements in $Z_e^{red}$. Let…

Let $F$ be a global field of characteristic different from $p$. Let $G$ be a connected reductive group, and let $G'$ be the split group over $F$ whose root datum is dual to that of…

Let $L$) be a number field and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_L)$ that is $L$-algebraic and regular. For a prime $p$ and an is…

Let $G$ be a connected reductive group over $\mathbb{Q}$, split over a number field $F$, and let $K$ be a level of $G$. Fix a prime $p$, let $f$ be a mod $p$ Hecke eigenform of lev…

Let $G_n=\operatorname{GSpin}_n$ and $G_{n+1}=\operatorname{GSpin}_{n+1}$, and let $\pi_n$ and $\pi_{n+1}$ be cuspidal automorphic representations with conjectural global Langlands…

Let $F$ be the function field of a smooth projective curve over a finite field, let $G$ be a split semisimple group over $F$, and let $u$ be a place. For a local representation…

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…

Let $E/K$ be a cyclic extension of global fields, and let $\rho: \Gamma_K \to \widehat{G}(\overline{\mathbb{Q}}_l)$ be a continuous, almost everywhere unramified homomorphism with…

The transfer conjecture. Suppose that $\xi$ induces an isomorphism