23 problems

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM, with $\pi_\infty$ essenti…

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $(\pi,\chi)$ be a RAESDC automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM. Symmetric power func…

Let $r\geq 1$. Let $F$ be a CM field, and let $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ be RAECSDC automorphic representations of $\mathrm{GL}_2({\mathbf A}_F)$ and…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $L$ be the Levi subgroup associated with a $\theta$-stable parabolic subalgebra $\mathfrak{q}$, let $\tau$ be a unit…

Let $bbalpha:bbG\tobbG'$ be an isomorphism of connected reductive groups, let $bbpi\inbbPi_{bbC}(bbG)$, and let ${}^{bbalpha}bbpi$ be the irreducible representation of $bbG'$ obtai…

Functoriality conjecture. The representation

Let $G=\mathrm{GL}_{2n}$, let $P\subsetneq G$ be a proper spin parabolic with associated parabolic $\cP\subset\mathrm{GSpin}_{2n+1}$, and let $\tilde\pi^P=(\pi,\alpha^P)$ be a $P$-…

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_4$, and let $P_{\tilde\pi}$ be its associated parabolic. A classical family is a family through $\tilde\pi$ consisting of cl…

Let $\tilde\pi$ be a $p$-refined RASCAR of $\mathrm{GL}_{2n}$, and let $P_{\tilde\pi}$ be the associated parabolic. A symplectic family is a family through $\tilde\pi$ in the eigen…

Let $A$ act on the rigid inner twists of a connected reductive group $G$, on tempered refined Langlands parameters, and on the set $\Pi_{\mathrm{temp}}$ of tempered representations…

Let $B\to A$ be a map of finite groups acting on a connected reductive group $G$, and write $G^A=G\rtimes A$ and $G^B=G\rtimes B$. Let $\pi^A$ correspond to $(\phi,\rho^A)$ for…

Let $F_1$, $F_2$, and $F_3$ be the functors defined above from enhanced log Higgs bundles to arithmetic Higgs bundles, with $F_3$ the direct construction and $F_2\circ F_1$ the con…

Functorial lifting conjecture. There is a self-dual automorphic representation $\widetilde\Pi$ of $GL_7(\mathbb A)$ satisfying all five listed properties: matching unramified Satak…

Let $K/k$ be a quadratic extension of number fields, let $\mathbb{H}^\circ$ be a $k$-quasisplit classical group, and let $T$ be the functorial lift to…

Functorial transfer conjecture. The representation $\iota(\pi)$ is cohomological if $\pi$ is one of the following: the trivial representation; a discrete series representation of…

Let $S$ be the surface supporting the tangled webs, let $\mathrm{S}\boldsymbol{\mathrm{Tan}}$ be the category of tangled webs and framed foams in four-dimensional space, and let…

Precise pullback conjecture. For every $(\phi,\rho)\in\Phi_e(G)$,

Borel's pullback conjecture. Suppose that a local Langlands correspondence exists for sufficiently large classes of representations of $G$ and $\tilde G$. Then

Let $\psi:P_F\rightarrow {^{L}\mathbf G}$ be a wild parameter, and let $\mathbf G_\psi$ be the associated tamely ramified group of ${\rm GL}$-type with…

Let $\mathbf{G}'$ and $\mathbf{G}$ be groups of ${\rm GL}$-type with an $L$-homomorphism $\xi:{^{L}\mathbf{G}'}\rightarrow {^{L}\mathbf{G}}$. Let…

Let $F$ be a number field, let $\mathbb{A}_F$ be its ring of adeles, let $G$ be a reductive group over $F$, and let ${}^{\lambda}G$ be the Zariski closure of the image of ${}^{L}G$…

Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbf{A}_F)$ that is not automorphically induced from any finite extension…

Let $G$ and $G'$ be split almost simple groups over $k$ whose dual groups satisfy $\check{G}\supset\check{G}'$ and appear in the same line of the paper's global-monodromy table. Le…