48 problems

Let $F$ be a number field, let $\mathbb A_F$ be its ring of adeles, and let $\pi\in\mathfrak{F}_{2}$ have central character $\omega_{\pi}$. For $m\geq 2$, let…

Let $F$ be a number field, let $\mathbb A_F$ be its ring of adeles, and let $\mathfrak{F}_{n}$ be the cuspidal automorphic representations of $\operatorname{GL}_{n}(\mathbb A_F)$ w…

Let $F$ be the global function field and let ${}^LG$ and ${}^LH$ be the $L$-groups of a reductive group ${\bf G}$ and a quasi-split reductive group ${\bf H}$ over $F$, respectively…

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…

Conjecture of base change. The representation of $GL(p+q,\mathbb{C})$ obtained from $\pi_1$ by base change has the same infinitesimal character as

Langlands functoriality conjecture. There exists an irreducible automorphic representation $\sigma$ of $G$ whose Hecke–Satake family agrees with $\iota(c(\pi))$ at almost all place…

Let $K/\mathbb Q$ be an imaginary quadratic extension, let $\chi$ be a conjugate-symplectic Hecke character of $K$ with $L(\frac12,\chi)\neq0$, and let $f$ be the associated dihedr…

Let $F$ be a number field, let $\mathbb A_F$ be its ring of adeles, and let $\pi\in\mathfrak{F}_{2}$. For each place $v$ of $F$, let $\pi_v$ be the local component of $\pi$, and le…

Let $\mathbf{F}_{4}$ be the split group under consideration, let $\mathbf{Spin}_{9}\subset\mathbf{F}_{4}$ be the maximal proper regular closed subgroup, and define the homogeneous…

Let $pi$ be an automorphic representation on $\operatorname{GL}_m$, and let $\operatorname{Ad}pi$ denote its adjoint lift. Adjoint-lift automorphy conjecture. For $m>2$, the adjoin…

Let $H$ and $G$ be groups with $L$-groups ${}^LH$ and ${}^LG$, and suppose that $G$ is quasi-split. A morphism … should induce a functorial lifting of representations from $H$ to…

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…

Fray functors' 2-functoriality conjecture. For each composition $\boldsymbol{\lambda}\vdash n$,

Langlands' symmetric power lifting conjecture. The representation $\operatorname{Sym}^{m}(\pi)$ is an automorphic representation of $\mathrm{GL}_{m+1}(\mathbb{A}_F)$, possibly not…

Functoriality conjecture. The representation

Let $B$ be a category, let $B^*$ denote its relevant indexing category, and let $\Phi,\Psi:B^*\to\operatorname{\mathbf{Cat}}$ be $\pi_2$-indexings. Write \textbf{pi2…

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…

Let $G$ be a group of type $G_2$, let $\sigma$ be a cuspidal automorphic representation of $G(\mathbf{A})$, and let $\pi$ be its lift to $\operatorname{PGSp}_6(\mathbf{A})$. Let…

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 a level 1 RACAR of $\mathrm{GL}_3$ mean a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_3$ of conductor $1$. A representation is self-dual if…