105 problems

Let $E/F$ be a finite extension, let $\rho:\operatorname{Gal}(E/F)\rightarrow GL_n(\mathbb{C})$ be a representation, and let $L(\rho,s)$ denote its Artin $L$-function. Langlands' f…

Let $G$ be a split classical group, and let $\pi$ be an isolated representation in the unitary dual of $G$. Automorphy conjecture. The representation $\pi$ is automorphic. The conj…

Shahidi Conjecture. For any quasi-split reductive group $G$, every tempered $L$-packet has a generic member.

Multiplicity and packet conjecture. For every $v\in S$, there is a representation $\Pi_v^-$ of $\mathrm{G}_2(\mathbb{Q}_v)$, different from $\mathcal{L}_\alpha(\pi_v,1/10)$, such t…

The local Langlands correspondence conjecture. There should be a local Langlands correspondence

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $\Gamma$ be a congruence subgroup, and let $\pi$ be an irreducible representation of $G$ occurring weakly in…

Assume the setup in which $G$ is simply connected and almost simple, $S$ is a set of places with at least two elements, and $T=\emptyset$. For each positive integer $m$, let…

Let $F$ be a non-Archimedean local field, let $G$ be the reductive group under consideration, and let $f^{EP}$ be the Euler–Poincaré function with respect to a Haar measure $dg$ on…

Prasad–Takloo-Bighash conjecture. Let $\pi$ be an irreducible admissible representation of $G$ with trivial central character. If $\pi$ is $H$-distinguished, then:

Sakellaridis–Venkatesh conjecture. A relative discrete series representation $\pi$ in $L^2(X)$ is contained in an Arthur packet corresponding to an $X$-distinguished $A$-parameter

Let $F$ be a non-archimedean local field, let $\mathbf{G}$ be a reductive group over $F$, and let $\mathbf{X}$ be a $\mathbf{G}$-spherical $F$-variety. Write $G$ for the group of…

Let $G$ be as above, and let $\pi=A_{\mathfrak q}$ be a cohomological representation of $G$ such that … where $\mathfrak l'_0$ is a compact Lie algebra and $\mathfrak g_1$ is a non…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$ obtained by restriction of scalars from a group of type ${}^3D_4$ or ${}^6D_4$ over a totally real number field, and supp…

Let $G$ be a semisimple group with discrete series. Let $\pi\in\widehat G$ be a cohomological representation, and consider the question of whether there is a lattice $\Gamma$ in…

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 $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra, and let $L\subset G$ be its associated Levi subgroup. Let…

Let $\mathcal O$ be a fixed closed orbit, and let $a_{n,\lambda}$ be the coefficients in the expansion associated with $\mathcal O$ and the representation parameter $\lambda$. Clos…

Let $(\pi,\pi')$ be fixed automorphic representations, let $\pi_{\lambda_i}$ denote the automorphic representation associated with the spectral parameter $\lambda_i$, and let $b_i$…

Fix a cocompact lattice $J\subset Z(\mathbbA)/Z(F)$ and an admissible $n$-tuple $\overline{\omega}$. Let … be the cuspidal intersection cohomology representation, and let…

Let $F$ be a non-Archimedean local field of characteristic zero, and let $G$ be a general connected reductive group defined over $F$. Let $\Pi_A(G)$ denote the set of Arthur repres…

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

Let $\varphi\colon W_F\to{{}^LG}({\overline{\mathbb{Q}}_\ell})$ be a semisimple $L$-parameter of Langlands–Shahidi type, and let ${\mathcal{F}}_\varphi$ be the corresponding Hecke…

Let $n$ be such that there is a cuspidal Hecke eigenform of weight $2n+12$ for $\operatorname{SL}_{2}(\mathbb{Z})$, and let $\pi$ be the associated level-one algebraic automorphic…

Mundy's conjecture. The representation $\Pi$ occurs in $L^2_{\rm disc}(G_2(\mathbb Q)\backslash G_2(\mathbb A))$ with multiplicity zero or one, and it occurs with multiplicity one…

Let ${\mathcal E}(\mathcal N^\bullet)$ be the eigenvariety constructed from the cohomological complex $\mathcal N^\bullet$ for $\operatorname{GL}_n/\mathbb Q$, and let a cuspidal c…