611 problems

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$, and let…

For every number field $F$, prime $p$, Hilbert modular variety $X/F$, and cohomological degree $i$, the locally analytic $p$-adically completed cohomology $\widetilde{H}^{i}(X)$ sh…

Let $G_n$ be an even orthogonal group over a non-Archimedean local field and let $\psi$ be a local Arthur parameter for $G_n$. Determine the local Arthur packet $\Pi_\psi(G_n)$ and…

For every degree $n$, the critical vectors occurring in the unconditional part of Chenevier's automorphic Hermite--Minkowski theorem are strictly positive in the associated orthogo…

For a nonarchimedean local field $F$ of characteristic zero and the relevant pair of representations $(\pi_{n+1},\pi_n)$ of $\mathrm{GL}_{n+1}(F)\times\mathrm{GL}_n(F)$, the associ…

For every integer $n\geq 2$ and every cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field, every unramified local component $\pi_v$ is tempered. Equiva…

Let $(G,X)$ and $(G',X')$ be Shimura data whose underlying groups are pure inner forms of one another, and let $\operatorname{Ig}(G,X)$ and $\operatorname{Ig}(G',X')$ denote the as…

For a family of tempered cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{Q})$ with bounded archimedean parameters and conductor $q(\pi)$ tending to infinity, d…

Let $F$ be a degree-$2$ Siegel cuspidal Hecke eigenform, with associated automorphic representation $\pi_F$ of $\mathrm{GSp}_4(\mathbb A)$, and let $g$ be an associated half-integr…

Let $S_k(N)$ be the space of Hilbert–Siegel cusp forms of weight $k$ and level $N$ on the split symplectic similitude group, let $S_k^B(N)$ be the corresponding space of algebraic…

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…

Weighted Langlands–Shelstad transfer conjecture. For each $G$ and $M$, there is a function $\hat{S}^{{\tilde G}'}_{{\tilde M}'}(\delta',f')$ on … such that

Selberg's eigenvalue conjecture. For every $N$,

Let $K$ be a number field, let $p$ be a rational prime unramified in $K$, and let $G_K$ denote the absolute Galois group of $K$. Let…

Exact dimension formula. The dimension is

Jacquet's local converse conjecture. If

Let $\mathbb{X}=\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, let $\{f_i\}$ be Hecke–Maass cusp forms normalized by … and define the Gaussian moments by…

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, a…

Kudla's arithmetic modularity conjecture. The formal generating function $Z(g,\varphi)_K$ converges absolutely and defines an element in

Lapid–Mao's local conjecture. For every $\eta\in E_v^\times$ such that $\bar\eta=-\eta$, one has

Sarnak–Xue conjecture. The multiplicity of $\pi$ satisfies

Let $F$ be a finite extension of $\mathbb Q$, let $\mathbb A_F$ be its ring of adeles, and fix a prime $p$ together with an isomorphism $\overline{\mathbb Q}_p\cong\mathbb C$. Cons…

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum with reflex field $E$, let $G$ be its associated reductive group, and let $\mu$ be the Hodge cocharacter. Let $(r_\mu,V)$ be the re…

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

Let $f(\tau)=\sum_{n=1}^{\infty}a_f(n)q^n\in S_\kappa(N,\omega)$ be a normalized elliptic newform of weight $\kappa\geq 2$, and let $\chi$ be a Dirichlet character. For a critical…