41 problems

Genericity conjecture. A pure $L$-packet $\Pi^{\text{pure}}_\phi(G)$ is generic if and only if $\phi$ is an open parameter for $G$.

Let $k$ be the number field and $G$ a $k$-split reductive group, with complex dual group $G^\vee(\mathbb{C})$. Let $\rho\colon G^\vee(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$…

Langlands' conjecture. Let $\rho$ be as in de Jong's conjecture. Then there exists a non-zero cuspidal Hecke eigen-form $f_\rho$ with eigenvalues corresponding to $\rho$.

Let $M=T^*X$ and $\check{M}=T^*\check{X}$ be a pair of dual polarized hyperspherical Hamiltonian varieties, where $X$ is a $G$-spherical variety and $\check{X}$ is the correspondin…

Ben-Zvi–Sakellaridis–Venkatesh duality conjecture. The dual of $\Delta$ is

Let $X$ be a curve with compactification $\ol{X}$, let $\sG$ be a reductive group, and let $G$ be its Langlands dual. Let … where $\Bun_G^{\on{level}}$ is the moduli space of $G$-b…

Automorphic Galois group conjecture.

Let $G$ be a reductive group, let $H$ be a subgroup equipped with a homomorphism $\boldsymbol{\rho}_H$ as in the BZSV setting, and let $\iota$ determine the dual homomorphism … For…

Let $G$ be a semisimple $\mathbb{Q}$-group admitting a reductive $\mathbb{Z}$-model. Let $\pi\in\Pi_{\mathrm{disc}}(G)$ and let $r:\widehat{G}\to\mathrm{SL}_{n,\mathbb{C}}$ be an a…

Let ${\mathbf G}^\omega({\mathsf k})$ be a reductive $p$-adic group, let $\pi(\lambda,s,{{\mathcal O}}^\vee,{{\mathcal L}}^\vee)$ be an irreducible representation in the $L$-packet…

Langlands correspondence conjecture. The alternatives and point-count identities above hold for every such eigenform.

Let $V$ be a representation of the Langlands dual group, and let $\scrZ^{\mathrm{mix}}(V)$ be the associated mixed perverse sheaf obtained from the mixed central functor. Its monod…

Let $\phi$ be a semi-simple $L$-parameter of weakly Langlands–Shahidi type with cuspidal support $(M,\phi_{M})$. Let $\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ be the corresponding Shim…

Let $V_f$ be the geometric Galois representation associated with a Hecke eigenform, and let $\phi_1,\ldots,\phi_{11}$ be the representations associated with the $11$ automorphic re…

Let $k$ be a number field, let $G$ be a $k$-split reductive group, and let $\rho\colon G^\vee(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$ be a finite-dimensional representation.…

Analytic continuation conjecture. La función $L(M,-)$ tiene una continuación meromorfa a todo $\mathbb{C}$.

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 $F$ 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 $K$ be a number field and let ${\mathfrak N}$ be an ideal of ${\mathcal O}_K$. Let ${\mathfrak f}$ be a non-trivial, new weight 2 complex eigenform over $K$ of level…

Adjoint $L$-function analytic continuation conjecture. The complete adjoint $L$-function $\Lambda(s,\pi,\operatorname{Ad})$ admits an analytic continuation to the whole complex pla…

Let $k$ be a global field and let $\Pi$ be a cuspidal automorphic representation of ${\rm GL}_n(\mathbb{A}_k)$. Symmetric-power automorphy conjecture. For every $n$, the symmetric-…

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 number field, let $\mathbb{A}$ be its ring of adeles, let $G$ be a connected linear algebraic group defined over $F$, let $\pi$ be an automorphic representation of…

Low-rank reduction conjecture. An irreducible admissible representation $\pi$ of $\operatorname{GL}_n({\mathbb R})$ occurs as a Jordan–Hölder factor of $Ps(\underline{\pi})$ if and…

Let $G$ be a reductive algebraic group over a local field $F$. For irreducible admissible representations of $G(F)$, define $\pi_1\geq\pi_2$ when $\pi_1$ appears as a Jordan–Hölder…