108 problems

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, and let $\chi$ be a non-trivial irreducible character of $\operatorname{Gal}(K/\mathbb Q)$. Write $\chi(1)$ fo…

Ben-Zvi–Sakellaridis–Venkatesh period conjecture. The period integral associated with $\Delta$ should be equal to the $L$-function associated with $\hat{\Delta}$, and conversely th…

Let $L$ be an automorphic L-function with associated representation $\rho$ of the conjectural Langlands group, and let its Sato–Tate group be the compact subgroup…

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…

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

Langlands conjecture. It is conjectured that $L^S(s,\pi,r)$ has an analytic continuation and satisfies a functional equation.

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 $F$ 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…

Langlands's conjecture concerns symmetric-power $L$-functions and certain automorphic $L$-functions. Langlands's conjecture. Some symmetric-power $L$-functions should extend to ent…

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…

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…

An Artin representation is a finite-dimensional complex representation of a Galois group, and an automorphic representation is a representation of an appropriate general linear gro…

Let $E$ be a finite extension of $\mathbb Q$ with Galois group $Gal~(E|\mathbb Q)$, and let $\sigma_{n+1}:Gal~(E|\mathbb Q)\to GL_{n+1}(\mathbb C)$ be an irreducible representation…

Arthur parameter conjecture. Every Langlands parameter $\sigma$ occurring in the canonical decomposition is of the form $\sigma_\psi$ for an elliptic Arthur parameter $\psi$. This…

Langlands correspondence conjecture. There exists a correspondence between these rank-$n$ $l$-adic representations and the Hecke-eigenvector automorphic forms on $GL_n(A)$.

Let $F$ be a local $p$-adic field and let $\mathbf{H}$ be a reductive group over $F$. Write $W_F$ for the Weil group of $F$ and $L_F$ for the corresponding Langlands group, and let…

Langlands reciprocity conjecture. The function $L(s,\sigma)$ is the $L$-function associated to an automorphic representation of $GL(n,{\mathbb A})$.

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 an arithmetic object, such as an elliptic curve, have an associated $L$-series. Arithmetic L-series conjecture. Such $L$-series are conjectured to be entire and to satisfy func…

Motives are expected to have associated zeta functions, and automorphic representations are related to zeta functions through the Langlands correspondence. Langlands conjecture. Th…

Let $L(\pi,s)$ be the automorphic $L$-function attached to an automorphic representation $\pi$ of ${\rm GL}_n(\mathbb A)^1$. A motivic $L$-function is an $L$-function arising from…

For every $m\geq 1$, let $\mathcal D_m$ be the class of Dirichlet series described above, and put $\mathcal D=\bigcup_{m\geq 1}\mathcal D_m$. An $L(s)\in\mathcal D$ is primitive if…

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…

Let $X$ be a smooth, proper Deligne--Mumford stack over $\operatorname{Spec} \mathbb{Z}$, such as $\overline{\mathcal{M}}_{g,n}$, and let $k\leq 22$. Write…