22 problems

Jacquet's nonvanishing conjecture. The central value $L(1/2,\pi_1\times\pi_2\times\pi_3)$ is non-zero if and only if there is a quaternion algebra $D$ over $F$ such that

Let $D$) be the quaternion algebra over $\mathbb{Q}$ that is non-split over $\mathbb{R}$ and $\mathbb{Q}_p$ for a prime $p$ and split at all other places. Let $G=GU_2(D)$ be the as…

Carayol's conjecture. In each setting, the construction realizes simultaneously the Jacquet–Langlands and Langlands correspondences: a tensor product…

Extended Jacquet–Langlands conjecture. Every smooth irreducible representation $c0'=Lg(c3')$ of $G'$ corresponds to the representation $c0=Lg(c3)$ of $G$ specified in the source.

Let $F$ be the $p$-adic field and let $G=\mathrm{GL}_n(F)$, with $D$ the central division algebra over $F$ of invariant $1/n$. For an admissible smooth $G$-representation $\pi$ ove…

Let $G=\operatorname{GL}_4\times\operatorname{GL}_2$, let $H=\operatorname{GL}_2\times\operatorname{GL}_2$ be embedded by $(a,b)\mapsto(\operatorname{diag}(a,b),b)$, and let…

Let $(G,s,\xi)$ be an endoscopic triple for $G_2$, let $\phi:W'_F\to{}^LG$ be an unramified Langlands parameter that is $\xi$-conormal, and let $\Theta^G_\phi$ denote the associate…

Let $E\subset\overline{\mathbb{Q}_p}$ be a finite extension of $\mathbb{Q}_p$, and let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}_2(E)$ be an odd continuous absolutely irreducible re…

Let $G$ be a reductive group, let $\fs$ be a supercuspidal Bernstein component of $G$ containing a pair $(\theta,\sigma)$, and let $\JL(\fs)$ denote its Jacquet–Langlands transfer.…

Fix a prime number $p$. Let $F$ be a totally real field, set $g=[F:{\mathbb Q}]$, and let ${\mathfrak p}$ be a prime ideal of $F$ above $p$. Let $f_\infty$ be a Hilbert eigen newfo…

Endo-class invariance conjecture. For any essentially square-integrable, irreducible complex representation $\pi$ of ${\rm G}$, the endo-classes of $\pi$ and ${}_{{\rm JL}}\pi$ are…

Let $\varphi:W_E\to\operatorname{GL}_n(\overline{\mathbb{Q}}_\ell)$ be a cuspidal discrete parameter, let $\pi$ be the corresponding square-integrable representation of…

Let $F$ be a totally real field, let ${\mathcal P}$ be a place at which both quaternion algebras split, and let $G_1$ and $G_2$ be the multiplicative groups of two quaternion algeb…

Let $D$ be a central division algebra over a number field $k$, let ${\mathbf{G}}={\mathrm{GL}}_2(D)$ and let ${\mathbf{G}}'={\mathrm{GL}}_{2n}$ be the corresponding inner form with…

Let $k$ be a number field with ring of adeles ${\mathbb {A}}$, let $D$ be a central division algebra over $k$ of index $d$, and set ${\mathbf{G}}={\mathrm{GL}}_{2m}(D)$ and…

Let $A$ be the coefficient ring of a global function field, let $\mathfrak n=\mathfrak p\mathfrak q$ be a product of two distinct prime ideals of $A$ with $\deg(\mathfrak p)\leq 2$…

Let $\mathfrak{m}$ be a maximal ideal of the Hecke algebra, with associated Galois representation $\bar{\rho}$. Let $\mathbf{F}[\mathfrak{m}]$ denote the corresponding Hecke-isotyp…

Let $\mathfrak{m}$ be a maximal ideal of a Hecke algebra corresponding to a modular mod $p$ Galois representation $\bar{\rho}$. Let $\sigma_{\mathfrak{m}}$ be the associated candid…

Let $Y(K_{\Sigma})$ and $Y'(K_{\Sigma})$ be a Jacquet–Langlands pair, let $\mathbf{T}$ be the relevant Hecke algebra, and let $\mathfrak{m}$ be a non-Eisenstein maximal ideal. Writ…

Let $G$ be the quaternionic inner form of $\operatorname{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$ of discriminant $\mathfrak d$, let…

Let ${\rm G}={\rm GL}_m(D)$ and ${\rm H}={\rm GL}_{md}(F)$, where $D$ is a central division algebra over the non-archimedean local field $F$ of reduced degree $d$. Let…

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…