21 problems

Prasad–Takloo-Bighash conjecture. If $\pi$ is $(H,\chi^{-1})$-distinguished, then the Langlands parameter of $\pi_0$ takes values in $\operatorname{GSp}_{2n}(\mathbb{C})$ with simi…

Let $F$ be a local field of characteristic zero, let $E/F$ be a quadratic étale algebra, and let $\psi$ be a non-trivial character of $F$. Let $\eta_{E/F}$ be the quadratic charact…

Let $p$ be an odd prime, let $L$ be a finite extension of $\mathbb Q_p$, and let $T_1$ and $T_2$ be symplectic self-dual $\mathcal O_L$-representations of $G_{\mathbb Q_p}$. Put ……

Geometric epsilon-factor conjecture. Under these hypotheses,

Assume $F$ is non-archimedean. Let $\phi\colon WD_F\to\mathrm{GL}_n(\mathbb{C})$ be an $L$-parameter, let $\Pi_\phi$ be the packet across all inner forms of $\mathrm{GL}_n(F)$, and…

The epsilon-character conjecture. The function $\omega_{\phi',\rho_X}$ is well defined, independent of the extended endoscopic triple and the lifting, descends to a function on…

Let $(G,H)$ be one of the smaller models $(\operatorname{GSpin}_{10}\times\operatorname{GSpin}_3,\operatorname{GSpin}_3\ltimes U)$,…

Let $G=\operatorname{GSp}_4\times\operatorname{GL}_2\times\operatorname{GL}_2$ with $H=(\operatorname{GL}_2\times\operatorname{GL}_2)^0$, let $(G_D,H_D)$ be its quaternionic inner…

Let $G=\operatorname{GL}_6$, $H=\operatorname{GL}_2\ltimes U$, and let $\chi'$ be defined by $\chi'(\operatorname{diag}(h,h,h)u)=\alpha(\det h)\xi(u)$. Let $\pi$ be irreducible tem…

Let $G=\operatorname{GL}_6$ with the model $H=\operatorname{GL}_2\ltimes U$, let $\rho_X=\wedge^3$, and let $\pi$ be an irreducible tempered representation with central character t…

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…

Fix a local field $F$, a model $(G,H)$ in the paper's table, a tempered $L$-parameter $\phi$, its component group $S_\phi$, and the representation $\rho_X$ attached to the spherica…

Langlands's conjecture. The global $\varepsilon$-factor $\varepsilon_{\overline{k}}(X,\mathcal{F})$ should split as a product of local contributions. This conjecture is the one-dim…

Sign conjecture. With the above notations, $G(Q)$ is $\pm1$.

Let $c_f$ be the constant relating the exterior square epsilon factor arising from the Jacquet–Shalika integral representation to the exterior square gamma factor defined using the…

Epsilon dichotomy conjecture.

Let $S$ be a regular Noetherian scheme over $\mathbb Z[1/\ell]$ and let $f\colon X\to S$ be a smooth proper morphism purely of relative dimension $n$. Let…

Let $F$ be a local field, let $\beta$ be a character of $F^{\times}$, and let $\Pi$ be an irreducible generic representation of $\mathrm{GL}_6(F)$ with central character $\beta^2$.…

Let $F$ be a local field, let $\pi$ be an irreducible admissible representation of $GL_6(F)$ with central character $\chi^2$, and let $\pi_D$ be its local Jacquet–Langlands corresp…

Let $F$ be a local field and let $\pi$ be an irreducible admissible representation of $\operatorname{GL}_6(F)$ satisfying the assumptions of Jiang's multiplicity-one conjecture, wi…

Let $F$ be a local field of characteristic $0$, let $\mathbb{M}$ denote the relevant metaplectic group, and let $c_\pi$ be the proportionality constant between the two local functi…