8 problems

Let $F$ be a CM field, let $\Sigma$ be a CM type, and let $F^+$ be the maximal totally real subfield of $F$. For each signature $I:\Sigma\to\{0,1,\ldots,n\}$, let $U_I$ be the corr…

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…

Assume that $\check{M}=T^*\check{X}$, where $\check{X}$ is a $\check{G}$-spherical variety. Let $\pi$ be a tempered, everywhere unramified automorphic representation of…

Let $(M_A,N_A)$ be a relevant pair of global A-parameters satisfying the automorphy conditions for the representation $\pi_\chi$, and let $F(\nu)$ be the regularized period integra…

Let $M_A\times N_A$ be a discrete A-parameter of the quasi-split group $G(\mathbb{A})=G(V)(\mathbb{A})\times G(W)(\mathbb{A})$, with decompositions … … Let $(I\times J)^\heartsuit$…

Let $G=G(V)\times G(W)$ over a number field, let $\pi\otimes\pi'$ be an irreducible automorphic representation occurring in $\mathcal{A}_{\rm aut}(G)$, and let $F(\nu)$ denote the…

Let $F$ be a totally real number field, let $n'=n$ or $n'=n-1$, and let $\pi=\otimes'_v\pi_v$ and $\pi'=\otimes'_v\pi'_v$ be irreducible cuspidal automorphic representations with t…

Let $E$ be a totally real field, let $J_E$ be its set of real embeddings, and let $\bot$ be a quaternion algebra over $E$. Let $$ be a cuspidal automorphic representation of…