27 problems

Let $V\in {\mathscr{V}}^{\circ}$ be an incoherent pair, and let $\pi$ be a distinguished, stable, ordinary, cuspidal automorphic representation of $\mathrm{G}^{V}(\mathbf{A})$, tri…

Let $L/\mathbb Q$ be an imaginary quadratic extension with discriminant $-D$, let $\Lambda$ be a character of $L^\times\backslash\mathbb A_L^\times$ trivial on $\mathbb A^\times$,…

Gan–Gross–Prasad conjecture. If $n-m$ is odd, a nonzero Bessel period $\mathcal{B}_{k,\psi}^{\varepsilon}$ on $\pi_1 \boxtimes \pi_2$ implies that $(M,N)$ is a relevant pair. For a…

Global Gan–Gross–Prasad conjecture. If $n-m$ is odd, then a nonzero Bessel period $\mathcal{B}$ on $\pi_1 \boxtimes \pi_2$ implies that $(M,N)$ is a relevant pair. When $(M,N)$ is…

Let $F$ be a $p$-adic field, let $E$ and $K$ be quadratic field extensions of $F$, and let $V$ be an $n$-dimensional skew-hermitian space relative to $E/F$. Set $V_K=V\otimes_F K$,…

Let $\sigma=\otimes_v\sigma_v$, $\Sigma=\otimes_v\Sigma_v$, and $\pi=\otimes_v\pi_v$ be the cuspidal representations corresponding to $\mathbf{f}$, $\mathbf{F}$, and $\mathbf{g}$,…

Let $E/F$ be a quadratic extension of number fields, let $V$ be an $n$-dimensional skew-Hermitian space over $E$, and let $\U(V)$ be its unitary group. Let $\pi$ be an irreducible…

Let $pi in \mathrm{Irr}(G_{n+1})$ and $pi' in \mathrm{Irr}(G_n)$. A pair $(pi,pi')$ is relevant if there exist multisegments $\mathfrak m,\mathfrak n$ such that…

Let $F$ be a non-Archimedean local field of characteristic $0$, let $E\neq K$ be distinct quadratic extensions of $F$, and put $L=E\otimes_F K$. Let $V$ be an $n$-dimensional skew-…

Unrefined arithmetic Gan–Gross–Prasad conjecture. The following are equivalent: (a) $\operatorname{FJ}^{\natural}_\varepsilon\neq0$; (b)…

Let $(W,V)$ be a pair of quadratic spaces over a local field $F$ of characteristic zero, with $W$ a subspace of $V$ such that $W^{\perp}$ is odd-dimensional and split. Let…

Let $F$ be a local field. For an Arthur type representation, let its associated Arthur parameter be a finite representation of $WD_F\times \mathrm{SL}_2(\mathbb C)$; two such param…

Assume the setup of the paper: $\Pi_0$ and $\Pi_1$ are the relevant representations under consideration, $F$ is the CM field, $E$ is their coefficient field, and the notation…

Let $F$ be a field with Weil group $W_F$. For $r\in\mathbb{Z}_{>0}$, let an Arthur parameter for $GL_r(F)$ be a completely reducible representation … where each $\psi_i$ is an irre…

Ichino–Ikeda conjecture. For every pure tensor $\phi=\bigotimes_v\phi_v\in\pi$,

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…

Semi-global arithmetic intersection conjecture. If $v_0$ is non-archimedean of hyperspecial type and $f'_{v_0}=\mathbf 1_{G'(O_{F_0,v_0})}$, then

Global arithmetic intersection conjecture at nontrivial level. One has

Global arithmetic intersection conjecture at trivial level. One has

Arithmetic Gan–Gross–Prasad conjecture. Conditions (a) $\dim\operatorname{Hom}_{{\mathscr H}_K}(\pi_f^K,{\mathcal Z}_{K,0})=1$ and (b) $\operatorname{ord}_{s=1/2}L(s,\pi,R)=1$, tog…

Let $\mathbb F$ be a number field, let $\mathbb E/\mathbb F$ be either trivial or quadratic, and let $\mathscr G(\mathcal V)$ and $\mathscr G(\mathcal W)$ be the groups attached to…

Let $\phi_n \in \Phi_\mathrm{temp}(\mathrm{U}_{n}(\mathbb{R}))$ and $\phi_{n+1} \in \Phi_\mathrm{temp}(\mathrm{U}_{n+1}(\mathbb{R}))$ have decompositions … … where…

Let $E/F$ be a quadratic extension of local fields of characteristic zero, let $W$ be an $n$-dimensional hermitian space over $E$, and let $V=W\oplus^\perp e$ with $(e,e)=1$. Set…

Special case of global Gan–Gross–Prasad. If there is an automorphic form $\phi$ in the space of $\pi$ such that $B(\phi,\Lambda)\neq0$, then

Let $G_n^*$ and $H_m^*$ be a relevant pair of $F$-quasisplit classical groups, let $R_{{\mathcal {O}}_\ell}(F)=H_m(F)\ltimes V_{\underline{p}_\ell}(F)$ be the Bessel subgroup attac…