34 problems

Let $\mu$ be a splitting character used to define the oscillator–Heisenberg representation, and let…

Let $G_n$ be embedded in $G_{n+1}$ by $g\mapsto \left(\begin{array}{cc}g&\\&1\end{array}\right)$. Let $\pi$ and $\sigma$ be irreducible smooth representations of $G_{n+1}$ and…

Assume that $L(\tfrac12,\Pi)\neq0$ for the Rankin–Selberg representation $\Pi$. Let $V_n$ and $V_{n+1}$ be hermitian spaces over $F$, totally positive definite at infinity, with…

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…

Let $F$ be a number field, let $G$ be either ${\mathbf{GU}}_{2,2}\times \mathbf{GL}_2$ or $\mathbf{GL}_4\times \mathbf{GL}_2$, and let $\pi\otimes\sigma$ be a globally generic cusp…

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 $V_K=V\otimes_F K$,…

Global Gan–Gross–Prasad conjecture. There exists a unique pair $(\pi_0,\sigma_0)$ in $\widetilde{\Pi}_{\phi_G\times\phi_H}[G_n^*\times H_m^*]$ such that

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

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 $\underline{\Pi}$ be a cuspidal Borel-ordinary family of forms on $\operatorname{GSp}_4$, let $\underline{\Sigma}=\underline{\sigma}_1\otimes\underline{\sigma}_2$ be a cuspidal…

The $p$-adic arithmetic Gan--Gross--Prasad conjecture. The following conditions are equivalent:

Let $F/F_0$ (resp. $E/F_0$) be a CM (resp. totally real) quadratic extension of number fields. Let $\mu$ be a conjugate symplectic automorphic character of…

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…

Non-tempered Gan–Gross–Prasad conjecture. The A-parameter $N_\sigma$ and $M_\tau(\mu^{-1})\otimes[2]$ form a relevant pair.

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)…

Global Gan–Gross–Prasad conjecture. Assuming the central epsilon factor is $1$, the global period $\mathcal{P}_\mu$ vanishes on the entire Vogan $L$-packet if and only if…

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…

Let $f_1$ and $f_2$ be cuspidal forms, and let the period integral be … where the integral is the one denoted by … should be related to the square root of the central value of the…

Gan–Gross–Prasad inequality. If $\sigma$ shares a global Vogan packet with $\sigma_0$ but is not equivalent to $\sigma_0$, then

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…

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