Arithmetic twisted GGP conjecture for Fourier–Jacobi periods

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm U(V_{E_0})(\mathbb A_{E_0})$, let $\Pi$ be a weak base change of $\pi$ to $\mathrm {GL}_n(\mathbb A_E)$, and let $\omega(\mu,W_f)$ and $\mathrm{FJ}(W_f)$ denote the finite-adelic Weil and Fourier--Jacobi representations appearing in the statement. Assume that $\pi$ is tempered and $\Pi$ is cuspidal. Moreover, assume that $\Pi$ is relevant: for every archimedean place $v$ of $E$, $\Pi_v$ is isomorphic to the irreducible principal series representation induced by the characters

$$(\arg^{1-n},\arg^{3-n},\ldots,\arg^{n-3},\arg^{n-1}),$$

where

$$\arg(z)=\left(\frac{z}{\bar z}\right)^{1/2}:\mathbb C^\times\to\mathbb C^\times.$$

Arithmetic twisted GGP conjecture. The following are equivalent:

1. $\mathrm {Hom}_{\mathrm U(V)(\mathbb A_{E_0,f})}(\pi_f,\mathrm {FJ}(W_f))\ne 0$.
2. $L'(1/2,\Pi,As_{E/F}\otimes\mu^{-1})\ne 0$ and $\mathrm {Hom}_{\mathrm U(V)(\mathbb A_f)}(\pi_f,\omega(\mu,W_f))\ne 0$. This conjecturally relates the finite-adelic Fourier--Jacobi period to the derivative of the twisted Asai L-function under the stated temperedness, cuspidality, and archimedean relevance hypotheses; no resolution is given in the supplied text.