Unrefined arithmetic Gan–Gross–Prasad conjecture for U(n) × U(n)

Let $\Pi_1$ and $\Pi_2$ be relevant representations of $\operatorname{GL}_n(\mathbf{A}_E)$, let $\mu\colon E^\times\backslash\mathbf{A}_E^\times\mathtt{o}\mathbf{C}^\times$ be a conjugate symplectic automorphic character of weight one, and let $\varepsilon$ be a $\mu$-admissible collection. Let $\mathbb{L}\subseteq\mathbf{C}$ contain $M_\mu$ and be a field of definition for $\Pi_1^\infty$ and $\Pi_2^\infty$. For pairs $(\mathbf{V},\pi_1^\infty)\in\Phi_{\Pi_1}$ and $(\mathbf{V},\pi_2^\infty)\in\Phi_{\Pi_2}$, let $\operatorname{FJ}^{\natural}_\varepsilon$ denote the associated Fourier–Jacobi cycle map.

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

$$\dim_{\mathbb{L}}\operatorname{Hom}_{\mathbb{L}[\mathbf{G}(\mathbf{A}_F^\infty)\times\mathbf{G}(\mathbf{A}_F^\infty)]}\bigl((\pi_1^\infty)^\vee\otimes_{\mathbb{L}}(\pi_2^\infty)^\vee,\operatorname{CH}^{n-1+[M_\mu:\mathbf{Q}]/2}_\mu(X_\infty\times X_\infty)_\mathbb{L}^{\natural}\bigr)=1;$$

(c) $L'(\tfrac12,\Pi_1\times\Pi_2\otimes\mu)\neq0$ and

$$\operatorname{Hom}_{\mathbb{L}[\mathbf{G}(\mathbf{A}_F^\infty)]}\bigl(\pi_1^\infty\otimes_{\mathbb{L}}\pi_2^\infty\otimes_{M_\mu}\Omega(\mu,\varepsilon),\mathbb{L}\bigr)\neq\{0\}.$$

This conjecture is an arithmetic refinement of Gan–Gross–Prasad: nonvanishing of the Fourier–Jacobi cycle should correspond to a nonzero central derivative and the expected one-dimensional multiplicity. The local multiplicity and refined local statements used in the formulation are known, while the global equivalence is open.