Gross–Prasad's global conjecture for special orthogonal groups

Let $F$ be a number field and $\mathbb{A}$ its ring of adeles. Let $(V_n,q_n)\subset (V_{n+1},q_{n+1})$ be an inclusion of quadratic spaces over $F$ of dimensions $n$ and $n+1$, with $n\geq 2$, $V_n$ not isomorphic to the hyperbolic plane, and $q_{n+1}|_{V_n}=q_n$. Write $\operatorname{SO}_n=\operatorname{SO}(V_n)$ and $\operatorname{SO}_{n+1}=\operatorname{SO}(V_{n+1})$. Let $\pi_n$ and $\pi_{n+1}$ be irreducible tempered cuspidal automorphic representations of $\operatorname{SO}_n(\mathbb{A})$ and $\operatorname{SO}_{n+1}(\mathbb{A})$. Gross–Prasad's conjecture. Assume that for every place $v$ of $F$, $\operatorname{Hom}_{\operatorname{SO}_n(F_v)}(\pi_{n+1,v}\otimes\pi_{n,v},\mathbb{C})\neq\{0\}$. Then there exist $\phi\in V_{\pi_{n+1}}$ and $f\in V_{\pi_n}$ such that

$$\int_{\operatorname{SO}_n(F)\backslash\operatorname{SO}_n(\mathbb{A})}\phi(g)f(g)\,dg\neq 0$$

if and only if $L(1/2,\pi_{n+1}\times\pi_n)\neq 0$. This is the original global Gross–Prasad conjecture, relating a period integral to the central value of a tensor-product $L$-function; its refinement by Ichino and Ikeda gives an explicit formula for the squared period.