The global Gross–Prasad conjecture for GSpin(4) × GSpin(5)

Let $\Pi$ and $\Sigma=\Sigma_1\boxtimes\Sigma_2$ be cuspidal automorphic representations of $G(\mathbf A_F)$ and $\operatorname{GL}_2(\mathbf A_F)\times\operatorname{GL}_2(\mathbf A_F)$, respectively. Assume $\chi_\Pi\chi_{\Sigma_1}\chi_{\Sigma_2}=1$, that $\Pi_v$ and $\Sigma_{i,v}$ are tempered for every place $v$, and that $\operatorname{Hom}_{H_v}(\Pi_v\times\Sigma_v,\mathbf C)$ is one-dimensional for every $v$. Put

$$[H]=H(F)Z_G(\mathbf A_F)\backslash H(\mathbf A_F)$$

and define the period

$$\mathcal P(\varphi,\sigma)=\int_{[H]}\varphi(\iota(h))\sigma(h)\\,\mathrm d h.$$

Global Gross–Prasad conjecture. The period is non-zero if and only if

$$\Lambda(\Pi\times\Sigma,\tfrac12)\ne0.$$

These conjectures are open in general, but have been proved for $\Pi$ of Yoshida or twisted Yoshida type.