First explicit reciprocity law for unitary Friedberg--Jacquet periods

Consider a unitary group $G({\mathbb{A}}_{F^{+}})=U_{2r}({\mathbb{A}}_{F^{+}})$ over a CM extension $F\mathrm{/}{F}^{+}$ with $G({\mathbb{A}}_{\mathrm{\infty }})$ compact. In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G({\mathbb{A}}_{F^{+}})\mathrm{.}$ We prove that if $\pi$ is distinguished by the unitary Friedberg–Jacquet period, then the Bloch–Kato Selmer group (with coefficients in a favorable field) of the motive of $\mathrm{\Pi }=\mathrm{B}\mathrm{C}(\pi )$ vanishes.