First explicit reciprocity law for unitary Friedberg--Jacquet periods

Abstract

Consider a unitary group $G({\mathbb{A}}_{F^{+}})=U_{2r}({\mathbb{A}}_{F^{+}})$ over a CM extension $F/F^{+}$ with $G({\mathbb{A}}_{\infty })$ compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G({\mathbb{A}}_{F^{+}}).$ 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 $\Pi =\mathrm{BC}(\pi )$ vanishes.