First explicit reciprocity law for unitary Friedberg--Jacquet periods

TL;DR

This paper establishes a new explicit reciprocity law linking unitary Friedberg--Jacquet periods to the Bloch--Kato conjecture, showing that certain automorphic representations imply the vanishing of associated Selmer groups.

Contribution

It introduces the first explicit reciprocity law for unitary Friedberg--Jacquet periods and connects it to the Bloch--Kato conjecture for motives.

Findings

Proves the vanishing of Bloch--Kato Selmer groups under distinguished automorphic representations.

Establishes a new explicit reciprocity law for unitary Friedberg--Jacquet periods.

Links automorphic period distinction to motivic Selmer group properties.

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.