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First explicit reciprocity law for unitary Friedberg--Jacquet periods

Murilo Corato-Zanarella

TL;DR

The paper proves a Beilinson--Bloch--Kato-type vanishing result for motives attached to unitary group automorphic representations distinguished by unitary Friedberg--Jacquet periods, under an explicit set of hypotheses. It develops a comprehensive framework combining weight spectral sequences, derived Kudla--Rapoport cycles, and p-adic uniformization to relate automorphic periods to arithmetic cycles via a first explicit reciprocity law. The approach hinges on level-raising congruences in the unitary Friedberg--Jacquet setting and leverages a derived-integral-model formalism to control reductions modulo inert primes, culminating in a vanishing of the Bloch--Kato Selmer group $H^1_f(F,\rho_{\Pi,\lambda}(r))$ for admissible primes $\lambda$ when the Friedberg--Jacquet period is nonvanishing. These results reinforce the Beilinson--Bloch--Kato conjecture in a new unitary context and lay groundwork for unitary Iwasawa main conjectures via an explicit reciprocity law and Euler-system arguments.

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 $π$ of $G(\mathbb{A}_{F^+}).$ We prove that if $π$ is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of $Π=\mathrm{BC}(π)$ vanishes.

First explicit reciprocity law for unitary Friedberg--Jacquet periods

TL;DR

The paper proves a Beilinson--Bloch--Kato-type vanishing result for motives attached to unitary group automorphic representations distinguished by unitary Friedberg--Jacquet periods, under an explicit set of hypotheses. It develops a comprehensive framework combining weight spectral sequences, derived Kudla--Rapoport cycles, and p-adic uniformization to relate automorphic periods to arithmetic cycles via a first explicit reciprocity law. The approach hinges on level-raising congruences in the unitary Friedberg--Jacquet setting and leverages a derived-integral-model formalism to control reductions modulo inert primes, culminating in a vanishing of the Bloch--Kato Selmer group for admissible primes when the Friedberg--Jacquet period is nonvanishing. These results reinforce the Beilinson--Bloch--Kato conjecture in a new unitary context and lay groundwork for unitary Iwasawa main conjectures via an explicit reciprocity law and Euler-system arguments.

Abstract

Consider a unitary group over a CM extension with compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations of We prove that if is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of vanishes.
Paper Structure (60 sections, 71 theorems, 364 equations, 2 figures)

This paper contains 60 sections, 71 theorems, 364 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy and new obstacles
  4. Outline of this article
  5. Conventions and notations
  6. Weight spectral sequence and potential map
  7. Semistable schemes and weight spectral sequence
  8. Potential map
  9. Description in terms of K-theory
  10. Unitary Rapoport--Zink spaces and special cycles
  11. Unitary Rapoport--Zink spaces
  12. Kudla--Rapoport cycles
  13. Case of signature (N-1,1)
  14. Bruhat--Tits stratification
  15. Inductive structure
  16. ...and 45 more sections

Key Result

Theorem 1.1.2

Suppose $F^+\neq\mathbb{Q}.$ Let $\pi$ be an irreducible cuspidal automorphic representation of $G(\mathbb{A}_{F^+})$ of weight $(0,\ldots,0)$ such that its base change $\Pi=\mathrm{BC}(\pi)$ is an irreducible cuspidal automorphic representation of $\mathrm{GL}_{2r}(\mathbb{A}_F).$ Let $E\subseteq\m

Figures (2)

  • Figure 1: Uniformization on the generic fiber and arithmetic level raising.
  • Figure 2: $p$-adic uniformization.

Theorems & Definitions (218)

  • Conjecture 1.1.1: Xiao--Zhang
  • Theorem 1.1.2: \ref{['ThmA']}
  • Remark 1.1.3
  • Theorem 1.1.4
  • Remark 1.1.5
  • Definition 2.1.1: LiuTriple
  • Definition 2.1.2
  • Remark 2.1.3
  • Definition 2.2.1
  • Lemma 2.2.2: LiuTriple
  • ...and 208 more