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Unitary Friedberg-Jacquet periods and their twists: Relative trace formulas

Spencer Leslie, Jingwei Xiao, Wei Zhang

TL;DR

The paper proves that for unitary symmetric pairs, the H-distinguished automorphic period is equivalent to the non-vanishing of central L-values after base change, using a novel relative trace formula that interleaves linear and unitary sides with endoscopic transfer. It develops linear and unitary relative trace formulas, establishes necessary transfer factors and fundamental lemmas, and stabilizes the geometric expansions to accommodate endoscopic terms. By combining weak base change, transfer compatibilities, and careful analysis of local relative characters, it derives a bi-directional link between period non-vanishing and L-value non-vanishing, with a positivity input from Lapid–Rallis ensuring the non-negativity of the relevant central values. The results extend Waldspurger-type theorems to higher rank unitary groups, providing a framework for twisted base change and endoscopic phenomena in relative trace formula comparisons. Overall, the work offers a global-to-local mechanism linking automorphic periods to special L-values in the presence of relative endoscopy, with potential applications to broader conjectures in the Gan–Gross–Prasad landscape.

Abstract

In a companion paper, we formulated a global conjecture for the automorphic period integral associated to the symmetric pairs defined by unitary groups over number fields, generalizing a theorem of Waldspurger's toric period for $\mathrm{GL}(2)$. In this paper, we introduce a new relative trace formula to prove our global conjecture under some local hypotheses. A new feature is the presence of relative endoscopy in the comparison. We also establish several local results on relative characters.

Unitary Friedberg-Jacquet periods and their twists: Relative trace formulas

TL;DR

The paper proves that for unitary symmetric pairs, the H-distinguished automorphic period is equivalent to the non-vanishing of central L-values after base change, using a novel relative trace formula that interleaves linear and unitary sides with endoscopic transfer. It develops linear and unitary relative trace formulas, establishes necessary transfer factors and fundamental lemmas, and stabilizes the geometric expansions to accommodate endoscopic terms. By combining weak base change, transfer compatibilities, and careful analysis of local relative characters, it derives a bi-directional link between period non-vanishing and L-value non-vanishing, with a positivity input from Lapid–Rallis ensuring the non-negativity of the relevant central values. The results extend Waldspurger-type theorems to higher rank unitary groups, providing a framework for twisted base change and endoscopic phenomena in relative trace formula comparisons. Overall, the work offers a global-to-local mechanism linking automorphic periods to special L-values in the presence of relative endoscopy, with potential applications to broader conjectures in the Gan–Gross–Prasad landscape.

Abstract

In a companion paper, we formulated a global conjecture for the automorphic period integral associated to the symmetric pairs defined by unitary groups over number fields, generalizing a theorem of Waldspurger's toric period for . In this paper, we introduce a new relative trace formula to prove our global conjecture under some local hypotheses. A new feature is the presence of relative endoscopy in the comparison. We also establish several local results on relative characters.

Paper Structure

This paper contains 65 sections, 60 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Main global results
  3. The relative trace formulas
  4. The organization of the paper
  5. Preliminaries
  6. Local fields
  7. Invariant theory and orbital integrals
  8. Unitary groups and Hermitian spaces
  9. Measure conventions
  10. Abelian $L$-functions and centralizers
  11. Geometric preliminaries
  12. Recollection of invariant theory
  13. The linear orbital integrals
  14. Transfer factors
  15. Orbital integrals on $\mathop{\mathrm{G}}\nolimits'(F)\times F_{n}$
  16. ...and 50 more sections

Key Result

Theorem 1.1

Let $\pi_{0}$ be a cuspidal automorphic representation of $\mathop{\mathrm{G}}\nolimits'(\mathbb A_{F})$ satisfying Assume that $\pi_0$ is of symplectic type. Then the following two assertions are equivalent:

Theorems & Definitions (121)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • ...and 111 more