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Unitary Friedberg-Jacquet periods and their twists: Fundamental lemmas

Spencer Leslie, Jingwei Xiao, Wei Zhang

TL;DR

This work develops a global framework for unitary Friedberg–Jacquet periods by formulating a global conjecture linking $H$-distinguished automorphic representations to central $L$-values via base change, and proceeds with a novel relative trace formula approach that incorporates relative endoscopy. The core technical achievement is the proof of three local relative fundamental lemmas (split-inert, inert-inert, and endoscopic) by reducing to Lie algebras and establishing a key Hironaka-transform-based bridge between orbital integrals on unitary and linear spaces. The results, combined with a companion extensive analysis in LXZ25, yield equivalences between non-vanishing of central $L$-values and the existence of $H$-distinguished inner forms after base change, in split-inert and inert-inert cases. Methodologically, the paper blends geometric invariant theory, orbital-integral transfer, mirabolic and Lie-algebra techniques, and new transfer conjectures to stabilize relative trace formulas, with potential to generalize Waldspurger-type results to higher rank unitary groups and related settings.

Abstract

We formulate 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 $\mathrm{GL}(2)$. We introduce a new relative trace formula to prove our global conjecture under some local hypotheses. A new feature is the presence of the relative endoscopy. In this paper we prove the main local theorem: a new relative fundamental lemma comparing certain orbital integrals of functions matched in terms of Hironaka and Satake transforms.

Unitary Friedberg-Jacquet periods and their twists: Fundamental lemmas

TL;DR

This work develops a global framework for unitary Friedberg–Jacquet periods by formulating a global conjecture linking -distinguished automorphic representations to central -values via base change, and proceeds with a novel relative trace formula approach that incorporates relative endoscopy. The core technical achievement is the proof of three local relative fundamental lemmas (split-inert, inert-inert, and endoscopic) by reducing to Lie algebras and establishing a key Hironaka-transform-based bridge between orbital integrals on unitary and linear spaces. The results, combined with a companion extensive analysis in LXZ25, yield equivalences between non-vanishing of central -values and the existence of -distinguished inner forms after base change, in split-inert and inert-inert cases. Methodologically, the paper blends geometric invariant theory, orbital-integral transfer, mirabolic and Lie-algebra techniques, and new transfer conjectures to stabilize relative trace formulas, with potential to generalize Waldspurger-type results to higher rank unitary groups and related settings.

Abstract

We formulate 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 . We introduce a new relative trace formula to prove our global conjecture under some local hypotheses. A new feature is the presence of the relative endoscopy. In this paper we prove the main local theorem: a new relative fundamental lemma comparing certain orbital integrals of functions matched in terms of Hironaka and Satake transforms.

Paper Structure

This paper contains 96 sections, 91 theorems, 515 equations.

Table of Contents

  1. Introduction
  2. The global conjecture
  3. Main global results
  4. The relative trace formulas
  5. Relative fundamental lemmas
  6. The organization of the paper
  7. Preliminaries
  8. Local fields
  9. Invariant theory and orbital integrals
  10. Contractions
  11. Unitary groups and Hermitian spaces
  12. Measure conventions
  13. Abelian $L$-functions and centralizers
  14. Orbital integrals and the fundamental lemmas
  15. Orbit calculations and matching
  16. ...and 81 more sections

Key Result

Theorem 1.5

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 (189)

  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Definition 2.1
  • Lemma 2.3
  • ...and 179 more