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The Relative Trace Formula for Galois Periods

Siddharth Mahendraker

TL;DR

The paper develops a framework for a coarse, regularized relative trace formula for Galois symmetric pairs with equal split rank, formalizing truncated geometric and spectral kernels and proving an exact coarse identity. Specializing to the SL$_2$-case with $G= ext{Res}_{E/F} ext{SL}_{2,E}$ and $G'= ext{SL}_{2,F}$, it proves convergence of the truncated geometric RTF and defines the regularized geometric distribution $J_{ ext{geom}}(f)$ as the constant term in the truncation parameter. It then provides a detailed fine geometric expansion, including explicit formulas for the regular semisimple and unipotent terms, and a geometric interpretation via Levi descent and Springer-type pictures to organize the unipotent data. The results lay groundwork toward stabilization and endoscopy in the relative trace formula for Galois periods and offer concrete tools for matching geometric and spectral data in rank-one settings, with potential extensions to higher-rank Galois symmetric spaces and to endoscopic transfer. Overall, the work advances the understanding of regularized GTF/R TF in the Galois-symmetric setting and provides explicit computable formulas essential for stabilization efforts.

Abstract

Let $E/F$ be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair $G \subset \mathrm{Res}_{E/F} G_E$, subject to the constraint that $G$ and $\mathrm{Res}_{E/F} G_E$ have the same split rank, and formulate a precise coarse RTF identity. Specializing to $SL_{2, F} \subset \mathrm{Res}_{E/F} SL_{2, E}$, we show that the truncated geometric RTF distribution converges, and is given by a linear polynomial in the truncation parameter. We then compute the fine geometric expansion explicitly, including the contribution of the regularized relative unipotent orbital integrals. We propose a geometric viewpoint which guided the computation of these unipotent terms.

The Relative Trace Formula for Galois Periods

TL;DR

The paper develops a framework for a coarse, regularized relative trace formula for Galois symmetric pairs with equal split rank, formalizing truncated geometric and spectral kernels and proving an exact coarse identity. Specializing to the SL-case with and , it proves convergence of the truncated geometric RTF and defines the regularized geometric distribution as the constant term in the truncation parameter. It then provides a detailed fine geometric expansion, including explicit formulas for the regular semisimple and unipotent terms, and a geometric interpretation via Levi descent and Springer-type pictures to organize the unipotent data. The results lay groundwork toward stabilization and endoscopy in the relative trace formula for Galois periods and offer concrete tools for matching geometric and spectral data in rank-one settings, with potential extensions to higher-rank Galois symmetric spaces and to endoscopic transfer. Overall, the work advances the understanding of regularized GTF/R TF in the Galois-symmetric setting and provides explicit computable formulas essential for stabilization efforts.

Abstract

Let be a quadratic extension of number fields. We introduce truncated geometric and spectral RTF distributions associated to a Galois symmetric pair , subject to the constraint that and have the same split rank, and formulate a precise coarse RTF identity. Specializing to , we show that the truncated geometric RTF distribution converges, and is given by a linear polynomial in the truncation parameter. We then compute the fine geometric expansion explicitly, including the contribution of the regularized relative unipotent orbital integrals. We propose a geometric viewpoint which guided the computation of these unipotent terms.

Paper Structure

This paper contains 69 sections, 59 theorems, 248 equations, 2 figures.

Table of Contents

  1. Motivation
  2. What is Proved?
  3. A Coarse RTF for Galois Symmetric Pairs
  4. Convergence of the Truncated Geometric RTF Distribution for $\mathop{\mathrm{SL}}\nolimits_2$
  5. The Fine Geometric Expansion
  6. Acknowledgements
  7. A Coarse RTF for Galois Symmetric Pairs
  8. Basic Notation and Conventions
  9. Notation
  10. Conventions
  11. A Review of Galois Symmetric Spaces
  12. Definition, Basic Properties
  13. Levi Galois Symmetric Spaces
  14. Rational Points, Adelic Points
  15. Integral Models and Basic Functions
  16. ...and 54 more sections

Key Result

Theorem A

With the notation as above, let $f \in C_c^{\infty}(X(\mathbb{A}))$ be a test function, and let $\{ \Phi^{\xi}_1 \}_{\xi \in \ker^1(F, X)}$ be the associated collection of test functions in $C_c^{\infty}(G(\mathbb{A})^1)$. Let $T \in \mathfrak{a}_{0'}$ be a truncation parameter. Assume that for $T \ Then the associated geometric and spectral RTF distributions agree:

Figures (2)

  • Figure 1: A biquadratic extension of $M/F$ with Galois group $\mathop{\mathrm{Gal}}\nolimits(M/F) = \{ 1, \sigma, \tau, \sigma\tau = \tau\sigma \}$. We say $L'$ is the reflection of $L$ through $E$.
  • Figure 2: A generalized biquadratic extension $M = E \mathop{\mathrm{\times}}\nolimits E$ with automorphism group $\mathop{\mathrm{Aut}}\nolimits(M/F) = \{ 1, \sigma, \tau, \sigma\tau = \tau\sigma \}$. Here $\sigma$ is component-wise Galois conjugation, and $\tau$ swaps coordinates. Note that while $M$ contains two copies of $E$, they are embedded differently (diagonally vs antidiagonally).

Theorems & Definitions (136)

  • Theorem A: Coarse RTF
  • Theorem B
  • Remark 1
  • Theorem C: Fine Geometric Expansion for $\mathop{\mathrm{SL}}\nolimits_2$
  • Definition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 126 more