The Relative Trace Formula for Galois Periods

TL;DR

This paper develops a relative trace formula framework for Galois periods over quadratic extensions, establishing convergence, explicit geometric expansions, and a new geometric perspective on unipotent orbital integrals.

Contribution

It introduces truncated geometric and spectral RTF distributions for Galois symmetric pairs, proves their convergence, and explicitly computes the geometric expansion including unipotent contributions.

Findings

Truncated geometric RTF distribution converges and is linear in the truncation parameter.

Explicit computation of the fine geometric expansion including unipotent orbital integrals.

A new geometric viewpoint aids in understanding unipotent terms.

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.