Rational Points in Regular Orbits attached to Infinitesimal Symmetric Spaces

TL;DR

This paper proves the existence of rational points on orbits in infinitesimal symmetric spaces and extends these results to global reductive symmetric spaces, motivated by problems in trace formulas and invariant theory.

Contribution

It introduces new methods to establish rational points in symmetric space orbits and extends local results to global settings.

Findings

Existence of rational points in specific infinitesimal symmetric space orbits

Extension of local orbit results to global reductive symmetric spaces

Applications to trace formulas and arithmetic invariant theory

Abstract

Motivated by problems arising in the relative trace formula and arithmetic invariant theory we prove the existence of rational points on orbits arising from certain infinitesimal symmetric spaces. As an application, we prove analogous results for orbits in certain global reductive symmetric spaces.