# On the fine expansion of the unipotent contribution of the Guo-Jacquet trace formula

## TL;DR

This paper refines the understanding of the unipotent contribution in the Guo-Jacquet trace formula for certain pairs of groups, providing a detailed expansion, linking to zeta integrals, and establishing homogeneity properties.

## Contribution

It introduces a fine expansion of the unipotent contribution, expresses nilpotent integrals via zeta integrals, and proves their homogeneity, using a novel truncation method.

## Key findings

- Fine expansion in terms of global nilpotent integrals
- Expression of nilpotent integrals as zeta integrals
- Homogeneity properties of these integrals

## Abstract

For a useful class of functions (containing functions whose one finite component is essentially a matrix coefficient of a supercuspidal representation), we establish three results about the unipotent contribution of the Guo-Jacquet relative trace formula for the pair $(GL_n(D),GL_n(E))$. First we get a fine expansion in terms of global nilpotent integrals. Second we express these nilpotent integrals in terms of zeta integrals. Finally we prove that they satisfy certain homogeneity properties. The proof is based on a new kind of truncation introduced in a previous article.