D\'eveloppement fin de la contribution unipotente \`a la formule des traces sur un corps global de caract\'eristique p>0, I

TL;DR

This paper develops a theory of unipotent F-strata in reductive groups over fields of positive characteristic, enabling detailed analysis of the unipotent contribution in the trace formula for global fields.

Contribution

It introduces a new framework for unipotent F-strata in positive characteristic, facilitating the expansion of unipotent contributions in the trace formula.

Findings

Established the fine expansion of the unipotent contribution in the trace formula.

Developed a theory of unipotent F-strata applicable to positive characteristic fields.

Provided tools analogous to geometric orbits in Arthur's work over number fields.

Abstract

For a field $F$ and a connected reductive group $G$ defined over $F$, we develop a theory of Kempf-Rousseau-Hesselink unipotent $F$-strata in $G(F)$ that should allow us to attack open problems in positive characteristic. As an application, we use this theory to establish the fine expansion of the unipotent contribution to the (non-twisted) trace formula over a global field of characteristic $p>0$. The unipotent $F$-strata play here the role of the unipotent geometric orbits in Arthur's work over a number field. The expansion in terms of products of local distributions is not discussed here; it will be the subject of further work.