Stabilisation et germes pour $SL(2)$ en toutes caract\'eristiques

TL;DR

This paper establishes the stabilization of local orbital integrals and the trace formula for SL(2) over global fields, including characteristic 2, providing new insights into germ expansions and unipotent contributions.

Contribution

It extends stabilization and germ expansion results for SL(2) to all characteristics, especially addressing challenges in characteristic 2.

Findings

Stabilization of local orbital integrals in all characteristics.

Asymptotic expansion near identity matches Shalika's germ expansion in characteristic not 2.

New pre-stabilization techniques for unipotent contributions in trace formula.

Abstract

We give the stabilisation of local orbital integrals and the trace formula over a global field for $SL(2)$ with proofs valid in any characteristic. New features appear in characteristic 2. We obtain, via the stabilisation, an asymptotic expansion near the identity of local orbital integrals which is equivalent, up to a Fourier transform, to the standard germ expansion due to Shalika when the characteristic is not 2 but it is new in characteristic 2. Similarly the fine expansion of the unipotent contribution to the trace formula cannot be obtained using Arthur's techniques and a pre-stabilisation is necessary.