La formule des traces tordue pour les corps de fonctions
Jean-Pierre Labesse, Bertrand Lemaire
TL;DR
This work extends the twisted Arthur–Selberg trace formula to global function fields (positive characteristic), following the Labesse–Waldspurger framework. It develops the geometric-combinatorial and spectral-truncation machinery—twisted spaces, Iwasawa reduction, Siegel sets, truncation operators, Eisenstein series, and operator-valued spectral families—in the new setting, obtaining a finite spectral expansion and a gross geometric description similar to the number-field case, while highlighting new geometric phenomena that arise in small characteristic. The study identifies key obstacles in positive characteristic (inseparability, primitive elements) that affect the fine geometric side but demonstrates that the spectral side remains tractable and mirrors the characteristic-zero theory to a large extent. It lays the groundwork toward a stabilized twisted trace formula over function fields, clarifying where stabilization and endoscopic transfer must be adapted to characteristic $p>0$ and outlining directions for further work. The results provide a concrete pathway to transfer and functoriality principles in the function-field context, contributing to the broader Langlands program in positive characteristic.
Abstract
We adapt to global fields of positive characteristic the contents of the book by Labesse and Waldspurger on the Twisted Trace Formula after the Friday Morning Seminar.