A motivic fundamental lemma

Forey, Arthur; Loeser, François; Wyss, Dimitri

doi:10.2140/gt.2026.30.2043

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A motivic fundamental lemma

Arthur Forey, François Loeser and Dimitri Wyss

Geometry & Topology 30 (2026) 2043–2096

DOI: 10.2140/gt.2026.30.2043

Abstract

In this paper we prove motivic versions of the Langlands–Shelstad fundamental lemma and Ngô’s geometric stabilization. To achieve this, we follow the strategy from the recent proof by Groechenig, Wyss and Ziegler which avoided the use of perverse sheaves using instead $p$ -adic integration and Tate duality. We make a key use of a construction of Denef and Loeser which assigns a virtual motive to any definable set in the theory of pseudofinite fields.

Keywords

fundamental lemma, geometric stabilization, Hitchin fibration, motivic integration, pseudofinite fields

Mathematical Subject Classification

Primary: 03C65, 11S37, 14D24, 14E18, 14H60

References

Publication

Received: 29 August 2023

Revised: 11 November 2024

Accepted: 12 March 2025

Published: 31 July 2026

Proposed: Marc Levine

Seconded: Dan Abramovich, Sándor J Kovács

Authors

Arthur Forey
Laboratoire Paul Painlevé Université de Lille Villeneuve-d’Ascq France

François Loeser
Institut Mathématique de Jussieu Sorbonne Université UMR 7586 CNRS Paris France

Dimitri Wyss
Institute of Mathematics Ecole polytechnique fédérale de Lausanne Lausanne Switzerland

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Volume 30, issue 6 (2026)