A motivic Fundamental Lemma

TL;DR

This paper proves motivic versions of the Langlands-Shelstad Fundamental Lemma and Ngô's Geometric Stabilization, employing $p$-adic integration and Tate duality instead of perverse sheaves, building on recent proof strategies.

Contribution

It introduces a motivic approach to fundamental lemmas in the Langlands program, extending previous methods with new motivic constructions and techniques.

Findings

Motivic versions of the Fundamental Lemma are established.

The approach avoids perverse sheaves, using $p$-adic integration and Tate duality.

A key use of Denef and Loeser's virtual motives is demonstrated.

Abstract

In this paper we prove motivic versions of the Langlands-Shelstad Fundamental Lemma and Ng\^o'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 pseudo-finite fields.