A motivic fundamental lemma

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.1. Introduction 2043 2. Motivic integration with pseudofinite residue field 2047 3. Geometric setup 2066 4. Definability of geometric stabilization 2070 5. Proof of the main theorems 2081 6.A motivic fundamental lemma 2088