A motivic Fundamental Lemma

TL;DR

The paper proves motivic versions of the Langlands-Shelstad Fundamental Lemma and Ngô’s Geometric Stabilization by extending Groechenig–Wyss–Ziegler’s p-adic approach to motivic integration over pseudo-finite residue fields. It encodes endoscopy, Hitchin fibrations, and orbifold volumes in the Denef–Loeser motivic framework, establishing a geometric-stabilization identity at the level of Chow motives and showing how the motivic identity specializes to the classical lemma for large residue characteristics. The methodology hinges on a robust definability theory in the Denef–Pas language, a careful treatment of Galois cohomology, twisted varieties, and isotypical components, and a reduction strategy via generic Hitchin fibers and Tate duality. The results illuminate a deep bridge between finite-field counts, motivic counts, and geometric objects like affine Springer and Hitchin fibers, with potential implications for transferring motivic information to classical automorphic forms.

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 pseudo-finite fields.

Theorems & Definitions (86)

• Lemma 2.2.1: LW19
• Proposition 2.2.2
• proof
• Proposition 2.2.3
• proof
• Lemma 2.3.1
• proof
• Definition 2.4.1
• Remark 2.6.1
• Proposition 2.6.2