Kloosterman identities over a quadratic extension. (English) Zbl 1071.11026

Jacquet, Hervé

Ann. Math. (2) 160, No. 2, 755-779 (2004).

This paper proves the fundamental lemma of the relative trace formula for quadratic base change from \(\text{GL} (m,F)\) to \(\text{GL} (m,E)\), where \(E/F\) is a quadratic extension of global fields. The lemma was formulated by H. Jacquet and Y. Ye [Bull. Soc. Math. Fr. 120, No. 3, 263–295 (1992; Zbl 0785.11032)]. It requires, at almost all places, a matching of certain orbital integrals: a Kloosterman integral on one side with a relative Kloosterman integral on the other side. Jacquet and Ye proved the lemma for \(\text{GL} (3)\) (loc.cit.). The proof for \(\text{GL} (m)\) over local fields of positive characteristic is due to Ngô Báo Châu [Duke Math. J. 96, No. 3, 473–520 (1999; Zbl 1047.11517)]. The paper under review proves the conjecture for unramified quadratic extensions of non-Archimedean local fields with the residual characteristic \(\neq 2\).

Reviewer: Dubravka Ban (Carbondale)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11L05	Gauss and Kloosterman sums; generalizations

Keywords:

Kloosterman identities; fundamental lemma; relative trace formula

Citations:

Zbl 0785.11032; Zbl 1047.11517