Kloosterman identities over a quadratic extension. II. (English) Zbl 1111.11028

Jacquet, Hervé

Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 609-669 (2005).

This work concerns the relative trace formula, an approach to study the quadratic base change from automorphic representations of \(\text{ GL} (n,F_{\mathbb{A}})\) to automorphic representations of \(\text{ GL} (n,E_{\mathbb{A}})\), where \(E/F\) is a quadratic extension of number fields. The main result is local, it deals with a quadratic unramified extension of a non-Archimedean local field of odd residual characteristic and characteristic 0. The author proves certain identities between Kloosterman integrals. They constitute the fundamental lemma of the relative trace formula for Hecke functions. This result for the unit element of the Hecke algebra was proved in the first part [H. Jacquet, Ann. Math. (2) 160, No.2, 755–779 (2004; Zbl 1071.11026)].
The author also discusses global applications, in particular, the following result. A cuspidal automorphic representation of \(\text{ GL} (n,E_{\mathbb{A}})\) is distinguished by some unitary group if and only if it is the base change of an automorphic cuspidal representation of \(\text{ GL} (n,F_{\mathbb{A}})\).

Reviewer: Dubravka Ban (Carbondale)

MSC:

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

Keywords:

relative trace formula; fundamental lemma

Citations:

Zbl 1071.11026

References:

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