Kloosterman integrals and base change for \(\text{GL}(2)\). (English) Zbl 0665.10020

Ye, Yangbo

J. Reine Angew. Math. 400, 57-121 (1989).

By calculating integrals of Selberg kernel functions based on certain double coset decompositions we establish a relative trace formula of a new kind. Using that formula, we give a new proof of the following characterization of the base change images for \(\text{GL}(2)\) over quadratic extensions of number fields.
Let \(\eta\) be the quadratic idele class character of a finite algebraic number field \(F\) attached to a quadratic extension \(E\). An automorphic irreducible cuspidal representation \(\Pi\) of \(Z_{\mathbb A}\setminus \mathrm{GL}(2,E_{\mathbb A})\) is the base change lifting of an automorphic irreducible cuspidal representation \(\pi\) of \(\mathrm{GL}(2,F_{\mathbb A})\) with central character \(\eta\) if and only if \(\Pi\) is distinguished.

Reviewer: Ye Yangbo

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

Kloosterman integrals; relative trace formula; base change; GL(2); automorphic irreducible cuspidal representation