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

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
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