## 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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