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On a result of Waldspurger. (Sur un résultat de Waldspurger.) (French) Zbl 0605.10015
J. L. Waldspurger’s result is contained in [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)]. It is a result on the non-vanishing of automorphic \(L\)-functions of forms of \(\mathrm{GL}_2\) in the symmetry center. The author gives a different proof for Waldspurger’s result. The theorem to be proved is formulated in terms of integrals over tori attached to automorphic representations of \(\mathrm{GL}_2\) and forms of \(\mathrm{GL}_2\). It is derived from an interesting identity (Theorem, p. 222) involving integrals of the cuspidal kernels over the square of a torus. The proof of the identity is based on a variant of the trace formula and involves an explicit parametrization of the double cosets modulo a torus and computation and transfer of orbital integrals.

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
Full Text: DOI Numdam EuDML
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[2] H. JACQUET et R. LANGLANDS , Automorphic Forms on GL(2) (Lecture Notes in Math., n^\circ 114, Springer Verlag, 1970 ). MR 401654 | Zbl 0236.12010 · Zbl 0236.12010 · doi:10.1007/BFb0058988
[3] R. LANGLANDS , Base Change for GL(2) (Annals of Mathematics Studies, 96, Princeton University Press). MR 574808 | Zbl 0444.22007 · Zbl 0444.22007
[4] J.-L. WALDSPURGER , Correspondance de Shimura (J. Math. pures et appl., vol. 59, 1980 , p. 1-133). MR 577010 | Zbl 0412.10019 · Zbl 0412.10019
[5] J.-L. WALDSPURGER , Correspondance de Shimura et Quaternions , prerint. · Zbl 0454.10015
[6] J.-L. WALDSPURGER , Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie , preprint < 19 p, 1 p 3 >. · Zbl 0567.10021
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