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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.

MSC:
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
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References:
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