an:03977088
Zbl 0605.10015
Jacquet, Herv??
On a result of Waldspurger
FR
Ann. Sci. ??c. Norm. Sup??r. (4) 19, No. 2, 185-229 (1986).
00165800
1986
j
11F67 11F70
non-vanishing of automorphic L-functions; symmetry center; automorphic representations of GL(2); forms of GL(2); trace formula; orbital integrals
\textit{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.
J. G. M. Mars (Utrecht)
Zbl 0567.10021