On the nonvanishing of someL-functions

The non-vanishing, at the centre of symmetry, of theL-function attached to an automorphic representation of GL(2) or its twists by quadratic characters has been extensively investigated, in particular by Waldspurger. The purpose of this paper is to outline a new proof of Waldspurger’s results. The automorphic representations of GL(2) and its metaplectic cover are compared in two different ways; one way is by means of a “relative trace formula”; the relative trace formula presented here is actually a generalization of the work of Iwaniec. This is the text of a lecture delivered in June 1987 in Paris at the Symposium in honour of R Godement. This work was supported in part by NSF grant DMS-85-02789.