Non-vanishing of quadratic twists of modular L-functions

Abstract

;kƒhave been the subject of much study, both because of their intrinsicinterest and because of the prominent role they have played inKolyvagin’s work on the Birch and Swinnerton-Dyer Conjecture (see[B-F-H], [I], [J], [Ko], [Ma-M], [M-M1], [O-S], [P-P]).Waldspurger proved a fundamental theorem [The´ore`me 1, W1]relating these central critical values to the Fourier coe†cients of half-integral weight cusp forms. For notational convenience, if D is afundamental discriminant of a quadratic number field, then define D