Central critical values of modular L -functions and coefficients of half-integral weight modular forms modulo ℓ

If F(z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χD,s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes . We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χD, λ) is not 0 (mod l), then there are infinitely many such D. These results depend on precise nonvanishing results for the Fourier coefficients of half-integral weight modular forms modulo l, which are of independent interest.