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 ℓ. We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ _ΧD, λ) in 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.