The algebraic parts of the central values of quadratic twists of modular L-functions modulo ℓ

Let F be a newform of weight 2k on Γ (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ, N) = 1. For each fundamental discriminant D, let χ_D be a quadratic character associated with quadratic field Q(D). Assume that for each D, the ℓ-adic valuation of the algebraic part of L(F⊗ χ_D, k) is non-negative. Let Wℓ+ (resp. Wℓ-) be the set of positive (resp. negative) fundamental discriminants D with (D, N) = 1 such that the ℓ-adic valuation of the algebraic part of L(F⊗ χ_D, k) is zero. We prove that for each sign ϵ, if Wℓϵ is a non-empty finite set, then Wℓϵ⊂{1,(-1)ℓ-12ℓ}.By this result, we prove that if ϵ is the sign of (- 1) ^k, then k≥ℓ-1ork=ℓ-12.These are applied to obtain a lower bound for #{D∈Wℓϵ:|D|≤X} and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q. To prove these results, first we refine Waldspurger’s formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations.