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

Abstract

Let F be a newform of weight 2k on Gamma(0)(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 chi(D) be a quadratic character associated with quadratic field Q(root D). Assume that for each D, the l-adic valuation of the algebraic part of L(F circle times chi(D), k) is non-negative. Let W-l(+) (resp. W-l(-)) 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 circle times chi(D), k) is zero. We prove that for each sign epsilon if W-l(epsilon) is a non-empty finite set, then W-l(c) subset of {1, (-1)(l-1/2)l}. By this result, we prove that if l is the sign of (-1)(k), then k >= l -1 or k = l-1/2 These are applied to obtain a lower bound for #{D is an element of W-l(epsilon) : vertical bar D vertical bar <= 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.