On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve

We study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductorN. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families FS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈FS L(1, Ed) alg, where L(1, Ed) alg denotes the algebraic part of the central L-value, L(1, Ed). Our main theorem relates the p-adic valuations of δS as S varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation between Waldspurger packets on f SL2 and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of p-indivisibility of L-values of quadratic twists.