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 conductor N. 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∈ℱSL(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 betweenWaldspurger packets on 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.

Newforms of half-integral weight

Two very different definitions of a newform of half-integral weight are present and continued to be developed in the literature. The first definition originated with Serre and Stark for forms of weight 1/2 [5], and is analogous to the definition of newform for integral weight forms, which uses forms of lower level and shifts of such forms to characterize the notion of old-forms. The second definition originated with Kohnen for half-integral weight forms of squarefree level [1], who used forms of lower level and their image under the Um2 operator to define the notion of oldforms. The choice of the Um2 operator over the shift operator Bd seems a propitious one, since the U operator commutes with the action of the Shimura lift, while the shift operator B does not. More to the point, Kohnen was able to develop a newform theory on a distinguished subspace of the full space of cusp forms (now referred to as the Kohnen subspace), and obtained a multiplicity-one result (with respect to Hecke eigenvalues) for half-integral weight newforms in this subspace. Even nicer, the multiplicity-one result was established by showing that there is a one-to-one correspondence between newforms of level AN in the subspace and the newforms of integral weight of level N.