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

Prasanna, Kartik

Canadian journal of mathematics, Tome 62 (2010) no. 2, pp. 400-414

Résumé

We study $p$ -indivisibility of the central values $L\left( 1,\,{{E}_{d}} \right)$ of quadratic twists ${{E}_{d}}$ 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 ${{\mathcal{F}}_{S}}$ , indexed by subsets $S$ of the primes dividing $N$ . Let ${{\delta }_{S}}={{\gcd }_{d\in {{\mathcal{F}}_{S}}}}L{{(1,{{E}_{d}})}^{\text{alg}}}$ , where $L{{(1,{{E}_{d}})}^{\text{alg}}}$ denotes the algebraic part of the central $L$ -value, $L(1,\,{{E}_{d}})$ . Our main theorem relates the $p$ -adic valuations of ${{\delta }_{S}}$ 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 $\widetilde{\text{S}{{\text{L}}_{2}}}$ 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.

Détail

DOI : 10.4153/CJM-2010-023-2

Mots-clés : 11G40, 11F67, 11G05