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

Published online by Cambridge University Press:  20 November 2018

Kartik Prasanna
Kartik Prasanna*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, USA, e-mail: kartik.prasanna@gmail.com
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Abstract

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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.

Keywords

11G4011F6711G05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 2 , 01 April 2010 , pp. 400 - 414
Copyright
Copyright © Canadian Mathematical Society 2010

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