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

  Published:2009-12-04
 Printed: Apr 2010
  • Kartik Prasanna
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Abstract

We study $p$-indivisibility of the central values $L(1,E_d)$ 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 $\delta_S= \gcd_{d\in \mathcal{F}_S} L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ 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 $\widetilde{\operatorname{SL}_2}$ 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.
MSC Classifications: 11F40, 11F67, 11G05 show english descriptions unknown classification 11F40
Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Elliptic curves over global fields [See also 14H52]
11F40 - unknown classification 11F40
11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
11G05 - Elliptic curves over global fields [See also 14H52]
 

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