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On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve |
Canad. J. Math. 62(2010), 400-414
Published:2009-12-04
Printed: Apr 2010
Printed: Apr 2010
- Kartik Prasanna
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 - 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] |