1. CJM 2009 (vol 62 pp. 400)
 Prasanna, Kartik

On pAdic Properties of Central LValues of Quadratic Twists of an Elliptic Curve
We study $p$indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semistable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
SwinnertonDyer 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 halfintegral weight. In
this context, we formulate a conjecture on congruences of
halfintegral weight forms and explain its relevance to the problem of
$p$indivisibility of $L$values of quadratic twists.
Categories:11F40, 11F67, 11G05 
