Published online by Cambridge University Press: 20 November 2018
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.