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A note on a question of J. Nekovár
and the Birch and Swinnerton-Dyer conjecture

Author: Ken Ono
Journal: Proc. Amer. Math. Soc. 126 (1998), 2849-2853
MSC (1991): Primary 11G40; Secondary 14G10
MathSciNet review: 1459142
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Abstract: If $D$ is a square-free integer, then let $E(D)$ denote the elliptic curve over $\mathbb{Q}$ given by the equation

\begin{equation*}E(D):\ \ \ Dy^{2}=4x^{3}-27. \tag{{1}} \end{equation*}

Let $L(E(D),s)$ denote the Hasse-Weil $L$-function of $E(D)$, and let $L^{*}(E(D),1)$ denote the `algebraic part' of the central critical value $L(E(D),1)$. Using a theorem of Sturm, we verify a congruence conjectured by J. Neková\v{r}. By his work, if $S(3,E(D))$ denotes the 3-Selmer group of $E(D)$ and $D\neq 1$ is a square-free integer with $|D|\equiv 1\pmod 3$, then we find that

\begin{equation*}L^{*}(E(D),1)\not \equiv 0 \pmod 3 \Longleftrightarrow S(3,E(D))=0. \end{equation*}

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Additional Information

Ken Ono
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Keywords: Elliptic curves, modular forms
Received by editor(s): March 13, 1997
Additional Notes: The author is supported by National Science Foundation grants DMS-9304580 and DMS-9508976, and NSA grant MSPR-YO12.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1998 American Mathematical Society