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

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. Nekovář. 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*}