Rank frequencies for quadratic twists of elliptic curves. (English) Zbl 1035.11025

Let \(E: y^2 = f(x)\) be an elliptic curve defined over \(\mathbb Q\), and let \(E_D: Dy^2 = f(x)\) denote its quadratic twists. J.-F. Mestre [C. R. Acad. Sci., Paris, Sér. I 314, 919–922 (1992; Zbl 0766.14023)] has shown that if \(j(E) \neq 0, 1728\), then there exists some \(g(t) \in \mathbb Q[t]\) such that \(E_{g(t)}\) has rank at least \(2\) over \(\mathbb Q[t]\), and C. L. Stewart and J. Top [J. Am. Math. Soc. 8, 943–973 (1995; Zbl 0857.11026)] found twists of rank at least \(3\). In this article, the authors construct new and more general examples with rank at least \(2\) and \(3\). These curves are then used for giving lower bounds for the number of twists \(E_D\) of \(E\) with squarefree integers \(0 < D < X\) that have rank at least \(2\) or \(3\).


11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry
Full Text: DOI arXiv EuDML


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