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

Rubin, Karl; Silverberg, Alice

Exp. Math. 10, No. 4, 559-569 (2001).

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 \(3\).

Reviewer: Franz Lemmermeyer (Bilkent)

MSC:

11G05	Elliptic curves over global fields
14G25	Global ground fields in algebraic geometry

Keywords:

elliptic curves; quadratic twists; Mordell-Weil rank

Citations:

Zbl 0766.14023; Zbl 0857.11026

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