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Note on the rank of quadratic twists of Mordell equations. (English) Zbl 1208.11072
Let \(E\) be the elliptic curve given by a Mordell equation \(y^2=x^3-A\) where \(A\in\mathbb Z\). M. Stoll [J. Reine Angew. Math. 501, 171–189 (1998; Zbl 0902.11024)] gave a precise formula for the size of a Selmer group of \(E\) for certain values of \(A\). For \(D\in\mathbb Z\), let \(E_D\) denote the quadratic twist \(Dy^2=x^3-A\). The author uses Stoll’s formula to show that for a positive square-free integer \(A\equiv 1\) or \(25\bmod 36\) and for a nonnegative integer \(k\), one can compute a lower bound for the proportion of square-free integers \(D\) up to \(X\) such that \(\text{rank}\,E_D\leq 2k\). He also computes an upper bound for a certain average rank of quadratic twists of \(E\) using a theorem of J. Nakagawa and K. Horie [Proc. Am. Math. Soc. 104, No. 1, 20–24 (1988; Zbl 0663.14023)].

MSC:
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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References:
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