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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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