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arXiv · 2001.02263

On $2$-Selmer groups and quadratic twists of elliptic curves

Daniel Barrera Salazar·Ariel Pacetti·Gonzalo Tornaría
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Abstract

Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.

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Daniel Barrera Salazar, Ariel Pacetti, Gonzalo Tornaría. 2020-01-07. On $2$-Selmer groups and quadratic twists of elliptic curves. https://arxiv.org/abs/2001.02263

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