Elliptic curves with a lower bound on 2-Selmer ranks of quadratic twists

For any number field $K$ with a complex place, we present an infinite family of elliptic curves defined over $K$ such that ${\mathrm{dim}_{\mathbb{F}_2}} {\mathrm{Sel}}_2(E^F/K) \ge {\mathrm{dim}_{\mathbb{F}_2}} E^F(K)[2] + r_2$ for every quadratic twist $E^F$ of every curve $E$ in this family, where $r_2$ is the number of complex places of $K$. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.

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Published 15 March 2013