A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

Details A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

2012-01-25

arxiv.org/abs/1201.5407v2

Mathematics

Number Theory

Latex, 9 pages. Added references and acknowledgment

Scientific paper

For any number field K with a complex place, we present an infinite family of
elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim
\mathbb{F}_2 E^F(K)[2] + r_2$ 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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