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

Daniel Barrera Salazar; Ariel Pacetti; Gonzalo Tornaría

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

Number Theory 2020-09-21 v3

Authors: Daniel Barrera Salazar , Ariel Pacetti , Gonzalo Tornaría

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.

Keywords

2-Selmer group elliptic curve quadratic twist narrow class group cubic extension 2-torsion number field

Cite

@article{arxiv.2001.02263,
  title  = {On $2$-Selmer groups and quadratic twists of elliptic curves},
  author = {Daniel Barrera Salazar and Ariel Pacetti and Gonzalo Tornaría},
  journal= {arXiv preprint arXiv:2001.02263},
  year   = {2020}
}

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20 pages

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

Abstract

Keywords

Cite

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