Mathematical Research Letters

Volume 28 (2021)

Number 6

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

Pages: 1633 – 1659

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n6.a1

Authors

Daniel Barrera Salazar (DMCC, Universidad de Santiago de Chile, Santiago, Chile)

Ariel Pacetti (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Portugal)

Gonzalo Tornaría (Universidad de la República, Montevideo, Uruguay)

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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To the memory of John Tate

D.B.S. was supported by the FONDECYT PAI 77180007.

A.P. was partially supported by FonCyT BID-PICT 2018-02073 and by the Portuguese Foundation for Science and Technology (FCT) within project UIDB/04106/2020 (CIDMA).

G.T. was partially supported by ANII–FCE 2017–136609.

Received 24 March 2020

Accepted 13 September 2020

Published 29 August 2022