Mathematical Research Letters
Volume 28 (2021)
On $2$-Selmer groups and quadratic twists of elliptic curves
Pages: 1633 – 1659
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.
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