Bounds for $2$-Selmer ranks in terms of seminarrow class groups

Hwajong Yoo and Myungjun Yu

Vol. 320 (2022), No. 1, 193–222

DOI: 10.2140/pjm.2022.320.193

Abstract

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is nice at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]∕(F(x))$, which we call the seminarrow class group of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime.

As an application, when $K$ is a real quadratic field, $E∕K$ is semistable and the discriminant of $F$ is totally negative, we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.

Keywords

elliptic curve, 2-Selmer rank, ideal class group, Mordell–Weil rank

Mathematical Subject Classification

Primary: 11G05, 11G07, 11R29

Milestones

Received: 3 January 2021

Revised: 28 September 2021

Accepted: 23 October 2021

Published: 16 October 2022