$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

TL;DR

This paper investigates the relationship between $ ext{Selmer}$ groups, ideal class groups, and $3$-Selmer ranks of a specific family of elliptic curves, providing bounds, constructions of high-rank curves, and statistical results over $ ext{Q}$.

Contribution

It establishes bounds on the $ ext{Selmer}$ group ranks in terms of class groups, constructs infinitely many curves with arbitrarily large $3$-Selmer ranks, and analyzes the distribution of root numbers and ranks over $ ext{Q}$.

Findings

Bounds on $ ext{Selmer}$ groups in terms of class groups.

Existence of infinitely many curves with arbitrarily large $3$-Selmer rank.

Positive proportion of curves with root number -1 and $ ext{Selmer}$ rank 1.

Abstract

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.