Statistics for 3-isogeny induced Selmer groups of elliptic curves

TL;DR

This paper investigates the distribution of Selmer groups associated with elliptic curves defined by sixth power free integers, using refined Davenport--Heilbronn methods to relate binary cubic form statistics to Selmer group dimensions.

Contribution

It provides explicit density results for the Selmer groups of elliptic curves with specific isogenies, advancing understanding of their arithmetic properties.

Findings

Lower density of sixth power free integers with Selmer group dimension ≤ 1

Refined analysis connecting binary cubic forms to Selmer group statistics

Explicit density formulas for Selmer groups over ree integers

Abstract

Given a sixth power free integer $a$, let $E_a$ be the elliptic curve defined by $y^2=x^3+a$. We prove explicit results for the lower density of sixth power free integers $a$ for which the $3$-isogeny induced Selmer group of $E_a$ over $\mathbb{Q}(\mu_3)$ has dimension $\leq 1$. The results are proven by refining the strategy of Davenport--Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for $3$-isogeny induced Selmer groups.