The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$

TL;DR

This paper studies the 3-isogeny Selmer groups of a family of elliptic curves defined by $y^2=x^3+n^2$, revealing that for most $n$, the Selmer group ranks are zero and depend on the prime factorization of $n$ and its residue class modulo 9.

Contribution

It provides a detailed analysis of the 3-isogeny Selmer groups for the family $E_n:y^2=x^3+n^2$, including rank determination based on prime factors and modular conditions.

Findings

For almost all $n$, the Selmer group rank is zero.

The rank of the dual Selmer group depends on prime factors of $n$ and $n mod 9.

The paper characterizes the Selmer group structure in terms of prime factorization and congruences.

Abstract

Consider the family of elliptic curves $E_n:y^2=x^3+n^2$, where $n$ varies over positive cubefree integers. There is a rational $3$-isogeny $\phi$ from $E_n$ to $\hat{E}_n:y^2=x^3-27n^2$ and a dual isogeny $\hat{\phi}:\hat{E}_n\rightarrow E_n$. We show that for almost all $n$, the rank of $\mathrm{Sel}_{\phi}(E_n)$ is $0$, and the rank of $\mathrm{Sel}_{\hat{\phi}}(\hat{E}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.