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

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 $\varphi$ from $E_n$ to ${\hat{E}}_n:y^2=x^3-27n^2$ and a dual isogeny $\hat{\varphi }:{\hat{E}}_n\to E_n$. We show that for almost all $n$, the rank of ${\mathrm{Sel}}_{\varphi }(E_n)$ is $0$, and the rank of ${\mathrm{Sel}}_{\hat{\varphi }}({\hat{E}}_n)$ is determined by the number of prime factors of $n$ that are congruent to $2\mathrm{mod}3$ and the congruence class of $n\mathrm{mod}9$.

23 pages

References in corpus (3)

• On 2-Selmer groups of twists after quadratic extension
• Integers expressible as the sum of two rational cubes
• Weak approximation on the norm one torus