The 3-Isogeny Selmer Groups of the Elliptic Curves y2=x3+n2

Abstract

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 $\operatorname {Sel}_{\phi }(E_{n})$ is $0$, and the rank of $\operatorname {Sel}_{\hat {\phi }}(\hat {E}_{n})$ is determined by the number of prime factors of $n$ that are congruent to $2\bmod 3$ and the congruence class of $n\bmod 9$.