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 $n\mathrm{mod}9$.

arXiv comment23 pages

References in corpus (9)

• The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$
• Elementary 3-descent with a 3-isogeny
• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
• On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-2p})$ for primes $p\equiv 1\mathrm{mod}4$
• On 2-Selmer groups of twists after quadratic extension
• The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion
• The Distribution of p-Torsion in Degree p Cyclic Fields
• Integers expressible as the sum of two rational cubes
• Weak approximation on the norm one torus