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arXiv · 2211.06062

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

Stephanie Chan
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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 $n\bmod 9$.

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Stephanie Chan. 2022-11-11. The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$. https://doi.org/10.1093/imrn%2Frnad266

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