$3$-Selmer group, ideal class groups and cube sum problem
Abstract
Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$ of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a rich history going back to the works of Cassels and the recent works of Bhargava et al. and Chao Li. Let $E/\mathbb Q$ be an elliptic curve with a rational $3$-isogeny. In this article, we give an upper bound and a lower bound of the rank of the Selmer group of $E$ over $\mathbb Q(\zeta_3)$ induced by the $3$-isogeny in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of Sylvester's conjecture on the rational cube sum problem and also exhibit infinitely many elliptic curves of arbitrary large $3$-Selmer rank over $\mathbb Q(\zeta_3)$. Our method also produces infinitely many imaginary quadratic fields and biquadratic fields with non-trivial $3$-class groups.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2022
- DOI:
- 10.48550/arXiv.2207.12487
- arXiv:
- arXiv:2207.12487
- Bibcode:
- 2022arXiv220712487J
- Keywords:
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- Mathematics - Number Theory;
- Primary 11G05;
- 11R29;
- 11R34;
- Secondary 11G40;
- 11S25