| Autor: |
Jha, Somnath, Majumdar, Dipramit, Shingavekar, Pratiksha |
| Rok vydání: |
2022 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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