$3$-Selmer group, ideal class groups and cube sum problem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917897284091904 |
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| author | Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha |
| author_facet | Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha |
| contents | Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12487 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $3$-Selmer group, ideal class groups and cube sum problem Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha Number Theory Primary 11G05, 11R29, 11R34, Secondary 11G40, 11S25 Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$). |
| title | $3$-Selmer group, ideal class groups and cube sum problem |
| topic | Number Theory Primary 11G05, 11R29, 11R34, Secondary 11G40, 11S25 |
| url | https://arxiv.org/abs/2207.12487 |