$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909473165017088 |
|---|---|
| author | Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha |
| author_facet | Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha |
| contents | We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic 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 $E_{a,b}$ over $K:=\mathbb{Q}(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks Jha, Somnath Majumdar, Dipramit Shingavekar, Pratiksha Number Theory We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic 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 $E_{a,b}$ over $K:=\mathbb{Q}(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$. |
| title | $\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.01069 |