Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.02514 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913604791435264 |
|---|---|
| author | Siksek, Samir Visser, Robin |
| author_facet | Siksek, Samir Visser, Robin |
| contents | Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+δ=1$, with $\varepsilon$, $δ\in \mathcal{O}_S^\times$, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$. Now instead of a number field, let $K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $\varepsilon+δ=1$, with $\varepsilon$, $δ\in \mathcal{O}_S^\times$, has infinitely many solutions for $\ell \in \{2,3,5,7\}$, where $S$ consists only of the totally ramified prime above $\ell$. Moreover, for every prime $\ell$, we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$. For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02514 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions Siksek, Samir Visser, Robin Number Theory 11G10 Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+δ=1$, with $\varepsilon$, $δ\in \mathcal{O}_S^\times$, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$. Now instead of a number field, let $K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $\varepsilon+δ=1$, with $\varepsilon$, $δ\in \mathcal{O}_S^\times$, has infinitely many solutions for $\ell \in \{2,3,5,7\}$, where $S$ consists only of the totally ramified prime above $\ell$. Moreover, for every prime $\ell$, we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$. For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes. |
| title | Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions |
| topic | Number Theory 11G10 |
| url | https://arxiv.org/abs/2302.02514 |