Fields generated by points on superelliptic curves
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909790647615488 |
|---|---|
| author | Beneish, Lea Keyes, Christopher |
| author_facet | Beneish, Lea Keyes, Christopher |
| contents | We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $d= \mathrm{deg}\ f (x) \geq m$, we find that for all degrees $n$ divisible by $\gcd(m, d)$ and sufficiently large, the number of such fields is asymptotically bounded below by $X^{δ_n}$, where $δ_n \to 1/m^2$ as $n \to \infty$. We then give geometric heuristics suggesting that for n not divisible by $\gcd(m, d)$, degree $n$ points may be less abundant than those for which $n$ is divisible by $\gcd(m,d)$ and provide an example of conditions under which a curve is known to have finitely many points of certain degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_16672 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Fields generated by points on superelliptic curves Beneish, Lea Keyes, Christopher Number Theory 11G30, 11D45, 12E05 We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $d= \mathrm{deg}\ f (x) \geq m$, we find that for all degrees $n$ divisible by $\gcd(m, d)$ and sufficiently large, the number of such fields is asymptotically bounded below by $X^{δ_n}$, where $δ_n \to 1/m^2$ as $n \to \infty$. We then give geometric heuristics suggesting that for n not divisible by $\gcd(m, d)$, degree $n$ points may be less abundant than those for which $n$ is divisible by $\gcd(m,d)$ and provide an example of conditions under which a curve is known to have finitely many points of certain degrees. |
| title | Fields generated by points on superelliptic curves |
| topic | Number Theory 11G30, 11D45, 12E05 |
| url | https://arxiv.org/abs/2103.16672 |