Torsion points of small order on cyclic covers of $\mathbb P^1$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913833242591232 |
|---|---|
| author | Bekker, Boris M. Zarhin, Yuri G. |
| author_facet | Bekker, Boris M. Zarhin, Yuri G. |
| contents | Let $d\geq 2$ be a positive integer, $K$ an algebraically closed field of characteristic not dividing $d$, $n\geq d+1$ a positive integer that is prime to $d$, $f(x)\in K[x]$ a degree $n$ monic polynomial without multiple roots, $C_{f,d}: y^d=f(x)$ the corresponding smooth plane affine curve over $K$, $\mathcal{C}_{f,d}$ a smooth projective model of $C_{f,d}$ and $J(\mathcal{C}_{f,d})$ the Jacobian of $\mathcal{C}_{f,d} $. We identify $\mathcal{C}_{f,d}$ with the image of its canonical embedding into $J(\mathcal{C}_{f,d})$ (such that the infinite point of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$).
Earlier the second named author proved that if $d=2$ and $n=2g+1 \ge 5$ then the genus $g$ hyperelliptic curve $\mathcal{C}_{f,2}$ contains no points of orders lying between $3$ and $n-1=2g$.
In the present paper we generalize this result to the case of arbitrary $d$. Namely, we prove that if $P$ is a point of order $m>1$ on $\mathcal{C}_{f,d}$, then either $m=d$ or $m\geq n$. We also describe all curves $\mathcal{C}_{f,d}$ having a point of order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion points of small order on cyclic covers of $\mathbb P^1$ Bekker, Boris M. Zarhin, Yuri G. Algebraic Geometry 14H40, 14G27, 11G10, 11G30 Let $d\geq 2$ be a positive integer, $K$ an algebraically closed field of characteristic not dividing $d$, $n\geq d+1$ a positive integer that is prime to $d$, $f(x)\in K[x]$ a degree $n$ monic polynomial without multiple roots, $C_{f,d}: y^d=f(x)$ the corresponding smooth plane affine curve over $K$, $\mathcal{C}_{f,d}$ a smooth projective model of $C_{f,d}$ and $J(\mathcal{C}_{f,d})$ the Jacobian of $\mathcal{C}_{f,d} $. We identify $\mathcal{C}_{f,d}$ with the image of its canonical embedding into $J(\mathcal{C}_{f,d})$ (such that the infinite point of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). Earlier the second named author proved that if $d=2$ and $n=2g+1 \ge 5$ then the genus $g$ hyperelliptic curve $\mathcal{C}_{f,2}$ contains no points of orders lying between $3$ and $n-1=2g$. In the present paper we generalize this result to the case of arbitrary $d$. Namely, we prove that if $P$ is a point of order $m>1$ on $\mathcal{C}_{f,d}$, then either $m=d$ or $m\geq n$. We also describe all curves $\mathcal{C}_{f,d}$ having a point of order $n$. |
| title | Torsion points of small order on cyclic covers of $\mathbb P^1$ |
| topic | Algebraic Geometry 14H40, 14G27, 11G10, 11G30 |
| url | https://arxiv.org/abs/2411.03508 |