Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III
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| Format: | Preprint |
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2026
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| _version_ | 1866909994412146688 |
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| author | Bekker, Boris M. Zarhin, Yuri G. |
| author_facet | Bekker, Boris M. Zarhin, Yuri G. |
| contents | Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$).
We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$.
Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$.
In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12643 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III Bekker, Boris M. Zarhin, Yuri G. Number Theory Algebraic Geometry 14H40, 14G27, 11G10 Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$. Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$. In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$. |
| title | Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III |
| topic | Number Theory Algebraic Geometry 14H40, 14G27, 11G10 |
| url | https://arxiv.org/abs/2601.12643 |