Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913432430706688 |
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| author | Yelton, Jeffrey |
| author_facet | Yelton, Jeffrey |
| contents | Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $Γ\lhd Γ_0 < \mathrm{PGL}_2(K)$ is a Schottky group normally contained in a larger group which is generated by order-$p$ elements each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a certain subset of the projective line $\mathbb{P}_K^1$ by the action of $Γ$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of fixed points is mapped bijectively modulo $Γ$ to the set $\mathcal{B}$ of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. A conjecture of Gerritzen and van der Put, in the case that $C$ is hyperelliptic and $K$ has residue characteristic $\neq 2$, compares the cluster data of $S$ with that of $\mathcal{B}$. We show that this conjecture requires a slight modification in order to hold and then prove a much stronger version of the modified conjecture that holds for any $p$ and any residue characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put Yelton, Jeffrey Number Theory 11G20, 11G07 Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $Γ\lhd Γ_0 < \mathrm{PGL}_2(K)$ is a Schottky group normally contained in a larger group which is generated by order-$p$ elements each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a certain subset of the projective line $\mathbb{P}_K^1$ by the action of $Γ$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of fixed points is mapped bijectively modulo $Γ$ to the set $\mathcal{B}$ of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. A conjecture of Gerritzen and van der Put, in the case that $C$ is hyperelliptic and $K$ has residue characteristic $\neq 2$, compares the cluster data of $S$ with that of $\mathcal{B}$. We show that this conjecture requires a slight modification in order to hold and then prove a much stronger version of the modified conjecture that holds for any $p$ and any residue characteristic. |
| title | Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put |
| topic | Number Theory 11G20, 11G07 |
| url | https://arxiv.org/abs/2407.11303 |