Branch points of split degenerate superelliptic curves I: construction of Schottky groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914871653695488 |
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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 modulo $Γ$ to the set of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. We produce an algorithm for determining whether an input even-cardinality subset $S \subset K \cup \{\infty\}$ consists of fixed points of generators of such a group $Γ_0$ and which, in the case of a positive answer, modifies $S$ into a subset $S^{\mathrm{min}} \subset K \cup \{\infty\}$ with particularly nice properties. Our results do not involve any restrictions on the prime $p$ or on the residue characteristic of $K$ and allow these to be the same. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17823 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Branch points of split degenerate superelliptic curves I: construction of Schottky groups 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 modulo $Γ$ to the set of branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. We produce an algorithm for determining whether an input even-cardinality subset $S \subset K \cup \{\infty\}$ consists of fixed points of generators of such a group $Γ_0$ and which, in the case of a positive answer, modifies $S$ into a subset $S^{\mathrm{min}} \subset K \cup \{\infty\}$ with particularly nice properties. Our results do not involve any restrictions on the prime $p$ or on the residue characteristic of $K$ and allow these to be the same. |
| title | Branch points of split degenerate superelliptic curves I: construction of Schottky groups |
| topic | Number Theory 11G20, 11G07 |
| url | https://arxiv.org/abs/2306.17823 |