Branch points of split degenerate superelliptic curves I: construction of Schottky groups

TL;DR

This paper presents an algorithm to identify and optimize sets of fixed points of Schottky groups related to superelliptic curves over fields with discrete valuation, without restrictions on prime or residue characteristic.

Contribution

It introduces a novel algorithm for recognizing and refining fixed point sets of Schottky groups associated with superelliptic curves, applicable in general prime and residue characteristic settings.

Findings

Algorithm effectively determines fixed point sets for Schottky groups.

The method produces minimal sets with desirable properties.

Results apply broadly without restrictions on prime or residue characteristic.

Abstract

Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $\Gamma \lhd \Gamma_0 < \mathrm{PGL}_2(K)$ 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 $\Gamma$ 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 $\Gamma$ 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 $\Gamma_0$ 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.