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

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.