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 $\Gamma ⊲{\Gamma }_0<{\mathrm{PGL}}_2(K)$ 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 $S^{\min }\subset K\cup \{\infty \}$ 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.

arXiv comment31 pages, 4 sections, 5 figures. Minor errors and oversights have been fixed since the last version (including in Definition 2.1, Remark 2.17, Lemma 3.16, and Example 4.7), and all instances of the algebraic closure of K have been replaced by the completion of the algebraic closure of K