Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put

TL;DR

This paper investigates the branch points of superelliptic curves arising from Schottky groups, modifies a conjecture by Gerritzen and van der Put, and proves a stronger version applicable to all primes and residue characteristics.

Contribution

It identifies necessary modifications to Gerritzen and van der Put's conjecture and proves a more general, stronger version valid for all primes and residue characteristics.

Findings

Modified the original conjecture to be universally valid.

Proved a stronger, more general version of the conjecture.

Established a link between cluster data and branch points for superelliptic curves.

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 bijectively modulo $\Gamma$ 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.