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

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 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 $\ne 2$, 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.

32 pages, 5 sections, no figures