Cyclic coverings of the $p$-adic projective line by Mumford curves

TL;DR

This paper establishes precise bounds for branch point positions in cyclic coverings of the p-adic projective line by Mumford curves, employing combinatorial and geometric methods.

Contribution

It introduces two novel approaches—using *-trees and explicit matrix representations—to determine branch point bounds in p-adic Mumford curve coverings.

Findings

Derived exact bounds for branch points

Applied Fumiharu Kato's *-trees in analysis

Provided explicit matrix group representations

Abstract

Exact bounds for the positions of the branch points for cyclic coverings of the $p$-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato's *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory.