Normal origamis of Mumford curves

TL;DR

This paper classifies normal p-adic origamis, which are coverings of Mumford curves with at most one branch point, and explores their invariants and geometric structures in the p-adic setting.

Contribution

It provides a complete classification of all normal non-trivial p-adic origamis and relates them to square-gluing constructions in the p-adic context.

Findings

Classification of all normal non-trivial p-adic origamis

Calculation of invariants for these origamis

Description of p-adic origamis via square-gluing methods

Abstract

An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit squares. By varying the complex structure of the torus one obtains easily accessible examples of Teichm\"uller curves in the moduli space of Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves. A p-adic origami is defined as a covering of Mumford curves with at most one branch point, where the bottom curve has genus one. A classification of all normal non-trivial p-adic origamis is presented and used to calculate some invariants. These can be used to describe p-adic origamis in terms of glueing squares.