Three examples of the relation between rigid-analytic and algebraic deformation parameters

TL;DR

This paper explores the relationship between algebraic and rigid-analytic deformation parameters in families of curves with group actions over non-archimedean fields, highlighting how these parameters relate via self-maps of the disk.

Contribution

It provides three explicit examples illustrating the connection between algebraic and rigid-analytic deformation parameters in equivariant curve families.

Findings

Algebraic and rigid-analytic parameters can be related through self-maps of the disk.

The examples demonstrate how group actions influence deformation parameters.

The study clarifies the interplay between algebraic equations and analytic structures in non-archimedean geometry.

Abstract

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.