Analytic connections on Riemann surfaces and orbifolds

TL;DR

This paper introduces a differential framework for understanding uniformizing representations on Riemann surfaces and orbifolds, emphasizing connections and their explicit construction via ODEs, linking classical and modern approaches.

Contribution

It provides a new differential description involving connection objects and their explicit construction, extending the classical uniformization theory with algorithmic ODEs.

Findings

Explicit construction of connections on hyperelliptic curves

Relation between classical Fuchsian equations and uniformizing τ-representations

Connections described by derivable ODEs with automorphic properties

Abstract

We give a differentially closed description of the uniformizing representation to the analytical apparatus on Riemann surfaces and orbifolds of finite analytic type. Apart from well-known automorphic functions and Abelian differentials it involves construction of the connection objects. Like functions and differentials, the connection, being also the fundamental object, is described by algorithmically derivable ODEs. Automorphic properties of all of the objects are associated to different discrete groups, among which are excessive ones. We show, in an example of the hyperelliptic curves, how can the connection be explicitly constructed. We study also a relation between classical/traditional `linearly differential' viewpoint (principal Fuchsian equation) and uniformizing $\tau$-representation of the theory. The latter is shown to be supplemented with the second (to the principal) Fuchsian equation.