On uniformizable representation for Abelian integrals

TL;DR

This paper explores a method to obtain uniformizable representations of Abelian integrals of nontrivial genus using Fuchsian differential equations and their connection to the sixth Painleve transcendent.

Contribution

It introduces a novel approach linking Abelian integrals with Fuchsian equations and Painleve transcendents to achieve uniformization.

Findings

Established a connection between Abelian integrals and Fuchsian equations.

Demonstrated the role of Painleve VI in uniformization.

Provided a new framework for representing Abelian integrals.

Abstract

We show how the unformizable representation for Abelian integrals of nontrivial genera arise. The technique makes use of the famous Chudnovsky's Fuchsian linear differential equations and their relation to the sixth Painleve transcendent.