A note on Chudnovsky's Fuchsian equations

TL;DR

This paper explores the algebraic and transcendental transformations connecting four special Fuchsian equations, known as Chudnovsky equations, and their elliptic curve counterparts, revealing their equivalence through explicit formulas.

Contribution

It demonstrates the algebraic and transcendental equivalences among Chudnovsky's Fuchsian equations and their elliptic curve versions, with explicit transformation descriptions.

Findings

Chudnovsky equations are mutually transformable via algebraic transformations.

Elliptic curve counterparts of these equations are equivalent through explicit transcendental transformations.

Transformations are expressed using elliptic and theta functions.

Abstract

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latter are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.