Arithmetic Differential Equations of Painleve' VI Type

TL;DR

This paper explores the translation of Painleve' VI differential equations into the framework of arithmetic differential equations using p-adic derivatives, connecting classical and modern mathematical theories.

Contribution

It introduces a novel approach to express Painleve' VI equations as arithmetic differential equations via Fermat quotient, bridging differential and arithmetic geometry.

Findings

Established a correspondence between Painleve' VI and arithmetic differential equations.

Demonstrated the use of Fermat quotient as a p-adic derivative analogue.

Extended classical differential equation concepts into the arithmetic setting.

Abstract

Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the role of derivative "in the $p$--adic direction" is played by a version of Fermat quotient