Arithmetic Differential Equations of Painleve' VI Type

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