International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Yuri I.Manin

Painlevé VI equations in p-adic time (joint with A. Buium)

Abstract

Painlevé equations have many applications in mathematical physics, in particular, in the theory of integrable systems and in the theory of gravitational instantons i.e., Einstein worlds with imaginary time.
We show that the description of Painlevé VI equations in terms of universal elliptic curve, going back to R. Fuchs, allows us to translate it into the realm of Arithmetic Differential Equations, where the role of derivative "in the $p$-adic direction" is played by a version of Fermat quotient.