Arithmetic aspects of the Jouanolou foliation

Abstract: We investigate the structure of the $p$-divisor for the Jouanolou foliation where we show, under some conditions, that it can be irreducible or has a $p$-factor. We study the reduction modulo $p$ of foliations on the projective plane and its applications to the problems of holomorphic foliations. We give new proof, via reduction modulo $2$, of the fact that the Jouanolou foliation on the complex projective plane of odd degree, under some arithmetic conditions, has no algebraic solutions.