Arithmetic aspects of the Jouanolou foliation

We investigate the structure of the [Formula: see text]-divisor associated with the Jouanolou foliation and we show, under certain conditions, that it can be irreducible or have a [Formula: see text]-factor. Furthermore, we study the reduction modulo [Formula: see text] of foliations and explore its applications to problems involving holomorphic foliations. As an application, we provide a new proof, via reduction modulo 2, of the fact that the Jouanolou foliation on the complex projective plane of odd degree, under specific arithmetic conditions, has no algebraic invariant curves.