For almost all prime numbers \(p\), the solutions of the differential equations induced by geometry, around a generic point \(t\), converge in the disk \(|x-t|_p<1\). B. Dwork conjectured that this is characteristic.
In order to test the conjecture, the author proves that the diagonal of a rational function satisfies a differential equation with coefficients in \(\mathbb{Q}(x)\). (Dwork has already proved that the diagonal of a rational function is actually involved in Apery’s famous prove of the irrationality of \(\zeta(3)\).) Provided the ring \(\mathbb{Q}(\lambda)[y_1,\ldots,y_{\mu}]\) satisfies a certain hypothesis (H) (this means a particular generation of the ring) the author shows that the conjecture is true. The results are given in terms of the Frobenius functor.