Cohomological divisibility and point count divisibility

We show that Ax-Katz divisibility on the number of rational points of a variety defined over a finite field by equations of low degrees comes from divisibility of the eigenvalues of the Frobenius action of the $\ell$-adic cohomology with compact supports. This completes the motivc picture, as we knew already that in characteristic 0, the Hodge type was the expected one. [version 2 as the title contained a typo..., else no changes]