de Mathématiques de Luminy
Aubry Yves, Perret Marc.
On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields
We give a formula for the number of rational points of projective algebraic curves defined over a finite field, and a bound "à la Weil" for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the étale -adic cohomology groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that, if Y > X is a finite flat morphism between two varieties over a finite field, then the characteristic polynomial of the Frobenius morphism on Hic(X,Q) divides Hic(Y,Q)'s one for any i. We are then enable to give an estimation for the number of rational points in a flat covering of curves.