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 ℓ-adic cohomology groups of the curve. Finally, as an analogue of Artinʹs 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 Hci(X,Qℓ) divides that of Hci(Y,Qℓ) for any i. We are then enable to give an estimate for the number of rational points in a flat covering of curves.