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 de ned over a nite eld, and a bound \ a la Weil" for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the etale `-adic cohomology groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that, if Y ! X is a nite at morphism between two varieties over a nite eld, then the characteristic polynomial of the Frobenius morphism on H c(X;Q`) divides H i c(Y;Q`)'s one for any i. We are then enable to give an estimation for the number of rational points in a at covering of curves.