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 "a la Weil" for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the etale l-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 aver a finite field, then the characteristic polynomial of the Frobenius morphism on H-c(i)(X, Q(l)) divides that of H-c(i)(Y, Q(l)) for any i. We are then enable to give an estimate for the number of rational points in a flat covering of curves. (C) 2003 Elsevier Inc. All rights reserved.