On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields

Abstract

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 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 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.