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

International audienceWe give a formula for the number of rational points of projective algebraic curves defined over a finite field, and a bound ''á 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 $Y\longrightarrow X$ is a finite flat morphism between two varieties over a finite field, then the characteristic polynomial of the Frobenius morphism on the i-th l-adic cohomology group with compact support of $X$ divides that of $Y$ for any i. We are then enable to give an estimate for the number of rational points in a flat covering of curves