Journal Articles Finite Fields and Their Applications Year : 2004

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 ''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.
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Dates and versions

hal-00978903 , version 1 (15-04-2014)

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Yves Aubry, Marc Perret. On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields. Finite Fields and Their Applications, 2004, 10 (3), pp.412--431. ⟨10.1016/j.ffa.2003.09.005⟩. ⟨hal-00978903⟩
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