## On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields.(English)Zbl 1116.14012

Summary: 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 $$\ell$$-adic cohomology groups of the curve. Finally, as an analogue of Artin’s holomorphy conjecture, we prove that, if $$Y \to X$$ is a finite flat morphism between two varieties over a finite field, then the characteristic polynomial of the Frobenius morphism on $$H_c^i(X,\mathbb Q_{\ell})$$ divides that of $$H_c^i(Y,\mathbb Q_{\ell})$$ for any $$i$$. We are then enable to give an estimate for the number of rational points in a flat covering of curves.

### MSC:

 14G15 Finite ground fields in algebraic geometry 11G20 Curves over finite and local fields 14H25 Arithmetic ground fields for curves 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

### Keywords:

algebraic curve; finite field; rational point; zeta function
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### References:

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