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

Aubry, Yves; Perret, Marc

Finite Fields Appl. 10, No. 3, 412-431 (2004).

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

References:

[1]	Y. Aubry, M. Perret, A Weil theorem for singular curves, in: Pellikaan, Perret, Vladuts (Eds.), Arithmetic, Geometry and Coding Theory (Luminy, 1993), Walter de Gruyter, Berlin, New York, 1996, pp. 1-7.; Y. Aubry, M. Perret, A Weil theorem for singular curves, in: Pellikaan, Perret, Vladuts (Eds.), Arithmetic, Geometry and Coding Theory (Luminy, 1993), Walter de Gruyter, Berlin, New York, 1996, pp. 1-7. · Zbl 0873.11037
[2]	Aubry, Y.; Perret, M., Coverings of singular curves over finite fields, Manuscripta Math., 88, 467-478 (1995) · Zbl 0862.11042
[3]	Y. Aubry, M. Perret, Divisibility of zeta functions of curves in a covering, Arch. Math., to appear.; Y. Aubry, M. Perret, Divisibility of zeta functions of curves in a covering, Arch. Math., to appear. · Zbl 1142.14306
[4]	Bach, E., Weil bounds for singular curves, Appl. Algebra Eng. Comm. Comput., 7, 4, 289-298 (1996) · Zbl 0877.11038
[5]	Foote, R.; Murty, K., Zeros and poles of Artin \(L\)-series, Math. Proc. Cambridge Philos. Soc., 105, 1, 5-11 (1989) · Zbl 0711.11043
[6]	A. Grothendieck (with M. Artin, J.-L. Verdier), SGA-4: Théorie des topos et cohomologie étale des schémas, Lectures Notes in Mathematics, Vols. 269, 270, 305, Springer, Heidelberg, 1972-1973.; A. Grothendieck (with M. Artin, J.-L. Verdier), SGA-4: Théorie des topos et cohomologie étale des schémas, Lectures Notes in Mathematics, Vols. 269, 270, 305, Springer, Heidelberg, 1972-1973.
[7]	S.L. Kleiman, Algebraic cycles and the Weil conjectures, in: Masson & Cie (Ed.), Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 359-386.; S.L. Kleiman, Algebraic cycles and the Weil conjectures, in: Masson & Cie (Ed.), Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 359-386. · Zbl 0198.25902
[8]	Leep, D.; Yeomans, C., The number of points on a singular curve over a finite field, Arch. Math., 63, 420-426 (1994) · Zbl 0819.11023
[9]	J.S. Milne, Étale cohomology, Princeton Mathematical Series, Vol. 33, Princeton University Press, Princeton, NJ, 1980.; J.S. Milne, Étale cohomology, Princeton Mathematical Series, Vol. 33, Princeton University Press, Princeton, NJ, 1980. · Zbl 0433.14012
[10]	Serre, J.-P, Sur le nombre de points rationnels d’une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér. I, 296, 397-402 (1983) · Zbl 0538.14015