Number of rational points of a singular curve

Galindo, W.

doi:10.1090/S0002-9939-98-04333-0

Number of rational points of a singular curve
HTML articles powered by AMS MathViewer

by W. A. Zúñiga Galindo PDF

Proc. Amer. Math. Soc. 126 (1998), 2549-2556 Request permission

Abstract:

In this paper, we give a bound for the number of rational points of a complete, geometrically irreducible, algebraic curve defined over a finite field. We compare it with other known bounds and discuss its sharpness. We also show that the asymptotic Drinfeld-Vladut bound can be generalized to the case of singular curves.

References

Yves Aubry and Marc Perret, A Weil theorem for singular curves, Arithmetic, geometry and coding theory (Luminy, 1993) de Gruyter, Berlin, 1996, pp. 1–7. MR 1394921
S. G. Vlèduts and V. G. Drinfel′d, The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen. 17 (1983), no. 1, 68–69 (Russian). MR 695100
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Jean-Pierre Serre, Zeta and $L$ functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 82–92. MR 0194396
Serre, J.P., Rational points on curves over finite fields, notes by Fernando Q. Gôuvea, unpublished.
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
Karl-Otto Stöhr, On the poles of regular differentials of singular curves, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 105–136. MR 1224302, DOI 10.1007/BF01231698