Number of rational points of a singular curve

Author:
W. A. Zúñiga Galindo

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2549-2556

MSC (1991):
Primary 11G20; Secondary 14H25

MathSciNet review:
1451803

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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.

**[A-P]**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****[D-V]**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****[H]**Heisuke Hironaka,*On the arithmetic genera and the effective genera of algebraic curves*, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.**30**(1957), 177–195. MR**0090850****[Ros1]**Maxwell Rosenlicht,*Equivalence relations on algebraic curves*, Ann. of Math. (2)**56**(1952), 169–191. MR**0048856****[Ros2]**Maxwell Rosenlicht,*Generalized Jacobian varieties*, Ann. of Math. (2)**59**(1954), 505–530. MR**0061422****[Ser1]**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 MR**0194396****[Ser2]**Serre, J.P.,*Rational points on curves over finite fields*, notes by Fernando Q. Gôuvea, unpublished.**[Ser3]**Jean-Pierre Serre,*Algebraic groups and class fields*, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988.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****[Sto]**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**, 10.1007/BF01231698

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Additional Information

**W. A. Zúñiga Galindo**

Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro-R.J., Brazil

Address at time of publication:
Universidad Autónoma de Bucaramanga, Laboratorio de Computo Especializado, A.A.1642, Bucaramanga, Colombia

Email:
wzuniga@bumanga.unab.edu.co

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04333-0

Received by editor(s):
October 17, 1995

Received by editor(s) in revised form:
January 31, 1997

Additional Notes:
The author thanks Prof. Karl-Otto Stöhr for many helpful discussions, and the referee for his or her useful comments. Supported by CNPq-Brazil and COLCIENCIAS-Colombia

Communicated by:
William W. Adams

Article copyright:
© Copyright 1998
American Mathematical Society