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Curves over Finite Fields Attaining the Hasse-Weil Upper Bound

  • Conference paper
European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 202))

Abstract

Curves over finite fields (whose cardinality is a square) attaining the Hasse-Weil upper bound for the number of rational points are called maximal curves. Here we deal with three problems on maximal curves:

  1. 1

    Determination of the possible genera of maximal curves.

  2. 2

    Determination of explicit equations for maximal curves.

  3. 3

    Classification of maximal curves having a fixed genus.

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References

  1. M. Abdán and F. Torres, On maximal curves in characteristic two, Manuscripta Math., 99 (1999), 39–53.

    Article  MathSciNet  Google Scholar 

  2. E. Bombieri, Hilbert’s 8th problem: An analogue, Proc. Symp. Pure Math., 28 (1976), 269–274.

    MathSciNet  Google Scholar 

  3. A. Cossidente, J. W. P. Hirschfeld, G. Korchm¨¢ros and F. Torres, On plane maximal curves, Compositio Math., 121 (2000), 163–181.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Cossidente, G. Korchm¨¢ros and F. Torres, On curves covered by the Hermitian curve, J. Algebra, 216 (1999), 56–76.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Cossidente, G. Korchmáros and F. Torres, Curves of large genus covered by the Hermitian curve, Comm Algebra, 28 (2000), 4707–4728.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. E. Dickson, History of the theory of numbers, Vol. II, Chelsea Publ. Comp., New York, (1971).

    Google Scholar 

  7. R. Fuhrmann, A. Garcia and F. Torres, On maximal curves, J. Number Theory, 67(1) (1997), 29–51.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Fuhrmann and F. Torres, The genus of curves over finite fields with many rational points, Manuscripta Math., 89 (1996), 103–106.

    Article  MathSciNet  MATH  Google Scholar 

  9. [9] A. Garcia and L. Quoos, A construction of curves over finite fields, to appear in Acta Arithmetica.

    Google Scholar 

  10. A. Garcia and H. Stichtenoth, On Chebyshev polynomials and maximal curves, Acta Arithmetica, 90 (1999), 301–311.

    MathSciNet  MATH  Google Scholar 

  11. A. Garcia, H. Stichtenoth and C. P. Xing, On subfields of the Hermitian function field, Compositio Math., 120 (2000), 137–170.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Garcia and F. Torres, On maximal curves having classical Weierstrass gaps, Contemporary Math., 245 (1999), 49–59.

    Article  MathSciNet  Google Scholar 

  13. G. van der Geer and M. van der Vlugt, Fibre products of Artin-Schreier curves and generalized Hamming weights of codes, J. Comb. Theory A, 70 (1995), 337–348.

    Google Scholar 

  14. G. van der Geer and M. van der Vlugt, Generalized Hamming weights of codes and curves over finite fields with many points, Israel Math. Conf. Proc., 9 (1996), 417–432.

    Google Scholar 

  15. V. D. Goppa, Geometry and Codes, Mathematics and its applications, 24, Kluwer Academic Publishers, Dordrecht-Boston-London (1988).

    Google Scholar 

  16. J. W. P. Hirschfeld, Projective geometries over finite fields, second edition, Oxford University Press, Oxford (1998).

    MATH  Google Scholar 

  17. Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokio, 28 (1981), 721–724.

    MathSciNet  MATH  Google Scholar 

  18. A. I. Kontogeorgis, The group of automorphisms of function fields of the curve xn + ytn +1 = 0, J Number Theory, 72 (1998), 110–136.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Lachaud, Sommes d’Eisenstein et nombre de points de certaines courbes alg¨¦briques sur les corps finis, C.R. Acad. Sci. Paris, 305, S¨¦rie I (1987), 729–732.

    Google Scholar 

  20. H. W. Leopoldt, Ober die Automorphismengruppe des Fermatkörpers, J. Number Theory, 56 (1996), 256–282.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of mathematics and its applications, 20 Addison-Wesley (1983).

    Google Scholar 

  22. C. J. Moreno, Algebraic curves over finite fields, Cambridge University Press, 97 (1991).

    Google Scholar 

  23. H. G. Rück and H. Stichtenoth, A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math., 457 (1994), 185–188.

    MATH  Google Scholar 

  24. F. K. Schmidt, Zur arithmetischen Theorie der algebraischen Funktionen II. Allgemeine Theorie der Weierstrasspunkte, Math. Z., 45 (1939), 75–96.

    Article  MathSciNet  Google Scholar 

  25. S. A. Stepanov, Arithmetic of algebraic curves, Monographs in contemporary mathematics, Consultants Bureau, New York - London (1994).

    Google Scholar 

  26. H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik, Arch. Math., 24 (1973), 527–544 and 615–631.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Stichtenoth and C. P. Xing, The genus of maximal function fields over finite fields, Manuscripta Math., 86 (1995), 217–224.

    Article  MathSciNet  MATH  Google Scholar 

  28. K. O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), 1–19.

    Article  MATH  Google Scholar 

  29. [29] A. Weil, Courbes alg¨¦briques et vari¨¦t¨¦s abeliennes,Hermann, Paris, (1971).IMPA,Estrada Dona Castorina 110 22.460–320, Rio de Janeiro, RJ BrasilE-mail address: garcia@impa.br

    Google Scholar 

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Author information

Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22.460-320, Brasil

    Arnaldo Garcia

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  1. Arnaldo Garcia
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Editor information

Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Carles Casacuberta  & Joan Verdera  & 

  2. Departament d’Algebra i Geometria Facultat de Matemàtiques, Universitat de Barcelona, 08007, Barcelona, Spain

    Rosa Maria MirĂł-Roig

  3. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, 08028, Barcelona, Spain

    SebastiĂ  XambĂł-Descamps

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© 2001 Springer Basel AG

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Garcia, A. (2001). Curves over Finite Fields Attaining the Hasse-Weil Upper Bound. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_15

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  • DOI: https://doi.org/10.1007/978-3-0348-8266-8_15

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9496-8

  • Online ISBN: 978-3-0348-8266-8

  • eBook Packages: Springer Book Archive

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