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A function field related to the Ree group

Part of the Lecture Notes in Mathematics book series (LNM,volume 1518)

Abstract

We construct an algebraic function field over a finite field of characteristic 3, which has the Ree group as automorphism group. In this way, we obtain an explicit construction of the Ree group. We also prove, that the function field has as many rational places as possible, and that the number for certain extensions of the ground field reaches the Hasse-Weil bound.

Keywords

  • algebraic function fields
  • Ree groups
  • Hasse-Weil bound

The author is with Mathematical Institute, Building 303, The Technical University of Denmark, DK-2800 Lyngby, Denmark.

This research was done while the author visited The Department of Discrete Mathematics, Eindhoven Technical University, The Netherlands. The visit was supported by a grant from The Danish Research Academy.

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References

  1. E. Bombieri, Thompson's Problem2=3), Inv. Math. 58, p. 77–100 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. R.W. Carter, “Finite Groups of Lie type”, John Wiley & Sons Ltd., (1985).

    Google Scholar 

  3. C. Chevalley, “Introduction to the Theory of Algebraic Functions of One Variable”, A.M.S., Providence, RI (1951).

    CrossRef  MATH  Google Scholar 

  4. P. Deligne and G. Lusztig, Representations of Reductive Groups over Finite Fields, Ann. Math. 103, p. 103–161 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. D. Gorenstein, “Finite Simple Groups”, Plenum Press, New York (1982).

    CrossRef  MATH  Google Scholar 

  6. J.P. Hansen, Deligne-Lusztig Varieties and Group Codes, to appear in these proceedings.

    Google Scholar 

  7. H.-W. Henn, Funktionenkörper mit großer Automorphismengruppe, J. Reine Angew. Math. 172, p. 96–115 (1978).

    MathSciNet  MATH  Google Scholar 

  8. A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41, p. 403–442 (1893).

    CrossRef  MathSciNet  Google Scholar 

  9. W.M. Kantor, M.E. O'Nan and G.M. Seitz, 2-Transitive Groups in which the Stabilizer of Two Points is Cyclic, J. Alg. 21, p. 17–50 (1972).

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. J. Lewittes, Places of Degree One in Function Fields over Finite Fields, J. Pure App. Alg. 69, p. 177–183 (1990).

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. G. Lusztig, Coxeter Orbits and Eigenspaces of Frobenius, Invent. Math. 38, p. 101–159 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. J.P. Pedersen and A.B. Sørensen, Codes from certain Algebraic Function Fields with many Rational Places, Mat-Report 1990-11, Tech. Uni. Denmark.

    Google Scholar 

  13. C. Peskine and L. Szpiro, Liaison des Variétés Algébriques I, Inv. Math. 26, p. 271–302 (1974).

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. R. Ree, A Family of Simple Groups associated with the Simple Lie Algebra of Type (G 2), Am. J. Math. 83, p. 432–462 (1961).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. H.L. Schmid, Über die Automorphismen eines algebraischen Funktionenkörpers von Primzahlcharacteristik, J. Reine Angew. Math. 179, p. 5–15 (1938).

    MathSciNet  MATH  Google Scholar 

  16. B. Segre, Forme e Geometrie Hermitiane, con particolare riguardo al Caso Finito, Ann. Mat. Pura Appl. 70, p. 1–201.

    Google Scholar 

  17. J.-P. Serre, Sur le Nombre des Points Rationnels d'une Courbe Algébrique sur un Corps Fini, C. R. Acad. Sci. Paris Sér. I Math. 296, p. 397–402 (1983).

    MATH  Google Scholar 

  18. J.-P. Serre, Algèbre et Géométrie, Ann. Collège de France 1983–1984, p. 79–84.

    Google Scholar 

  19. H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharacteristik, Teil I: Eine Abschätzung der Ordnung der Automorphismengruppe, Archiv Math. 24, p. 527–544 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharacteristik, Teil II: Ein spezieller Typ von Funktionenkörpern, Archiv Math. 24, p. 615–631 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1992 Springer-Verlag

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Pedersen, J.P. (1992). A function field related to the Ree group. In: Stichtenoth, H., Tsfasman, M.A. (eds) Coding Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087997

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  • DOI: https://doi.org/10.1007/BFb0087997

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55651-0

  • Online ISBN: 978-3-540-47267-4

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