Explicit Towers of Drinfeld Modular Curves

  • Noam D. Elkies
Part of the Progress in Mathematics book series (PM, volume 202)


We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.


Finite Field Elliptic Curf Function Field Quadratic Extension Explicit Equation 
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  1. 1.
    S. G. Drinfeld and S. G. Vl¨¢dutt: The number of points of an algebraic curve.Functional Anal. Appl.17 (1983), #1, 53–54 (translated from the Russian paper inFunktsional. Anal. i Prilozhen). Google Scholar
  2. 2.
    N. D. Elkies: Linearized algebra and finite groups of Lie type, I: Linear and symplectic groups. Pages 77–108 inApplications of Curves over Finite Fields(1997 AMS-IMSSIAM Joint Summer Research Conference, July 1997, Washington, Seattle; M. Fried, ed.; Providence: AMS, 1999) =Contemp. Math.245.Google Scholar
  3. 3.
    N. D. Elkies: Explicit modular towers. Pages 23–32 in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (1997, T. Basar, A. Vardy, eds.), Univ. of Illinois at Urbana-Champaign 1998.Google Scholar
  4. 4.
    A. Garcia and H. Stichtenoth: A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vl¨¢,dut bound.Invent. Math.121 (1995), #1, 211–233.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A. Garcia and H. Stichtenoth: On the asymptotic behaviour of some towers of function fields over finite fields.J. Number Theory61 (1996), #2, 248–273.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. Garcia and H. Stichtenoth: Asymptotically good towers of function fields over finite fields.C. R. Acad. Sci. Paris 1322 (1996), #11, 1067–1070.Google Scholar
  7. 7.
    A. Garcia, H. Stichtenoth and M. Thomas: On towers and composita of towers of function fields over finite fields.Finite Fields and their Appl.3 (1997), #3, 257–274.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    E.-U. Gekeler: Drinfeld-Moduln und modulare Formen iiber rationalen Funktionenkörpern. Bonner Math. Schriften 119, 1980.Google Scholar
  9. 9.
    E.-U. Gekeler:Drinfeld Modular Curves.Berlin: Springer-Verlag, 1980 (Lecture Notes in Math. 1231).Google Scholar
  10. 10.
    E.-U. Gekeler: Über Drinfeld’sche Modulkurven vom Hecke-Typ.Compositio Math.57 (1986), #2, 219–236.Google Scholar
  11. 11.
    E.-U. Gekeler and U. Nonnengardt: Fundamental domains of some arithmetic groups over function fields.International J. Math.6 (1995), #5, 689–708.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    V. D. Goppa: Codes on algebraic curves.Soviet Math. Dokl.24 (1981), #1, 170–172.Google Scholar
  13. 13.
    D. Goss: 0x70-adic Eisenstein series for function fields.Compositio Math. 41(1980), #1, 3–38.Google Scholar
  14. 14.
    Y. Hamahata: Tensor products of Drinfeld modules and v-adic representations.Man’usc. Math. 79(1993), #3–4, 307–327.Google Scholar
  15. 15.
    Y. Ihara: Congruence relations and Shimura curves. Pages 291–311 ofAutomorphic Forms Representations,and L-functions (A. Borel and W. Casselman, eds.; Providence: AMS, 1979; Part 2 of Vol. 33 of Proceedings of Symposia in Pure Mathematics).Google Scholar
  16. 16.
    Y. Ihara: Some remarks on the number of rational points of algebraic curves over finite fields.J. Fac. Sci. Tokyo 28(1981), #3, 721–724.Google Scholar
  17. 17.
    M. A. Tsfasman and S. G. Vlá dut:Algebraic-Geometric Codes.Dordrecht: Kluwer, 1991.zbMATHCrossRefGoogle Scholar
  18. 18.
    M. A. Tsfasman, S. G. V1á,dutt and T. Zink: Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound.Math. Nachr. 109(1982), 21–28.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Basel AG 2001

Authors and Affiliations

  • Noam D. Elkies
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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