Number Theory pp 229-250 | Cite as

Rational Desingularization of a Curve Defined Over a Finite Field

  • A. T. Vasquez
Conference paper


In 1983, the IEEE’s “paper of the year” in Information Theory was Modular Curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound [18]—its authors are Tsfasman, Vladut, and Zink. As the title suggests, the paper established the existence of families of codes with asymptotically “better” parameters than had been thought possible.


Prime Ideal Maximal Ideal Algebraic Curf Valuation Ring Minimal Polynomial 
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  1. [1]
    E. Artin. Elements of Algebraic Geometry. Notes. Courant Institute of Mathematical Sciences, NYU, 1955.Google Scholar
  2. [2]
    G. A. Bliss. Algebraic Functions. Dover Publications, Inc., 1966.zbMATHGoogle Scholar
  3. [3]
    D. Le Brigand & J. J. Risler. Algorithme de Brill-Noether et codes de Goppa. Bull. Soc. Math. France 116 (1988), 231–253.MathSciNetzbMATHGoogle Scholar
  4. [4]
    C. Chevalley. Introduction to the theory of algebraic functions of one variable, Mathematical Surveys 6. American Mathematical Society, 1951.CrossRefzbMATHGoogle Scholar
  5. [5]
    C. Chevalley. Fundamental concepts of algebra, Pure and applied mathematics; a series of monograghs and textbooks 7. Academic Press, 1956.zbMATHGoogle Scholar
  6. [6]
    W. Fulton. Algebraic Curves. Benjamin, 1969.zbMATHGoogle Scholar
  7. [7]
    V. D. Goppa. Codes on algebraic curves. Soviet Math. Dokl. 24 #1 (1981), 170–172.zbMATHGoogle Scholar
  8. [8]
    V. D. Goppa. Geometry and Codes. Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers, Dordrecht, the Netherlands, 1988.CrossRefzbMATHGoogle Scholar
  9. [9]
    R. Harts Horne. Algebraic Geometry, Graduate Texts in Mathematics 52. Springer-Verlag, 1977.CrossRefGoogle Scholar
  10. [10]
    S. Lang. Introduction to Algebraic Geometry, Interscience Tracts in Pure and Applied Mathematics 5. Interscience Publishers, Inc., 1958.zbMATHGoogle Scholar
  11. [11]
    S. Lang.Algebra. Addison-Wesley, 2nd edn., 1984.zbMATHGoogle Scholar
  12. [12]
    M.D. Fried & M. Jarden. Field Arithmetic, Ergebn. der Math, und ihrer Grenz.: 3. Folge 11. Springer-Verlag, 1986.CrossRefzbMATHGoogle Scholar
  13. [13]
    M.F. Atiyah & I.G. MacDonald. Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Addison-Wesley, 1969.zbMATHGoogle Scholar
  14. [14]
    M. Pohst & H. Zassenhaus. Algorithmic algebraic number theory. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 1989.CrossRefzbMATHGoogle Scholar
  15. [15]
    A. Seidenberg. Elements of the Theory of Algebraic Curves. Addison-Wesley, 1969.Google Scholar
  16. [16]
    J. H. Silverman. The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Springer-Verlag, 1986.CrossRefzbMATHGoogle Scholar
  17. [17]
    B. M. Trager. Integration of Algebraic Functions. Ph.D. thesis, MIT, 1984.Google Scholar
  18. [18]
    M. A. Tsfasman, S. G. Vladut, & Th. Zink. Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound. Math. Nachr. 109 (1982), 21–28.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    B. L. van der Waerden. Modern Algebra. Frederick Ungar Publishing Co., revised english edition edn., 1953.Google Scholar
  20. [20]
    J. H. van Lint & G. van der Geer. Introduction to Coding Theory and Algebraic Geometry, DMV seminar Bd. 12. Birkhäuser Verlag, 1988.CrossRefzbMATHGoogle Scholar
  21. [21]
    S. G. Vléduts & Yu. I. Manin. Linear codes and modular curves. J. of Sov. Math. 30 #6 (1985), 2611–2643.CrossRefzbMATHGoogle Scholar
  22. [22]
    R. J. Walker. Algebraic Curves. Princeton University Press, 1950.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • A. T. Vasquez
    • 1
  1. 1.Mathematics Department, Graduate School and University CenterCity University of New YorkNew YorkUSA

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