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Construction of Modular Curves and Computation of Their Cardinality over Fp

  • Cédric Tavernier
Conference paper

Abstract

Following [3], and in using several results, we describe an algorithm which compute with a level N given the cardinality over F p of the Jacobian of elliptic curves and hyperelliptic curves of genus 2 which come from X 0(N). We will also sketch how to get a plane affine model for these curves.

Keywords

Modular Form Elliptic Curf Abelian Variety Hyperelliptic Curve Representative System 
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References

  1. 1.
    John Cremona. Arithmetic of modular elliptique curves. Cambridge University Press, 1992.Google Scholar
  2. 2.
    Bas Edixhoven. The modular curves X 0(N). In Trieste, ICTP, Summer school on elliptic curves, 1997.Google Scholar
  3. 3.
    Gerhard Frey and Michael Müller. Arithmetic of modular curves and application. In Matzat, B.H; Greuel, G.M.; Hiss, G. editor, Algorithmic Algebra and Number Theory, Springer-Verlag, 1999.Google Scholar
  4. 4.
    G. Shimura. Introduction to the arithmetic theory of automorphic Functions, Princeton University Press, 1971.MATHGoogle Scholar
  5. 5.
    Loïc Merel. Universal fourier expansions of modular forms. Lecture Notes in Mathematics, 1994.Google Scholar
  6. 6.
    Jean-Francois Mestre. Construction de courbes de genre 2 à partir de leur modules. Effective Methods in Algebraic Geometry, 1991.Google Scholar
  7. 7.
    Joseph Milne. Elliptic curves. Available on http://www.jmilne.org/math/CourseNotes/math679.html 1996.
  8. 8.
    Jean-Pierre Serre. Cours d’arithmétique. Presses Univ. France, 1970.Google Scholar
  9. 9.
    William A. Stein. The modular forms database. Available on, 1999. http://modular.fas.harvard.edu/Tables/index.htm1
  10. 10.
    Xiangdong Wang. 2-dimensional simple factors of J 0(N). Manuscripta Mathematica, 1995.Google Scholar
  11. 11.
    Xiangdong Wang. The Hecke operators on the cusp-forms of Γ 0(N). In G. Frey, editor, On Artin’s Conjecture for Odd 2-dimensional Representations, number 1585 in Lecture notes in Mathematics, pages 59–94. Springer-Verlag, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Cédric Tavernier
    • 1
  1. 1.INRIA RocquencourtLe ChesnayFrance

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