Space Elliptic Curves

  • Takashi Ono
Part of the The University Series in Mathematics book series (USMA)


In Chapter 3, we found a group structure on the plane cubic C(M, N) by a geometric construction (the cord-and-tangent method), which depends on properties peculiar to plane cubics. Since the space curve E(M, N), the solution space of equations of the Fibonacci-Fermat type, is biregularly equivalent to C(M, N), it carries a group structure, too. It certainly is nice to recognize a group structure on the set of solutions of a system of Diophantine equations. However, we soon find that it is extremely impractical to try to copy the group structure of C(M, N) on E(M, N) via the biregular equivalence.


Group Structure Elliptic Curf Elliptic Function Theta Function Abelian Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York (1964).Google Scholar
  2. 2.
    Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York (1966).Google Scholar
  3. 3.
    P. Du Val, Elliptic Functions and Elliptic Curves, Cambridge Univ. Press, Cambridge, UK (1973).zbMATHGoogle Scholar
  4. 4.
    A. Hurwitz and R. Courant, Funktionentheorie, Die Grund, d. math. Wiss., 3, Springer (1964).zbMATHGoogle Scholar
  5. 5.
    A. W. Knapp, Elliptic Curves, Princeton Univ. Press, Princeton, NJ (1992).zbMATHGoogle Scholar
  6. 6.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Grad. Texts in Math., Springer, New York (1984).zbMATHCrossRefGoogle Scholar
  7. 7.
    S. Lang, Diophantine Geometry, Interscience, New York (1962).zbMATHGoogle Scholar
  8. 8.
    S. Lang, Abelian Varieties, Interscience, New York (1959).Google Scholar
  9. 9.
    H. Rademacher, Topics in Analytic Number Theory, Die Grund, d. math. Wiss., 169, Springer, Berlin (1973).zbMATHCrossRefGoogle Scholar
  10. 10.
    J. H. Silverman, Arithmetic of Elliptic Curves, Grad. Texts in Math., 106, Springer, Heidelberg (1986).zbMATHCrossRefGoogle Scholar
  11. 11.
    J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergrad. Texts in Math., Springer (1992).zbMATHGoogle Scholar
  12. 12.
    J. Tannery and J. Molk, Eléments de la théorie des fonctions elliptiques, 4 vols., Gauthiers-Villars, Paris (1896).zbMATHGoogle Scholar
  13. 13.
    E. T. Whittaker and G. N. Watson, A Course of Modem Analysis, 4th ed., Cambridge Univ. Press, Cambridge, UK (1958).Google Scholar

Copyright information

© Takashi Ono 1994

Authors and Affiliations

  • Takashi Ono
    • 1
  1. 1.The Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations