A Geometric Procedure with Prover9

  • Ranganathan Padmanabhan
  • Robert Veroff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7788)


Here we give an automated proof of the fact that a cubic curve admits at most one group law. This is achieved by proving the tight connection between the chord-tangent law of composition and any potential group law (as a morphism) on the curve. An automated proof of this is accomplished by implementing the rigidity lemma and the Cayley-Bacharach theorem of algebraic geometry as formal inference rules in Prover9, a first-order theorem prover developed by Dr. William McCune.


Elliptic Curve Inference Rule Elliptic Curf Theorem Prover 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.
    Knapp, A.: Elliptic Curves. Princeton University Press (1992)Google Scholar
  2. 2.
    McCune, W.: Otter 3.0 Reference Manual and Guide. Tech. Report ANL-94/6, Argonne National Laboratory, Argonne, IL (1994),
  3. 3.
    McCune, W.: Prover9, version 2009-02A,
  4. 4.
    Mumford, D.: Abelian varieties. Tata Institute of Fundamental Research. Studies in Mathematics, vol. 5. Oxford University Press, London (1970)zbMATHGoogle Scholar
  5. 5.
    Padmanabhan, R.: Logic of equality in geometry. Discrete Mathematics 15, 319–331 (1982)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Padmanabhan, R., McCune, W.: Automated reasoning about cubic curves. Computers and Mathematics with Applications 29(2), 17–26 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Padmanabhan, R., McCune, W.: Uniqueness of Steiner laws on cubic curves. Beiträge Algebra Geom. 47(2), 543–557 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Padmanabhan, R., Veroff, R.: A geometric procedure with Prover9 (Web support) (2012),
  9. 9.
    Padmanabhan, R., Veroff, R.: A gL clause generator (2012),
  10. 10.
    Silverman, J., Tate, J.: Rational Points on Elliptic Curves. Springer (1992)Google Scholar
  11. 11.
    Veroff, R.: Solving open questions and other challenge problems using proof sketches. J. Automated Reasoning 27, 157–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ranganathan Padmanabhan
    • 1
  • Robert Veroff
    • 2
  1. 1.University of ManitobaWinnipegCanada
  2. 2.University of New MexicoAlbuquerqueUSA

Personalised recommendations