Rocky Mountain Journal of Mathematics

Infinite Descent on Elliptic Curves

Samir Siksek

Full-text: Open access

Article information

Rocky Mountain J. Math. Volume 25, Number 4 (1995), 1501-1538.

First available in Project Euclid: 5 June 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11Y16: Algorithms; complexity [See also 68Q25]

Elliptic curves Diophantine equations computational number theory Mordell-Weil group


Siksek, Samir. Infinite Descent on Elliptic Curves. Rocky Mountain J. Math. 25 (1995), no. 4, 1501--1538. doi:10.1216/rmjm/1181072159.

Export citation


  • B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves I, J. Reine Angew. Math. 212 (1963), 7-25.
  • J.P. Buhler, B.H. Gross and D.B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), 473-481.
  • A. Bremner, On the equation $Y^2=X(X^2+p)$, in Number theory and applications (R.A. Mollin, ed.), Kluwer, Dordrecht, 3-23, 1989.
  • A. Bremner and J.W.S. Cassels, On the equation $Y^2=X(X^2+p)$, Math. Comp. 42 (1984), 257-264.
  • J.W.S. Cassels, Lectures on elliptic curves, LMS Student Texts, Cambridge University Press, 1991.
  • --------, Rational quadratic forms, LMS Monographs, Academic Press, London, 1978.
  • --------, Introduction to the geometry of numbers, Springer-Verlag, 1959.
  • H. Cohen, A course in computational algebraic number theory, GTM 138 -1993.
  • J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 1992.
  • V.A. Dem'janenko, An estimate of the remainder term in Tate's formula, Mat Zametki 3 (1968), 271-278, in Russian.
  • J. Gebel and H.G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over $Q$, in Elliptic curves and related topics (H. Kisilevsky and M. Ram. Murty, ed.), CRM Proceedings and Lecture Notes Volume 4, Amer. Math. Soc., 1994.
  • J. Gebel, A. Pethő and H.G. Zimmer, Computing integral points on elliptic curves, Acta Arith., to appear.
  • D. Husemoller, Elliptic curves, Springer-Verlag, 1987.
  • T.J. Kretschmer, Construction of elliptic curves with large rank, Math. Comp. 46 (1986), 627-635.
  • J.-F. Metre, Construction d'une courbe elliptique de rang $\ge12$, C.R. Acad. Sci. Paris 295 (1982), 643-644.
  • C. Batut, D. Bernardi, H. Cohen and M. Olivier, User's guide to PARI-GP (version 1.38.62), 1994.
  • H.A. Priestley, Introduction to complex analysis, Oxford University Press, 1985.
  • C.L. Siegel, Lectures on the geometry of numbers, Springer-Verlag, 1988.
  • S. Siksek, Descents on curves of genus 1, Ph.D. thesis, Exeter University, 1995.
  • J.H. Silverman, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743.
  • --------, The arithmetic of elliptic curves, GTM 106 -1986.
  • --------, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339-358.
  • N.P. Smart, $S$-integral points on elliptic curves, Proc. Camb. Phil. Soc. 116 (1994), 391-399.
  • N.P. Smart and N.M. Stephens, Integral points on elliptic curves over number fields, to appear.
  • R.J. Stroeker and J. Top, On the equation $Y^2=(X+p)(X^2+p^2)$, Rocky Mountain J. Math. 24 (1994), 1135-1161.
  • R.J. Stroeker and N. Tzanakis, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177-196.
  • H.G. Zimmer, On the difference of the Weil height and the Neron-Tate height, Math. Z. 147 (1976), 35-51.