Skip to main content
SpringerLink
Log in

Rational Points Near Curves and Small Nonzero | x 3 − y 2| via Lattice Reduction

  • Conference paper
Algorithmic Number Theory (ANTS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1838))

Included in the following conference series:

Abstract

We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of | x 3 + y 3 −z 3| < M with 0 < x ≤y < z < N in heuristic time ≪ (logO(1) N ) M provided M ≫N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M < N 3), the computational costs are essentially as low as possible. Moreover the algorithm readily parallelizes. It not only yields new numerical examples but leads to theoretical results, difficult open questions, and natural generalizations. We also adapt our algorithm to investigate Hall’s conjecture: we find all integer solutions of 0 < |x 3 − y 2| ≪x 1/2 with x < X in time O(X 1/2logO(1) X). By implementing this algorithm with X = 1018 we shattered the previous record for x 1/2/|x 3 − y 2|. The O(X 1/2logO(1) X) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of (cx)3/2 for any positive rational c.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bernstein, D.J.: Enumerating solutions to p(a)+q(b)=r(c)+s(d). Math. of Computation (to appear)

    Google Scholar 

  2. Bremner, A.: Sums of three cubes. In: Number Theory (Halifax, Nova Scotia, 1994) (CMS Conf. Proc.15), pp. 87–91. AMS, Providence (1995)

    Google Scholar 

  3. Birch, B.J., Chowla, S., Hall Jr., M., Schinzel, A.: On the difference x3 – y2. Norske Vid. Selsk. Forh. 38, 65–69 (1965)

    MATH  MathSciNet  Google Scholar 

  4. Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59(2), 337–357 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruce, J.W., Wall, C.T.C.: On the classification of cubic surfaces. J. LondonMath. Soc. 19(2)(2), 245–256 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cremona, J.E.: Algorithms for modular elliptic curves. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  7. Colin, H.L.: New Bounds on Sphere Backings. Ph.D. thesis, Harvard (2000)

    Google Scholar 

  8. Crux Mathematicorum 8 (1982)

    Google Scholar 

  9. Conway, J.H., Sloane, N.J.A.: Sphere Backings, Lattices and Groups. Springer, New York (1993)

    Google Scholar 

  10. Conn, W., Vaserstein, L.N.: On sums of integral cubes. The Bademacher legacy to mathematics (University Park, 1992). Contemp. Math. AMS 166, 285–294 (1994)

    MathSciNet  Google Scholar 

  11. Danilov, L.V.: The Diophantine equation x3 – y2 = k and Hall’s conjecture. Math. Notes Acad. Set. USSB 32, 617–618 (1982)

    Article  Google Scholar 

  12. Davenport, H.: On f3(t) - g2(t). Norske Vid. Selsk. Forh. 38, 86–87 (1965)

    MATH  MathSciNet  Google Scholar 

  13. Elkies, N.D.: On A4 + B4 + C4 = D4. Math. of Computational 51(184), 825–835 (1988)

    MATH  MathSciNet  Google Scholar 

  14. Elkies, N.D.: ABC implies Mordell. International Math. Besearch Notices 7, 99–109 (1991)

    Article  MathSciNet  Google Scholar 

  15. Elkies, N.D.: Heegner point computations. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 122–133. Springer, Heidelberg (1994)

    Google Scholar 

  16. Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Berspectives on Number Theory: Points Near Curves and Hall’s Conjecture via Lattice Reduction 63 Proceedings of a Conference in Honor of A.O.L. Atkin, pp. 21–76. AMS/International Press (1998)

    Google Scholar 

  17. Elkies, N.D.: Shimura curve computations. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 1–47. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  18. Fricke, R.: Ueber eine einfache Gruppe von 504 Operationen. Math. Ann. 52, 321–339 (1899)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fulton, W., Harris, J.: Representation Theory: A First Course, GTM 129. Springer, New York (1991)

    Google Scholar 

  20. Gebel, J., Petho, A., Zimmer, H.G.: On Mordell’s equation. Compositio Math. 110, 335–367 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Guy, R.K.: Unsolved Problems in Number Theory. Springer, New York (1981)

    MATH  Google Scholar 

  22. Hall, M.: The Diophantine equation x3 – y2 = k. In: Atkin, A., Birch, B. (eds.) Computers in Number Theory, pp. 173–198. Academic Press, London (1971)

    Google Scholar 

  23. Heath-Brown, D.R.: The density of zeros of forms for which weak approximation fails. Math. of Computation 59(200), 613–623 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Heath-Brown, D.R.: The density of rational points on projective hypersurfaces (2000) (preprint)

    Google Scholar 

  25. Heath-Brown, D.R., Lioen, W.M., te Riele, H.J.J.: On solving the Diophantine equation x3 + y3 + z3 = k on a vector computer. Math. of Computation 61(203), 235–244 (1993)

    MATH  Google Scholar 

  26. Keller, W., Kulesz, L.: Courbes algebriques de genre 2 et 3 possedant de nombreux points rationnels. C. R. Acad. Sci. Paris, Ser. I Math. 321(11), 1469–1472 (1995)

    MATH  MathSciNet  Google Scholar 

  27. Koyama, K., Tsuruoka, U., Sekigawa, H.: On searching for solutions of the Diophantine equation x3 + y3 + z3 = n. Math. of Computation 66(218), 841–851 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lang, S.: Old and new conjectured diophantine inequalities. Bull. Amer. Math. Society 23, 37–75 (1990)

    Article  MATH  Google Scholar 

  29. Macbeath, A.M.: On a curve of genus. Proc. London Math. Soc. 7(15), 527–542 (1965)

    Article  MathSciNet  Google Scholar 

  30. Mahler, K.: Lectures on Transcendental Numbers. Lecture Notes in Math., vol. 546. Springer, Berlin (1976)

    MATH  Google Scholar 

  31. Mason, R.C.: Diophantine Equations over Function Fields. London Math. Soc. Lecture Notes Series 96. Cambridge Univ. Press, Cambridge (1984); See also pp. 149–157 in Springer LNM 1068 [=proceedings of Journees Arithmetiques 1983, Noordwijkerhout] (1984)

    MATH  Google Scholar 

  32. Oesterlé, J.: Nouvelles approches du theorémé de Fermat. Sem. Bourbaki 2(88), 694 (exposé)

    Google Scholar 

  33. Pila, J.: Geometric postulation of a smooth function and the number of rational points. Duke Math. J. 63, 449–463 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  34. Payne, G., Vaserstein, L.N.: Sums of three cubes. In: The Arithmetic of Function Fields, pp. 443–454. de Gruyter, Berlin (1992)

    Google Scholar 

  35. Stahlke, C.: Algebraic curves over Q with many rational points and minimal automorphism group. International Math. Research Notices 1, 1–4 (1997)

    Article  MathSciNet  Google Scholar 

  36. Takeuchi, K.: Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo 24, 201–212 (1977)

    MATH  Google Scholar 

  37. Weil, A.: Abstract versus classical algebraic geometry. In: Proceedings of the International Congress of Mathematicians, Amsterdam, vol. III, pp. 550–558 (1954)

    Google Scholar 

  38. Wildanger, K.: Uber das Losen von Einheiten- und Lndexformgleichungen in algebraischen Zahlkorpern mit einer Anwendung auf die Bestimmung alter ganzen Punkte einer Mordellschen Kurve. Ph.D. Thesis, TU Berlin, Berlin (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Noam D. Elkies

Authors
  1. Noam D. Elkies
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematisch Instituut, Universiteit Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Wieb Bosma

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Elkies, N.D. (2000). Rational Points Near Curves and Small Nonzero | x 3 − y 2| via Lattice Reduction. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_2

Download citation

  • DOI: https://doi.org/10.1007/10722028_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67695-9

  • Online ISBN: 978-3-540-44994-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation