Skip to main content
SpringerLink
Log in

Determined Sequences, Continued Fractions, and Hyperelliptic Curves

  • Conference paper
Algorithmic Number Theory (ANTS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 4076))

Included in the following conference series:

Abstract

In this report I sanitise (in the sense of ‘bring some sanity to’) the arguments of earlier reports detailing the correspondence between sequences (M+hS) − − ∞ < h < ∞  of divisors on elliptic and genus two hyperelliptic curves, the continued fraction expansion of quadratic irrational functions in the relevant elliptic and hyperelliptic function fields, and certain integer sequences satisfying relations of Somos type. I note that one may often readily determine the coefficients in those relations by elementary linear algebra.

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. Adams, W.W., Razar, M.J.: Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc. 41, 481–498 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berry, T.G.: A type of hyperelliptic continued fraction. Monatshefte Math. 145(4), 269–283 (2005)

    Article  MATH  Google Scholar 

  3. Braden, H.W., Enolskii, V.Z., Hone, A.N.W.: Bilinear recurrences and addition formulæ for hyperelliptic sigma functions. J. Nonlin. Math. Phys. 12(suppl. 2), 46–62 (2005), Also at: http://www.arxiv.org/math.NT/0501162

  4. Cantor, D.G.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48(177), 95–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cantor, D.G.: On the analogue of the division polynomials for hyperelliptic curves. J. für Math (Crelle) 447, 91–145 (1994)

    MATH  MathSciNet  Google Scholar 

  6. Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence Sequences. In: Mathematical Surveys and Monographs. American Mathematical Society, vol. 104, pp. xiv+318 (2003)

    Google Scholar 

  7. Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. in Appl. Math. 28, 119–144 (2002), 21p. Also at: http://www.arxiv.org/math.CO/0104241

  8. Gale, D.: The strange and surprising saga of the Somos sequences. The Mathematical Intelligencer 13(1), 40–42 (1991); Somos sequence update. Ibid. 13(4), 49–50

    Article  MathSciNet  Google Scholar 

  9. Hone, A.N.W.: Elliptic curves and quadratic recurrence sequences. Bull. London Math. Soc. 37, 161–171 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lauter, K.E.: The equivalence of the geometric and algebraic group laws for Jacobians of genus 2 curves. In: Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD, 2001). Contemp. Math., vol. 324, pp. 165–171. Amer. Math. Soc, Providence, RI (2001)

    Google Scholar 

  11. van der Poorten, A.J.: Periodic continued fractions and elliptic curves. In: van der Poorten, A., Stein, A. (eds.) High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications, vol. 42, pp. 353–365. American Mathematical Society (2004)

    Google Scholar 

  12. van der Poorten, A.J.: Elliptic sequences and continued fractions. Journal of Integer Sequences 8(05.2.5), 1–19 (2005)

    Google Scholar 

  13. van der Poorten, A.J.: Curves of genus 2, continued fractions, and Somos sequences. Journal of Integer Sequences 8(05.3.4), 1–9 (2005)

    Google Scholar 

  14. van der Poorten, A.J., Swart, C.S.: Recurrence relations for elliptic sequences: every Somos 4 is a Somos k. Bull. London Math. Soc. (to appear), At: http://arxiv.org/math.NT/0412293

  15. Propp, J.: The Somos Sequence Site, http://www.math.wisc.edu/~propp/somos.html

  16. Shipsey, R.: Elliptic divisibility sequences, Phd Thesis, Goldsmiths College, University of London (2000), At: http://homepages.gold.ac.uk/rachel/

  17. Sloane, N.: On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/

  18. Swart, C.: Elliptic curves and related sequences. PhD Thesis, Royal Holloway, University of London (2003), http://www.isg.rhul.ac.uk/alumni/thesis/swart_c.pdf

  19. Swart, C., Hone, A.: Integrality and the Laurent phenomenon for Somos 4 sequences, 18p. At: http://www.arxiv.org/math.NT/0508094

  20. Ward, M.: Memoir on elliptic divisibility sequences. Amer. J. Math. 70, 31–74 (1948)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Number Theory Research, 1 Bimbil Place, Killara, NSW 2071, Australia

    Alfred J. van der Poorten

Authors
  1. Alfred J. van der Poorten
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Technische Universität Berlin, Germany

    Florian Hess

  2. Institut für Mathematik, MA 8–1, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Sebastian Pauli

  3. Institut für Mathematik, Technische Universität Berlin, Fakultät II, MA 8-1, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Michael Pohst

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

van der Poorten, A.J. (2006). Determined Sequences, Continued Fractions, and Hyperelliptic Curves. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_28

Download citation

  • DOI: https://doi.org/10.1007/11792086_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-36075-9

  • Online ISBN: 978-3-540-36076-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation