Advertisement

On the Periodicity of Continued Fractions in Hyperelliptic Fields

  • Gleb V. Fedorov
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 69)

Abstract

Let L be a function field of a hyperelliptic curve defined over an arbitrary field characteristic different from 2. We construct an arithmetic of continued fractions of an arbitrary quadratic irrationality in field of formal power series with respect to linear finite valuation. The set of infinite valuation and finite linear valuation of L is denoted by S. As an application, we have found a relationship between the issue of the existence of nontrivial S-units in L and periodicity of continued fractions of some key elements of L.

References

  1. 1.
    Adams, W.W., Razar, M.J.: Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc. 41(3), 481–498 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benyash-Krivets, V.V., Platonov, V.P.: Continued fractions and S-units in hyperelliptic fields. Russ. Math. Surv. 63(2), 357–359 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benyash-Krivets, V.V., Platonov, V.P.: Groups of S-units in hyperelliptic fields and continued fractions. Sb. Math. 200(11), 1587–1615 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berry, T.G.: On periodicity of continued fractions in hyperelliptic function fields. Arch. Math. 55, 259–266 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cantor, D.G.: Computing in the Jacobian of a Hyperelliptic Curve. Math. comput. 48(177), 95–101 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fedorov, G.V., Platonov, V.P.: S-Units and Periodicity of Continued Fractions in Hyperelliptic Fields. Dokl. Math. 92(3), 1–4 (2015)zbMATHGoogle Scholar
  7. 7.
    Koblitz, N.: “Algebraic aspect of cryptography”, with an appendix on Hyperelliptic curves. In: Menezes, A.J., Yi-Hong Wu., Zuccherato, R.J. (eds.) Algorithms and Computation in Mathematics, vol. 3, Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Lang, S.: Introduction to Diophantine Approximations. Columbia University, New York (1966)zbMATHGoogle Scholar
  9. 9.
    Platonov, V.P.: Arithmetic of quadratic fields and torsion in Jacobians. Dokl. Math. 81(1), 55–57 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Platonov, V.P.: Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field. Russ. Math. Surv. 69(1), 1–34 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mechanics and Mathematics FacultyMoscow State UniversityMoscowRussia
  2. 2.Research Institute of System Development, Russian Academy of SciencesMoscowRussia

Personalised recommendations