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Groups of S-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

  • V. P. PlatonovEmail author
  • M. M. Petrunin
Article
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Abstract

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form \(\sqrt f /d{h^s}\), where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element \(\sqrt f \) is periodic. We also analyze the continued fraction expansion of the key element \(\sqrt f /{h^{g + 1}}\), which defines the set of quasiperiodic elements of a hyperelliptic field.

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Scientific Research Institute for System Analysis of the Russian Academy of SciencesMoscowRussia

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