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On S-Units for Linear Valuations and the Periodicity of Continued Fractions of Generalized Type in Hyperelliptic Fields

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Abstract

An equivalence theorem is proved for the following conditions: the periodicity of continued fractions of generalized type for key elements of hyperelliptic field \(L\), the existence of nontrivial \(S\)-units in \(L\) for sets \(S\) consisting two valuations of degree one, and the existence of a torsion of certain type in the Jacobian variety associated with hyperelliptic field \(L\). In practice, this theorem allows using continued fractions of generalized type to effectively search for fundamental \(S\)-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, which shows all three equivalent conditions in the indicated theorem.

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Authors and Affiliations

  1. Scientific Research Institute for System Analysis, Russian Academy of Sciences, 117218, Moscow, Russia

    V. P. Platonov & G. V. Fedorov

  2. Steklov Mathematical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    V. P. Platonov

  3. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991, Moscow, Russia

    G. V. Fedorov

Authors
  1. V. P. Platonov
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  2. G. V. Fedorov
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Correspondence to V. P. Platonov or G. V. Fedorov.

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Translated by I. Ruzanova

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Platonov, V.P., Fedorov, G.V. On S-Units for Linear Valuations and the Periodicity of Continued Fractions of Generalized Type in Hyperelliptic Fields. Dokl. Math. 99, 277–281 (2019). https://doi.org/10.1134/S1064562419030116

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  • DOI: https://doi.org/10.1134/S1064562419030116

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