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 Mat. Sb., 2018, Volume 209, Number 4, Pages 54–94 (Mi msb8998)

On the problem of periodicity of continued fractions in hyperelliptic fields

V. P. Platonov, G. V. Fedorov

Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow

Abstract: We present new results concerning the problem of periodicity of continued fractions which are expansions of quadratic irrationalities in a field $K((h))$, where $K$ is a field of characteristic different from 2, $h \in K[x]$, $\deg h=1$.
Let $f \in K[h]$ be a square-free polynomial and suppose that the valuation $v_h$ of the field $K(x)$ has two extensions $v_h^-$ and $v_h^+$ to the field $L=K(h)(\sqrt{f})$. We set $S_h=\{v_h^-,v_h^+\}$. A deep connection between the periodicity of continued fractions in the field $K((h))$ and the existence of $S_h$-units made it possible to make great advances in the study of periodic and quasiperiodic elements of the field $L$, and also in problems connected with searching for fundamental $S_h$-units. Using a new efficient algorithm to search for solutions of the norm equation in the field $L$ we manage to find examples of periodic continued fractions of elements of the form $\sqrt{f}$, which is a fairly rare phenomenon. For the case of an elliptic field $L=\mathbb{Q}(x)(\sqrt{f})$, $\deg f=3$, we describe all square-free polynomials $f \in \mathbb{Q}[h]$ with a periodic expansion of $\sqrt{f}$ into a continued fraction in the field $\mathbb{Q}((h))$.
Bibliography: 16 titles.

Keywords: hyperelliptic fields, continued fractions, periodicity, $S$-units, problem of torsion in Jacobian.

 Funding Agency Grant Number Russian Science Foundation 16-11-10111 This work was supported by the Russian Science Foundation under grant no. 16-11-10111.

DOI: https://doi.org/10.4213/sm8998

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English version:
Sbornik: Mathematics, 2018, 209:4, 519–559

Bibliographic databases:

UDC: 511.6
MSC: Primary 11R58; Secondary 11J70, 11R27

Citation: V. P. Platonov, G. V. Fedorov, “On the problem of periodicity of continued fractions in hyperelliptic fields”, Mat. Sb., 209:4 (2018), 54–94; Sb. Math., 209:4 (2018), 519–559

Citation in format AMSBIB
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• https://doi.org/10.4213/sm8998
• http://mi.mathnet.ru/eng/msb/v209/i4/p54

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This publication is cited in the following articles:
1. V. P. Platonov, M. M. Petrunin, “Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields”, Proc. Steklov Inst. Math., 302 (2018), 336–357
2. V. P. Platonov, V. S. Zhgoon, G. V. Fedorov, “On the periodicity of continued fractions in hyperelliptic fields over quadratic constant field”, Dokl. Math., 98:2 (2018), 430–434
3. V. P. Platonov, G. V. Fedorov, “An infinite family of curves of genus 2 over the field of rational numbers whose Jacobian varieties contain rational points of order 28”, Dokl. Math., 98:2 (2018), 468–471
4. V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, Yu. N. Shteinikov, “On the finiteness of hyperelliptic fields with special properties and periodic expansion of $\sqrt f$”, Dokl. Math., 98:3 (2018), 641–645
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