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 Tr. Mat. Inst. Steklova, 2018, Volume 302, Pages 354–376 (Mi tm3923)

Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields

V. P. Platonov, M. M. Petrunin

Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Nakhimovskii pr. 36, korp. 1, Moscow, 117218 Russia

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 $\sqrt f/dh^s$, 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.

 Funding Agency Grant Number Russian Science Foundation 16-11-10111 This work is supported by the Russian Science Foundation under grant 16-11-10111.

DOI: https://doi.org/10.1134/S0371968518030184

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 336–357

Bibliographic databases:

UDC: 511.6

Citation: V. P. Platonov, M. M. Petrunin, “Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 354–376; Proc. Steklov Inst. Math., 302 (2018), 336–357

Citation in format AMSBIB
\Bibitem{PlaPet18} \by V.~P.~Platonov, M.~M.~Petrunin \paper Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields \inbook Topology and physics \bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday \serial Tr. Mat. Inst. Steklova \yr 2018 \vol 302 \pages 354--376 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3923} \crossref{https://doi.org/10.1134/S0371968518030184} \elib{http://elibrary.ru/item.asp?id=36503451} \transl \jour Proc. Steklov Inst. Math. \yr 2018 \vol 302 \pages 336--357 \crossref{https://doi.org/10.1134/S0081543818060184} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000454896300018} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85059460834} 

• http://mi.mathnet.ru/eng/tm3923
• https://doi.org/10.1134/S0371968518030184
• http://mi.mathnet.ru/eng/tm/v302/p354

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This publication is cited in the following articles:
1. 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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