RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy MIAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 1 scientific paper (total in 1 paper)

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

Full text: PDF file (311 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 511.6
Received: April 10, 2018

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}


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

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  crossref  zmath  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
    Number of views:
    This page:71
    References:1
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019