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Izv. RAN. Ser. Mat., 2020, Volume 84, Issue 2, Pages 197–242 (Mi izv8888)  

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

On $S$-units for valuations of the second degree in hyperelliptic fields

G. V. Fedorovab

a Lomonosov Moscow State University
b Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow

Abstract: In this paper we propose a new effective approach to the problem of finding and constructing non-trivial $S$-units of a hyperelliptic field $L$ for a set $S=S_h$ consisting of two conjugate valuations of the second degree. The results obtained are based on a deep connection between the problem of torsion in the Jacobians of hyperelliptic curves and the quasiperiodicity of continued $h$-fractions, that is, generalized functional continued fractions of special form constructed with respect to a valuation of the second degree. We find algorithms for searching for fundamental $S_h$-units which are comparable in effectiveness with known fast algorithms for two linear valuations.

Keywords: generalized continued fractions, hyperelliptic curves, fundamental $S$-units, divisor class group, torsion group of a Jacobian variety.

DOI: https://doi.org/10.4213/im8888

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English version:
Izvestiya: Mathematics, 2020, 84:2, 392–435

Bibliographic databases:

UDC: 511.6
MSC: 11R58
Received: 04.12.2018
Revised: 29.03.2019

Citation: G. V. Fedorov, “On $S$-units for valuations of the second degree in hyperelliptic fields”, Izv. RAN. Ser. Mat., 84:2 (2020), 197–242; Izv. Math., 84:2 (2020), 392–435

Citation in format AMSBIB
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\by G.~V.~Fedorov
\paper On $S$-units for valuations of the second degree in hyperelliptic fields
\jour Izv. RAN. Ser. Mat.
\yr 2020
\vol 84
\issue 2
\pages 197--242
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\crossref{https://doi.org/10.4213/im8888}
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\transl
\jour Izv. Math.
\yr 2020
\vol 84
\issue 2
\pages 392--435
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. P. Platonov, G. V. Fedorov, “O probleme klassifikatsii mnogochlenov $f$ s periodicheskim razlozheniem $\sqrt{f}$ v nepreryvnuyu drob v giperellipticheskikh polyakh”, Izv. RAN. Ser. matem., 85:5 (2021), 152–189  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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