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This article is cited in 3 scientific papers (total in 3 papers)
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.
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
Linking options:
https://www.mathnet.ru/eng/im8888https://doi.org/10.1070/IM8888 https://www.mathnet.ru/eng/im/v84/i2/p197
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Abstract page: | 295 | Russian version PDF: | 47 | English version PDF: | 23 | References: | 33 | First page: | 10 |
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