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 Chebyshevskii Sb., 2018, Volume 19, Issue 3, Pages 282–297 (Mi cheb695)

Periodic continued fractions and $S$-units with second degree valuations in hyperelliptic fields

G. V. Fedorov

Scientific Research Institute of System Analysis (SRISA/NIISI RAS), Moscow

Abstract: Based on the method of continued fractions by now the problem of the existence and construction of nontrivial $S$-units is deeply studied in hyperelliptic fields in the case when the set $S$ consists of two linear valuations. This article is devoted to a more general problem, namely the problem of the existence and construction of fundamental $S$-units in hyperelliptic fields for sets $S$ containing valuations of the degree $2$. The key case when the set $S = S_h$ consists two conjugate valuations, connected with an irreducible polynomial $h$ of the degree $2$. The main results were obtained using the theory of generalized functional continued fractions in conjunction with the geometric approach to the problem of torsion in Jacobian varieties of hyperelliptic curves.
We have developed a theory of generalized functional continued fractions and the divisors of the hyperelliptic field associated with them, constructed with the help of valuations of the degree $2$. This theory allowed us to find new effective methods for searching and constructing fundamental $S_h$-units in hyperelliptic fields.
As a demonstration of the results, we consider in detail algorithm to search for fundamental $S_h$-units for hyperelliptic fields of genus $3$ over the field of rational numbers and give explicit computational examples of hyperelliptic fields $L = \mathbb{Q}(x)(\sqrt{f})$ for polynomials $f$ of degree $7$, possessing fundamental $S_h$-units of large powers.

Keywords: continued fractions, fundamental units, $S$-units, torsion in the Jacobians, hyperelliptic curves, divisors, the group of divisor classes.

 Funding Agency Grant Number Russian Science Foundation 16-11-10111

DOI: https://doi.org/10.22405/2226-8383-2018-19-3-282-297

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UDC: 511.6
Accepted:15.10.2018

Citation: G. V. Fedorov, “Periodic continued fractions and $S$-units with second degree valuations in hyperelliptic fields”, Chebyshevskii Sb., 19:3 (2018), 282–297

Citation in format AMSBIB
\Bibitem{Fed18} \by G.~V.~Fedorov \paper Periodic continued fractions and $S$-units with second degree valuations in hyperelliptic fields \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 3 \pages 282--297 \mathnet{http://mi.mathnet.ru/cheb695} \crossref{https://doi.org/10.22405/2226-8383-2018-19-3-282-297} \elib{https://elibrary.ru/item.asp?id=39454404} 

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• http://mi.mathnet.ru/eng/cheb/v19/i3/p282

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This publication is cited in the following articles:
1. G. V. Fedorov, “On $S$-units for valuations of the second degree in hyperelliptic fields”, Izv. Math., 84:2 (2020), 392–435
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