Abstract
The article proves the following statement: in any hyperelliptic field L defined over the field of algebraic numbers K which having non-trivial units of the ring of integer elements of the field L, there is an element for which the period length of the continued fraction is greater any pre-given number.
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REFERENCES
V. V. Benyash-Krivets and V. P. Platonov, “Groups of S-units in hyperelliptic fields and continued fractions,” Sb. Math. 200 (11), 1587–1615 (2009).
V. P. Platonov, “Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field,” Russ. Math. Surv. 69 (1), 1–34 (2014).
W. M. Schmidt, “On continued fractions and diophantine approximation in power series fields,” Acta Arith. 95 (2), 139–166 (2000).
S. Lang, Introduction to Diophantine Approximations: New Expanded Edition (Springer Science and Business Media, 2012).
A. Lasjaunias, “A survey of diophantine approximation in fields of power series,” Monatsh. Math. 130, 211–229 (2000).
V. P. Platonov and G. V. Fedorov, “On the classification problem for polynomials f with a periodic continued fraction expansion of \(\sqrt f \) in hyperelliptic fields,” Izv. Math. 85 (5), 972–1007 (2021).
V. P. Platonov, “Arithmetic of quadratic fields and torsion in Jacobians,” Dokl. Math. 81 (1), 55–57 (2010).
U. Zannier, “Hyperelliptic continued fractions and generalized Jacobians,” Am. J. Math. 141 (1), 1–40 (2019).
G. V. Fedorov, “On the period length of a functional continued fraction over a number field,” Dokl. Math. 102 (3), 513–517 (2020).
V. P. Platonov and G. V. Fedorov, “On the problem of periodicity of continued fractions in hyperelliptic fields,” Sb. Math. 209 (4), 519–559 (2018).
V. P. Platonov and G. V. Fedorov, “On the problem of classification of periodic continued fractions in hyperelliptic fields,” Russ. Math. Surv. 75 (4), 785–787 (2020).
G. V. Fedorov, “Continued fractions and the classification problem for elliptic fields over quadratic fields of constants,” Math. Notes 114 (6), 1203–1219 (2023).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. (Oxford Univ. Press, Oxford, 1979).
Funding
This work was supported by the Ministry of Science and Higher Education of the Russian Federation as part of the state assignment, project no. FNEF-2024-0001.
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Translated by I. Ruzanova
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Platonov, V.P., Fedorov, G.V. Continued Fractions in Hyperelliptic Fields with an Arbitrarily Long Period. Dokl. Math. (2024). https://doi.org/10.1134/S1064562424701928
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DOI: https://doi.org/10.1134/S1064562424701928