Fedorov, G. V. On fundamental \(S\)-units and continued fractions constructed in hyperelliptic fields using two linear valuations. (English. Russian original) Zbl 1475.11133 Dokl. Math. 103, No. 3, 151-156 (2021); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 498, 65-70 (2021). Summary: In this paper, for elements of hyperelliptic fields, the theory of functional continued fractions of generalized type associated with two linear valuations has been formulated for the first time. For an arbitrary element of a hyperelliptic field, the continued fraction of generalized type converges to this element for each of the two selected linear valuations of the hyperelliptic field. Denote by \(S\) the set consisting of these two linear valuations. We find equivalent conditions describing the relationship between the quasi-periodicity of a continued fraction of generalized type, the existence of a fundamental \(S\)-unit, and the existence of a class of divisors of finite order in the divisor class group of a hyperelliptic field. The last condition is equivalent to the existence of a torsion point in the Jacobian of a hyperelliptic curve. These results complete the algorithmic solution of the periodicity problem in the Jacobians of hyperelliptic curves of genus two. MSC: 11J70 Continued fractions and generalizations 11R58 Arithmetic theory of algebraic function fields 11R27 Units and factorization Keywords:continued fraction; fundamental \(S\)-unit; hyperelliptic field; divisor class group PDFBibTeX XMLCite \textit{G. V. Fedorov}, Dokl. Math. 103, No. 3, 151--156 (2021; Zbl 1475.11133); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 498, 65--70 (2021) Full Text: DOI References: [1] Platonov, V. P., Russ. Math. Surv., 69, 1-34 (2014) · Zbl 1305.11096 · doi:10.1070/RM2014v069n01ABEH004877 [2] Mazur, B., Ivent. Math., 44, 129-162 (1978) · Zbl 0386.14009 [3] Platonov, V. P.; Fedorov, G. V., Sb. Math., 209, 519-559 (2018) · Zbl 1445.11135 · doi:10.1070/SM8998 [4] Platonov, V. P.; Fedorov, G. V., Russ. Math. Surv., 75, 785-787 (2020) · Zbl 1451.11127 · doi:10.1070/RM9962 [5] Artin, E., Math. Z., 19, 153-246 (1924) · JFM 50.0107.01 · doi:10.1007/BF01181074 [6] Adams, W. W.; Razar, M. J., Proc. London Math. Soc., 41, 481-498 (1980) · Zbl 0403.14002 · doi:10.1112/plms/s3-41.3.481 [7] Platonov, V. P.; Petrunin, M. M., Proc. Steklov Inst. Math., 302, 336-357 (2018) · Zbl 1440.11127 · doi:10.1134/S0081543818060184 [8] Fedorov, G. V., Dokl. Math., 102, 513-517 (2020) · Zbl 1476.11016 · doi:10.1134/S1064562420060101 [9] Berry, T. G., Arch. Math., 55, 259-266 (1990) · Zbl 0728.14027 · doi:10.1007/BF01191166 [10] Zhgoon, V. S., Chebyshev. Sb., 18, 208-220 (2017) · Zbl 1434.11132 · doi:10.22405/2226-8383-2017-18-4-208-220 [11] Zannier, U., Am. J. Math., 141, 1-40 (2019) · Zbl 1422.11156 · doi:10.1353/ajm.2019.0000 [12] Fedorov, G. V., Izv. Math., 84, 392-435 (2020) · Zbl 1457.11159 · doi:10.1070/IM8888 [13] D. Mumford, Tata Lectures on Theta (Birkhäuser, Boston, 1983, 1984), Vols. 1, 2. · Zbl 0509.14049 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.