On some properties of continued periodic fractions with small length of period related with hyperelliptic fields and $S$-units

Yu. V. Kuznetsov ; Yu. N. Shteinikov

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 261-268.

Résumé

Let $\mathbb{Q}$ be a field of rational numbers, let $\mathbb{Q}(x)$ be the field of rational functions of one variable, and let $ f\in \mathbb{Q}[x]$ be a squarefree polynomial of odd degree which is equal to $2g+1, g>0$. Suppose that for a polynomial $h$ of degree $1$ the discrete valuation $\nu_{h} $ which is uniquely defined on $\mathbb{Q}(x)$ has two nonequivalent extensions to the field $L = \mathbb{Q}(x)(\sqrt{f})$ and $\nu'_{h}$ is one of these extensions. We put $S =\{\nu'_{h},\nu_{\infty}\}$, where $\nu_{\infty}$ is an infinite valuation of the field $L$. In the paper [4] V. P. Platonov and M. M. Petrunin (see also [2]) obtained that a $S$-unit in $L$ exists if and only if the element $\frac{\sqrt{f}}{h^{g+1}}$ expands into the periodic infinite continuous functional fraction. In this paper we study continuous periodic fractions connected with this expansion. For some small values of the length of period and quasiperiod we obtained estimates for the degrees of corresponding fundamental $S$-units and some necessary conditions with which the elements of these fractions have to satisfy. In the proof we use the results obtained by V. P. Platonov and M. M. Petrunin in the paper [4].

Détail

Keywords: continued fractions, hyperelliptic fields, $S$-units, valuation.

Bibliographie

[1] Platonov V. P., Petrunin M. M., “Fundamental $S$-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves”, Dokl. Math., 92:3 (2015), 667–669 | DOI | MR | Zbl

[2] Platonov V. P., Petrunin M. M., “S-units and periodicity of continued fractions in hyperelliptic fields”, Dokl. Math., 92:3 (2015), 752–756 | DOI | MR | Zbl

[3] Platonov V. P., Petrunin M. M., “S-Units and periodicity in quadratic function fields”, Russian Math. Surveys, 71:5 (2016), 973–975 | DOI | DOI | MR | Zbl

[4] Platonov V. P., Petrunin M. M., “S-units in hyperelliptic fields and periodicity of continued fractions”, Dokl. Math., 94:2 (2016), 532–537 | DOI | MR | Zbl

[5] V. P. Platonov, V. S. Zhgoon, G. V. Fedorov, “Continued Rational Fractions in Hyperelliptic Fields and the Mumford Representation”, Dokl. Math., 94:3 (2016), 692–696 | DOI | MR | Zbl

[6] Platonov V. P., “Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field”, Russian Math. Surveys, 69:1 (2014), 1–34 | DOI | DOI | MR | Zbl

[7] Benyash-Krivets V. V., Platonov V. P., “Groups of S-units in hyperelliptic fields and continued fractions”, Sb. Math., 200:11 (2009), 1587–1615 | DOI | DOI | MR | Zbl

[8] Schmidt W. M., “On continued fractions and diophantine approximation in power series fields”, Acta Arithm., 95:2 (2000), 139–166 | DOI | MR | Zbl

[9] Benyash-Krivets V. V., Platonov V. P., “S-units in hyperelliptic fields”, Russian Math. Surveys, 62:4 (2007), 784–786 | DOI | DOI | MR | Zbl

[10] Benyash-Krivets V. V., Platonov V. P., “Groups of $S$-units in hyperelliptic fields”, Dokl. Math., 417:4 (2007), 446–450 | Zbl

[11] Benyash-Krivets V. V., Platonov V. P., “Continued fractions and S-units in hyperelliptic fields”, Russian Math. Surveys, 63:2 (2008), 357–359 | DOI | DOI | MR | Zbl

[12] Platonov V. P., Fedorov G. V., “On the periodicity of continued fractions in hyperelliptic fields”, Dokl. Math., 95:3 (2017), 254–258 | DOI | MR | Zbl

[13] Platonov V. P., Fedorov G. V., “On the periodicity of continued fractions in elliptic fields”, Dokl. Math., 475:2 (2017), 133–136 | DOI | Zbl

[14] Adams W. W., Razar M. J., “Multiples of point on elliptic curves and continued fractions”, Proc. London Math. Soc., 41:3 (1980), 481–498 | DOI | MR | Zbl

[15] Leprevost F. Points rationnels de torsion de jacobiennes de certaines courbes de genre 2, C.R. Acad. Sci. Paris, 316:8 (1993), 819–821 | MR | Zbl