The criterion of periodicity of continued fractions of key elements in hyperelliptic fields

V. P. Platonov   ; G. V. Fedorov

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 248-260

Résumé

The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $L = \mathbb{Q}(x)(\sqrt {f})$ has a more complex nature, than the periodicity of the numerical continued fractions of the elements of a quadratic fields. It is known that the periodicity of a continued fraction of the element $\sqrt{f}/h^{g + 1}$, constructed by valuation associated with a polynomial $h$ of first degree, is equivalent to the existence of nontrivial $S$-units in a field $L$ of the genus $g$ and is equivalent to the existence nontrivial torsion in a group of classes of divisors. This article has found an exact interval of values of $s \in \mathbb{Z}$ such that the elements $\sqrt {f}/h^s $ have a periodic decomposition into a continued fraction, where $f \in \mathbb{Q}[x] $ is a squarefree polynomial of even degree. For polynomials $f$ of odd degree, the problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s $ are discussed in the article [5], and it is proved that the length of the quasi-period does not exceed degree of the fundamental $S$-unit of $L$. The problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s$ for polynomials $f$ of even degree is more complicated. This is underlined by the example we found of a polynomial $f$ of degree $4$, for which the corresponding continued fractions have an abnormally large period length. Earlier in the article [5] we found examples of continued fractions of elements of the hyperelliptic field $L$ with a quasi-period length significantly exceeding the degree of the fundamental $S$-unit of $L$.

Détail

Keywords: continued fractions, fundamental units, $S$-units, torsion in the Jacobians, hyperelliptic fields, divisors, divisor class group.

Bibliographie

[1] N. H. Abel, “Uber die Integration der Differential-Formel $\rho \text{d}x / \sqrt{R}$, wenn $R$ und $\rho$ ganze Functionen sind”, J. Reine Angew. Math., 1826, no. 1, 185–221 | MR | Zbl

[2] P. L. Chebychev, “Sur l'integration de la differential $\frac{x+A}{\sqrt{x^4 + \alpha x^3 + \beta x^2 + \gamma}}\text{d}x$”, J. Math. Pures Appl., 2:9 (1864), 225–246

[3] 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 | MR | Zbl

[4] T. G. Berry, “On periodicity of continued fractions in hyperelliptic function fields”, Arch. Math., 55 (1990), 259–266 | MR | Zbl

[5] Platonov V. P., Fedorov G. V., “On the problem of periodicity of continued fractions in hyperelliptic fields”, Sb. Math., 209:4 (2018), 519–559 | MR | Zbl

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

[7] Fedorov G. V., “Periodic continued fractions and S-units with second degree valuations in hyperelliptic fields”, Chebyshevskii Sbornik, 19:3 (2018) (In Russ.) | Zbl

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

[9] Platonov V. P., Fedorov G. V., “On the periodicity of continued fractions in elliptic fields”, Dokl. Math., 96:1 (2017), 332–335 | MR | Zbl

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

[11] Platonov V. P., Petrunin M. M., “Groups of S-units and the problem of periodicity of continued fractions in hyperelliptic fields”, Proc. Steklov Inst. Math., 302 (2018), 336–357 | MR | Zbl

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

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

[14] V. S. Zhgoon, “On generalized jacobians and rational continued fractions in the hyperelliptic fields”, Chebyshevskii Sbornik, 18:4 (2017), 208–220 (In Russ.) | MR

[15] Platonov V. P., Fedorov G. V., “S-Units and Periodicity of Continued Fractions in Hyperelliptic Fields”, Dokl. Math., 92:3 (2015), 752–756 | MR | Zbl

[16] D. S. Kubert, “Universal bounds on the torsion of elliptic curves”, Proc. London Math. Soc. (3), 33:2 (1976), 193–237 | MR | Zbl