Periodic elements $\sqrt{f}$ in elliptic fields with a field of constants of zero characteristic

V. P. Platonov ; M. M. Petrunin ; Yu. N. Shteinikov

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 273-296.

Résumé

A study of the periodicity problem of functional continued fractions of elements of elliptic and hyperelliptic fields was begun about 200 years ago in the classical papers of N. Abel and P. L. Chebyshev. In 2014 V. P. Platonov proposed a general conceptual method based on the deep connection between three classical problems: the problem of the existence and construction of fundamental $S$-units in hyperelliptic fields, the torsion problem in Jacobians of hyperelliptic curves, and the periodicity problem of continued fractions of elements of hyperelliptic fields. In 2015-2019, in the papers of V. P. Platonov et al. was made great progress in studying the problem of periodicity of elements in hyperelliptic fields, especially in the effective classification of such periodic elements. In the papers of V. P. Platonov et al, all elliptic fields $\mathbb{Q}(x)(\sqrt{f})$ were found such that $\sqrt{f}$ decomposes into a periodic continued fraction in $\mathbb{Q}((x))$, and also futher progress was obtained in generalizing the indicated result, as to other fields of constants, and to hyperelliptic curves of genus $2$ and higher. In this article, we provide a complete proof of the result announced by us in 2019 about the finiteness of the number of elliptic fields $k(x)(\sqrt{f})$ over an arbitrary number field $k$ with periodic decomposition of $\sqrt{f}$, for which the corresponding elliptic curve contains a $k$-point of even order not exceeding $18$ or a $k$-point of odd order not exceeding $11$. For an arbitrary field $k$ being quadratic extension of $\mathbb{Q}$ all such elliptic fields are found, and for the field $k = \mathbb{Q}$ we obtained new proof about of the finiteness of the number of periodic $\sqrt{f}$, not using the parameterization of elliptic curves and points of finite order on them.

Détail

Keywords: elliptical field, hyperelliptic field, periodicity, continued fractions, period length, fundamental units, $S$-units, resultant, Gröbner basis, quadratic irrationality.

Bibliographie

[1] Abel N. H., “Ueber die integration der differential-formel $\rho dx/\sqrt{R}$ wenn r und $\rho$ ganze functionen sind”, Journal für die reine und angewandte Mathematik, 1 (1826), 185–221 | MR | Zbl

[2] Tchebicheff P., “Sur l'intégration des différentielles qui contiennent une racine carrée d'un polynome du troisieme ou du quatrieme degré'”, Journal des math. pures et appl., 2 (1857), 168–192

[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 | DOI | MR | Zbl

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

[5] Adams W. W., Razar M. J., “Multiples of points on elliptic curves and continued fractions”, Proceedings of the London Mathematical Society, 41 (1980), 481–498 | DOI | MR | Zbl

[6] 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

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

[8] Petrunin M. M., “S-units and periodicity of square root in hyperelliptic fields”, Dokl. Math., 95:3 (2017), 222–225 | DOI | 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 | DOI | MR | Zbl

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

[11] 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

[12] Platonov V. P., Zhgoon V. S., Petrunin M. M., Shteinikov Yu. N., “On the finiteness of hyperelliptic fields with special properties and periodic expansion of $\sqrt{f}$”, Dokl. Math., 98 (2018), 603–608 | Zbl

[13] Platonov V. P., Zhgoon V. S., Fedorov G.V., “Continued rational fractions in hyperelliptic fields and the Mumford representation”, Dokl. Math., 482:2 (2018), 137–141

[14] 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 | MR | Zbl

[15] Kenku M. A., Momose F., “Torsion points on elliptic curves defined over quadratic fields”, Nagoya Mathematical Journal, 109 (1988), 125–149 | DOI | MR | Zbl

[16] Kamienny Sheldon, “Torsion points on elliptic curves and q-coefficients of modular forms”, Inventiones mathematicae, 109:1 (1992), 221–229 | DOI | MR | Zbl

[17] Kamienny S., Najman F., “Torsion groups of elliptic curves over quadratic fields”, Acta Arithmetica, 3:152 (2012), 291–305 | DOI | MR | Zbl

[18] 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 | DOI | MR | Zbl

[19] Platonov V. P., Petrunin M. M., Shteinikov Yu. N., “On the finiteness of the number of elliptic fields with given degrees of S-units and periodic expansion of $\sqrt{f}$”, Dokl. Math., 100:2 (2019), 1–5 | DOI | MR | Zbl

[20] Wieb Bosma, John Cannon, Catherine Playoust, “The Magma algebra system. I. The user language”, J. Symbolic Comput., 24 (1997), 235–265 | DOI | MR | Zbl

[21] SageMath, the Sage Mathematics Software System (Version 9.0), , The Sage Developers, 2020 https://www.sagemath.org

[22] Mazur B., “Rational isogenies of prime degree”, Inventiones Mathematicae, 44:2 (1978), 129–162 | DOI | MR | Zbl