New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields

V. P. Platonov ; M. M. Petrunin

Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 278-286.

Résumé

We study the problem of describing square-free polynomials $f(x)$ of odd degree with periodic expansion of $\sqrt {f(x)}$ into a functional continued fraction in $k((x))$, where $k\subseteq \overline {\mathbb Q}$. We obtain a complete description of such polynomials $f(x)$ that does not depend on the field $k$ and the degree of a polynomial, provided that the degree $U$ of the fundamental $S$-unit of the corresponding hyperelliptic field $k(x)(\sqrt {f(x)})$ either does not exceed $12$ or is even and does not exceed $20$.

Bibliographie

[1] Abel N.H., “Ueber die Integration der Differential-Formel $\frac {\rho dx}{\sqrt R}$, wenn $R$ und $\rho $ ganze Functionen sind”, J. reine angew. Math., 1 (1826), 185–221 | MR

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

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

[4] 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 (2014), 1–34 | DOI | DOI | MR | Zbl

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

[6] V. P. Platonov and G. V. Fedorov, “On the problem of classification of periodic continued fractions in hyperelliptic fields”, Russ. Math. Surv., 75:4 (2020), 785–787 | DOI | DOI | MR | Zbl

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

[8] V. P. Platonov and M. M. Petrunin, “On the finiteness of the number of expansions into a continued fraction of $\sqrt {f}$ for cubic polynomials over algebraic number fields”, Dokl. Math., 102:3 (2020), 487–492 | DOI | DOI | MR | Zbl

[9] V. P. Platonov, M. M. Petrunin, and Yu. N. Shteinikov, “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

[10] V. P. Platonov, M. M. Petrunin, and Yu. N. Shteinikov, “On the periodicity problem for the continued fraction expansion of elements of hyperelliptic fields with fundamental $S$-units of degree at most 11”, Dokl. Math., 104:5 (2021), 258–263 | DOI | DOI | MR | Zbl

[11] V. P. Platonov, M. M. Petrunin, and V. S. Zhgoon, “On the problem of periodicity of continued fraction expansions of $\sqrt {f}$ for cubic polynomials over number fields”, Dokl. Math., 102:1 (2020), 288–292 | DOI | DOI | MR | Zbl

[12] V. P. Platonov, V. S. Zhgoon, and M. M. Petrunin, “On the problem of periodicity of continued fraction expansions of $\sqrt {f}$ for cubic polynomials $f$ over algebraic number fields”, Sb. Math., 213:3 (2022), 412–442 | DOI | DOI | MR | Zbl

[13] V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, and Yu. N. Shteinikov, “On the finiteness of hyperelliptic fields with special properties and periodic expansion of $\sqrt {f}$”, Dokl. Math., 98:3 (2018), 641–645 | DOI | MR | Zbl

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

[15] Sutherland A.V., “Constructing elliptic curves over finite fields with prescribed torsion”, Math. Comput., 81:278 (2012), 1131–1147 | DOI | MR | Zbl

[16] Tchebichef P., “Sur l'intégration des différentielles qui contiennent une racine carrée d'un polynome du troisième ou du quatrième degré”, J. math. pures appl., 2 (1857), 1–42