On the Problem of Periodicity of Continued Fraction Expansions of $\sqrt f $ for Cubic Polynomials over Number Fields

Academician of the RAS V. P. Platonov^a^,^b^,*, M. M. Petrunin^a^,**, and V. S. Zhgoon^a^,***

^aScientific Research Institute for System Analysis, Russian Academy of Sciences, Moscow, 117218 Russia

^bSteklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia

Correspondence to: *e-mail: platonov@mi-ras.ru
Correspondence to: **e-mail: petrushkin@yandex.ru
Correspondence to: ***e-mail: zhgoon@mail.ru

Received 17 June, 2020

Abstract—We obtain a complete description of fields $\mathbb{K}$ that are quadratic extensions of $\mathbb{Q}$ and of cubic polynomials $f \in \mathbb{K}[x]$ for which a continued fraction expansion of $\sqrt f $ in the field of formal power series $\mathbb{K}((x))$ is periodic. We also prove a finiteness theorem for cubic polynomials $f \in \mathbb{K}[x]$ with a periodic expansion of $\sqrt f $ over cubic and quartic extensions of $\mathbb{Q}$.

Keywords: elliptic field, S-units, continued fractions, periodicity, modular curves, points of finite order

DOI: 10.1134/S1064562420040249