On the problem of periodicity of continued fractions in hyperelliptic fields

V. P. Platonov ; G. V. Fedorov

Sbornik. Mathematics, Tome 209 (2018) no. 4, pp. 519-559

Résumé

We present new results concerning the problem of periodicity of continued fractions which are expansions of quadratic irrationalities in a field $K((h))$, where $K$ is a field of characteristic different from 2, $h \in K[x]$, $\deg h=1$. Let $f \in K[h]$ be a square-free polynomial and suppose that the valuation $v_h$ of the field $K(x)$ has two extensions $v_h^-$ and $v_h^+$ to the field $L=K(h)(\sqrt{f})$. We set $S_h=\{v_h^-,v_h^+\}$. A deep connection between the periodicity of continued fractions in the field $K((h))$ and the existence of $S_h$-units made it possible to make great advances in the study of periodic and quasiperiodic elements of the field $L$, and also in problems connected with searching for fundamental $S_h$-units. Using a new efficient algorithm to search for solutions of the norm equation in the field $L$ we manage to find examples of periodic continued fractions of elements of the form $\sqrt{f}$, which is a fairly rare phenomenon. For the case of an elliptic field $L=\mathbb{Q}(x)(\sqrt{f})$, $\deg f=3$, we describe all square-free polynomials $f \in \mathbb{Q}[h]$ with a periodic expansion of $\sqrt{f}$ into a continued fraction in the field $\mathbb{Q}((h))$. Bibliography: 16 titles.

Détail

Keywords: hyperelliptic fields, continued fractions, periodicity, $S$-units
Mots-clés : problem of torsion in Jacobian.