On estimates for the period length of functional continued fractions over algebraic number fields

G.V. Fedorov

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 162-189

Résumé

The paper investigates upper bounds on the period length of functional continued fractions for key elements of hyperelliptic fields over number fields. In the case when the hyperelliptic field is given by a polynomial of odd degree, the finite period length is trivially estimated from above twice the power of the fundamental $S$-unit. A more interesting and complicated case is when the hyperelliptic field is given by a polynomial of even degree. In 2019 V.P. Platonov and G.V. Fedorov for hyperelliptic fields $\mathcal{L} = \mathbb{Q}(x)(\sqrt{f})$, $\deg f = 2g+2$, over the field $\mathbb{Q}$ of rational numbers the exact interval of values $s \in \mathbb{Z}$ is found such that the continued fractions of elements of the form $\sqrt{f}/x^s \in \mathcal{L} \setminus \mathbb{Q}(x)$ are periodic. In this article, we find a generalization of this result for an arbitrary field as a field of constants. Based on this result, sharp upper estimates for the lengths of the periods are found functional continued fractions of elements of hyperelliptic fields over number fields $K$, depending only on the genus $g$ of the hyperelliptic field, the degree of extension $k = [K:\mathbb{Q}]$ and order $m$ of the Jacobian torsion subgroup of the corresponding hyperelliptic curve.

Détail

Keywords: Continued fractions, period length, hyperelliptic field, fundamental $S$-units, torsion problem in Jacobians.