On boundedness of period lengths of continued fractions of key elements hyperelliptic fields over the field of rational numbers

G. V. Fedorov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 357-370

Résumé

The problem of the periodicity of functional continued fractions of elements of a hyperelliptic field is closely related to the problem of finding and constructing fundamental $S$-units of a hyperelliptic field and the torsion problem in the Jacobian of the corresponding hyperelliptic curve. For elliptic curves over a field of rational numbers, the torsion problem was solved by B. Mazur in 1978. For hyperelliptic curves of genus 2 and higher over the field of rational numbers, the above three problems remain open. The theory of functional continued fractions has become a powerful arithmetic tool for studying these problems. In addition, tasks arising in the theory of functional continued fractions have their own interest. Sometimes these tasks have analogues in the numerical case, but tasks that are significantly different from the numerical case are especially interesting. One such problem is the problem of estimating from above the lengths of periods of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers. In this article, we find upper bounds on the period lengths for key elements of a hyperelliptic field over a field of rational numbers. In the case when the hyperelliptic field is defined by an odd degree polynomial, the period length of the elements under consideration is either infinite or does not exceed twice the degree of the fundamental $S$-unit. A more interesting and complicated case is when a hyperelliptic field is defined by a polynomial of even degree. In 2019, V. P. Platonov and G. V. Fedorov for hyperelliptic fields $L = \mathbb{Q}(x)(\sqrt{f})$, $\deg f = 2g + 2$, found the exact interval values $s \in \mathbb{Z}$ such that continued fractions of elements of the form $\sqrt{f}/h^s \in L \setminus \mathbb{Q}(x) $ are periodic. Using this result in this article, we find exact upper bounds on the period lengths of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers, depending only on the genus of the hyperelliptic field and the order of the torsion group of the Jacobian of the corresponding hyperelliptic curve.

Détail

Keywords: continued fractions, period length, fundamental units, $S$-units, torsion in the Jacobians, hyperelliptic fields, divisors, divisor class group.