Groups of S-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form f/dhs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt f /d{h^s}$$\end{document}, where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt f $$\end{document} is periodic. We also analyze the continued fraction expansion of the key element f/hg+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt f /{h^{g + 1}}$$\end{document}, which defines the set of quasiperiodic elements of a hyperelliptic field.