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 $\sqrt f/dh^s$, 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 $\sqrt f$ is periodic. We also analyze the continued fraction expansion of the key element $\sqrt f/h^{g+1}$, which defines the set of quasiperiodic elements of a hyperelliptic field.