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 / d h 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 f is periodic. We also analyze the continued fraction expansion of the key element f / h g + 1 , which defines the set of quasiperiodic elements of a hyperelliptic field.