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

Abstract

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 $$\sqrt f /d{h^s}$$ , 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 $$\sqrt f $$ 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.