On S-Units for Linear Valuations and the Periodicity of Continued Fractions of Generalized Type in Hyperelliptic Fields

Abstract

An equivalence theorem is proved for the following conditions: the periodicity of continued fractions of generalized type for key elements of hyperelliptic field \(L\), the existence of nontrivial \(S\)-units in \(L\) for sets \(S\) consisting two valuations of degree one, and the existence of a torsion of certain type in the Jacobian variety associated with hyperelliptic field \(L\). In practice, this theorem allows using continued fractions of generalized type to effectively search for fundamental \(S\)-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, which shows all three equivalent conditions in the indicated theorem.