On S-units for linear valuations and the periodicity of continued fractions of generalized type in hyperelliptic fields

Abstract

This article proves the equivalence theorem for the following conditions: the periodicity of continued fractions of a generalized type for key elements hyperelliptic field L, the existence in the hyperelliptic field L of nontrivial S-units for sets S, consisting two valuations of degree one, and the existence of the torsion of a certain type in the Jacobian variety, associated with the hyperelliptic field L. This theorem allows in practice using continued fractions of a generalized type effectively search for fundamental S-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, showing all three equivalent conditions in the indicated theorem.