On -units for valuations of the second degree in hyperelliptic fields

Abstract

Abstract In this paper we propose a new effective approach to the problem of finding and constructing non-trivial -units of a hyperelliptic field for a set consisting of two conjugate valuations of the second degree. The results obtained are based on a deep connection between the problem of torsion in the Jacobians of hyperelliptic curves and the quasiperiodicity of continued -fractions, that is, generalized functional continued fractions of special form constructed with respect to a valuation of the second degree. We find algorithms for searching for fundamental -units which are comparable in effectiveness with known fast algorithms for two linear valuations.