Explicit Form of Fundamental Units of High Degrees in Hyperelliptic Fields of Genus 2 over the Field of Rational Numbers

Abstract

Earlier, polynomials [ ] f x  were obtained and described, for which the hyperelliptic field ( )( ) x f contains fundamental units of high degrees. This is equivalent to the existence of -torsion points of high order in the Jacobians of the corresponding hyperelliptic curves. For a series of problems, such as the classification of components of complex curves with pairs of conjugate torsion points, not only the polynomial f is required, but also the explicit form of the fundamental unit of the corresponding hyperelliptic field. In this paper, the explicit form of fundamental units for hyperelliptic fields of genus 2 with torsion points in the Jacobian of order 33, 36, and 48 is presented for the first time.