On a unit group generated by special values of Siegel modular functions

Abstract

There has been important progress in constructing units and S-units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyper-elliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field k6 of Q(exp(2πi/5)) modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that k6 = Q(exp(2πi/15), 5√-24). Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.