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 hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field $k_6$ of $\mathbb {Q}(\exp (2\pi i/5))$ modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that $k_6=\mathbb {Q}(\exp (2\pi i/15), \sqrt [5]{-24} )$. 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.