Shintani cones for complex cubic number fields

Abstract

In 1979, Shintani showed the existence of a fundamental domain, consisting of a finite number of [Formula: see text]-rational polyhedral cones, for the action of the units of a number field [Formula: see text] under its geometric embedding. Explicit construction of such cones is only known for totally real fields of degree [Formula: see text]. Here, we extend this to complex cubic fields using Espinoza’s work on signed Shintani domains for such fields.