Abstract

We give a signed fundamental domain for the action on Rr+1 × C∗r2 of the totally positive units E+ of a number field k of degree n = r1 + 2r2 which we assume is not totally complex. Here r1 and r2 denote the number of real and complex places of k and R+ denotes the positive real numbers. The signed fundamental domain consists of n-dimensional k-rational cones Cα, each equipped with a sign µα = ±1, with the property that the net number of intersections of the cones with any E+-orbit is 1. The cones Cα and the signs µα are explicitly constructed from any set of fundamental totally positive units and a set of 3r2 “twisters”, i.e. elements of k whose arguments at the r2 complex places of k are sufficiently varied. Introducing twisters gives us the right number of generators for the cones Cα and allows us to make the Cα turn in a controlled way around the origin at each complex embedding.