Twisters and signed fundamental domains for number fields

Abstract

We give a signed fundamental domain for the action on R(+)(r1 )x C*(r2) of the totally positive units E+ of a number field k of degree n = r(1 )+ 2r(2 )which we assume is not totally complex. Here r(1) and r(2) 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 C-alpha, each equipped with a sign mu(alpha) = +/- 1, with the property that the net number of intersections of the cones with any E+-orbit is 1.

The cones C(alpha )and the signs mu(alpha)( )are explicitly constructed from any set of fundamental totally positive units and a set of 3(r2) "twisters", i.e. elements of k whose arguments at the r(2) complex places of k are sufficiently varied. Introducing twisters gives us the right number of generators for the cones C-alpha and allows us to make the C(alpha )turn in a controlled way around the origin at each complex embedding.