Colmez cones for fundamental units of totally real cubic fields

Using work of Colmez, we give a quick algorithm for obtaining a clean fundamental domain for the action on of the totally positive units of a totally real cubic field. The fundamental domain consists of two infinite solid cones in R3, one generated by 1,ε1 and ε1ε2, the other by 1,ε2 and ε1ε2. Here ε1,ε2 are certain fundamental totally positive units, included in by the usual geometric embedding, which we show to be easily computable from any set of fundamental units of k. Similar cones were found by Thomas and Vasquez in 1980, and by Halbritter and Pohst in 2000, but their methods did not result in practical algorithms.