Attractor-repeller construction of Shintani domains for totally complex quartic fields

Abstract

The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields.To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields.