Signed Shintani cones for number fields with one complex place

Abstract

We give a signed fundamental domain for the action on ${\mathbb{C}}^{\ast }\times {\mathbb{R}}_{+}^{n-2}$ of the totally positive units $E(k{)}_{+}$ of a number field $k$ of degree $n$ and having exactly one pair of complex embeddings. This signed fundamental domain, built of $k$-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of $k$.

37 pages, 2 figures. arXiv admin note: text overlap with arXiv:1303.3989 by other authors