Signed fundamental domains for totally real number fields

Abstract

We give a signed fundamental domain for the action on ${\mathbb{R}}_{+}^n$ of the totally positive units $E_{+}$ of a totally real number field $k$ of degree $n$. The domain $\{(C_{\sigma },w_{\sigma }){\}}_{\sigma }$ is signed since the net number of its intersections with any $E_{+}$-orbit is 1, i. e. for any $x\in {\mathbb{R}}_{+}^n$, $$\sum _{\sigma \in S_{n-1}}\sum _{\epsilon \in E_{+}}{w}_{\sigma }{\chi }_{C_{\sigma }}^{}(\epsilon x)=1.$$ Here ${\chi }_{C_{\sigma }}$ is the characteristic function of $C_{\sigma }$, $w_{\sigma }=\pm 1$ is a natural orientation of the $n$-dimensional $k$-rational cone $C_{\sigma }\subset {\mathbb{R}}_{+}^n$, and the inner sum is actually finite. Signed fundamental domains are as useful as Shintani's true ones for the purpose of calculating abelian $L$-functions. They have the advantage of being easily constructed from any set of fundamental units, whereas in practice there is no algorithm producing Shintani's $k$-rational cones. Our proof uses algebraic topology on the quotient manifold ${\mathbb{R}}_{+}^n/E_{+}$. The invariance of the topological degree under homotopy allows us to control the deformation of a crooked fundamental domain into nice straight cones. Crossings may occur during the homotopy, leading to the need to subtract some cones.

To appear in Proc. London Math. Soc

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