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 $\big\{(C_\sigma,w_\sigma) \big\}_\sigma$ is signed since the net number of its intersections with any $E_+$-orbit is 1, i. e. for any $x\in \mathbb{R}^n_+$, $$ \sum_{\sigma\in S_{n-1}} \sum_{\varepsilon\in E_+} w_\sigma\chi^{\phantom{1}}_{C_\sigma}(\varepsilon x) = 1. $$ Here $\chi_{C_\sigma}$ is the characteristic function of $C_\sigma$, $w_\sigma=\pm1$ 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.