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Signed fundamental domains for totally real number fields. (English) Zbl 1325.11117
In his work on special values of abelian \(L\)-functions attached to a totally real number feld \(k\), T. Shintani found a fundamental domain for the action of the totally positive units \(E_{+}\) of \(k\) on \(\mathbb R^{[k:Q]}_{ +}\) [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393–417 (1976; Zbl 0349.12007)] consisting of a finite number of \(k\)-rational cones of varying dimensions. For quadratic fields and the totally real cubic fields the situation is almost simple. In the general case, the best result is due to P. Colmez [Invent. Math. 91, No. 2, 371–389 (1988; Zbl 0651.12010); ibid. 95, No. 1, 161–205 (1989; Zbl 0666.12008)]. Given independent totally positive units \(\varepsilon_{1},\ldots,\varepsilon_{n-1}\), he defined \((n-1)!\) explicit \(k\)-rational cones \(C_{s}=C_{s}(\varepsilon_{1},\ldots,\varepsilon_{n-1})\). When these units satisfy certain geometric conditions, Colmez proved that the union \(\{C_{s}\}_{s}\) of his cones is a fundamental domain for the action on \(\mathbb R^{n}_{+}\) of the group generated by the \(\varepsilon_{i}\).
In this paper, the authors 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 \({(C_{s},w_{s})}s\) is signed since the net number of its intersections with any \(E_{+}\)-orbit is \(1\), that is, for any \(x\in \mathbb R^{n}_{+}\), \[ \sum_{\sigma\in S_{n-1}}\sum_{ \varepsilon\in E_{+}} w_{\sigma}\chi_{C_{\sigma}}(\varepsilon x)=1. \] Here, \(\chi_{C_{\sigma}}\) is the characteristic function of \(C_{\sigma}\), \(w_{\sigma} =\pm\) 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. The authors 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.

MSC:
11R27 Units and factorization
11Y40 Algebraic number theory computations
11R42 Zeta functions and \(L\)-functions of number fields
11R80 Totally real fields
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