全实数域的符号基本区
Signed fundamental domains for totally real number fields
摘要：
给出了次数为$n$的全实数域$k$的全正单位$E_+$在$\mathbb{R}^n_+$上作用的一个带符号基本整环.域$\big\{(C_σ,w_σ) \big\}_σ$是有符号的，因为它与任何$E_+$ -轨道相交的净数目是1，i. e.对于任何$x\in \mathbb{R}^n_+$，$$ \sum_{σ\in S_{n-1}} \sum_{\varepsilon\in E_+} w_σχ^{\phantom{1}}_{C_σ}(\varepsilon x) = 1. $$这里$χ_{C_σ}$是$C_σ$的特征函数，$w_σ=\pm1$是$n$ -维$k$ -有理锥$C_σ\subset\mathbb{R}^n_+$的自然定向，并且内部和实际上是有限的。符号基本区是有用的Shintani的真正的计算交换$L$ -功能的目的。它们的优点是可以很容易地从任何一组基本单位中构造出来，而在实践中，没有算法可以生成Shintani的$k$ -有理锥。我们的证明使用代数拓扑上的商流形$\mathbb{R}^n_+/E_+$。同伦下拓扑度的不变性允许我们控制弯曲基本域变形为漂亮的直锥。在同伦过程中可能会发生交叉，导致需要减去一些锥。
英文摘要：
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 $\big\{(C_σ,w_σ) \big\}_σ$ is signed since the net number of its intersections with any $E_+$-orbit is 1, i. e. for any $x\in \mathbb{R}^n_+$, $$ \sum_{σ\in S_{n-1}} \sum_{\varepsilon\in E_+} w_σχ^{\phantom{1}}_{C_σ}(\varepsilon x) = 1. $$ Here $χ_{C_σ}$ is the characteristic function of $C_σ$, $w_σ=\pm1$ is a natural orientation of the $n$-dimensional $k$-rational cone $C_σ\subset\mathbb{R}^n_+$, 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.
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