数域的扭域和带符号基本区
Twisters and signed fundamental domains for number fields
摘要：
本文给出了$n=r_1+2r_2$次数域$k$的全正单位$E_+$在$\mathbb{R}^{r_1}_+\times{\mathbb{C}^*}^{r_2}$上作用的一个带符号基本整环，并假定该数域不是全复的.这里$r_1$和$r_2$表示$k$的实数和复数位数，$\mathbb{R}_+$表示正实数。带符号的基本域由$n$ -维$k$ -有理锥$C_α$组成，每个锥都配备有符号$μ_α=\pm1$，具有锥与任何$E_+$ -轨道的交叉的净数量为1的性质。锥$C_α$和符号$μ_α$是从任何一组基本的全正单位和一组$3^{r_2}$“扭转”，即$k$的元素，其参数在$k$的$r_2$复杂的地方是充分变化的。引入扭转器为我们提供了锥$C_α$的正确数量的生成器，并允许我们以受控的方式在每个复数嵌入处围绕原点旋转$C_α$。
英文摘要：
We give a signed fundamental domain for the action on $\mathbb{R}^{r_1}_+\times{\mathbb{C}^*}^{r_2}$ of the totally positive units $E_+$ of a number field $k$ of degree $n=r_1+2r_2$ which we assume is not totally complex. Here $r_1$ and $r_2$ denote the number of real and complex places of $k$ and $\mathbb{R}_+$ denotes the positive real numbers. The signed fundamental domain consists of $n$-dimensional $k$-rational cones $C_α$, each equipped with a sign $μ_α=\pm1$, with the property that the net number of intersections of the cones with any $E_+$-orbit is 1. The cones $C_α$ and the signs $μ_α$ are explicitly constructed from any set of fundamental totally positive units and a set of $3^{r_2}$ "twisters", i.e. elements of $k$ whose arguments at the $r_2$ complex places of $k$ are sufficiently varied. Introducing twisters gives us the right number of generators for the cones $C_α$ and allows us to make the $C_α$ turn in a controlled way around the origin at each complex embedding.
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