论文标题
PSL（2，K）中的共轭类
Conjugacy classes in PSL(2, K)
论文作者
论文摘要
我们首先描述了与2不同的特征k在谎言组PGL（2，K）和PSL（2，K）的lie lie代数SL（2，K）上的伴随作用的轨道。虽然前者是众所周知的，但后者导致了表征相应轨道的广义pell-fermat方程的分辨率。合成方法使基本场更改，我们与四面体和二十面体组的几何形状有关三个和五个元素的田地上的这张图。尽管结果可能看起来很熟悉，但现有文献似乎并未被这样的普遍性或细节所覆盖。 我们将此讨论应用于将积分二进制二次形式的PSL（2，z）类别的集合分为PSL（2，K）类别的组。当k = c时，我们获得给定判别物的类组。然后，我们将其分区的完整描述从Hilbert符号方面，将其分配到PSL（2，Q）类中，并将其与分区相关联。结果是经典的，但是我们的几何方法具有独立的兴趣，因为它可能会对高斯组成的几何形状产生新的见解，并在功能场上统一图像。 最后，我们在两个或两个封闭的大地测量学对应于PSL（2，K）等效二次形式的模块化孔PSL（2，z）\ H中提供几何解释，就这些模块化循环之间的双曲线距离和角度而言。这些几何量与链接模块结的数量有关。可以使用二次晶格（SL（2，Z），det）的几何形状来研究它们的分布特性，但此处未进行此类研究。
We first describe, over a field K of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PGL(2, K) and PSL(2, K) on their Lie algebra sl(2, K). While the former are well known, the latter lead to the resolution of generalised Pell-Fermat equations which characterise the corresponding orbit. The synthetic approach enables to change the base field, and we illustrate this picture over the fields with three and five elements, in relation with the geometry of the tetrahedral and icosahedral groups. While the results may appear familiar, they do not seem to be covered in such generality or detail by the existing literature. We apply this discussion to partition the set of PSL(2, Z)-classes of integral binary quadratic forms into groups of PSL(2, K)-classes. When K = C we obtain the class groups of a given discriminant. Then we provide a complete description of their partition into PSL(2, Q)-classes in terms of Hilbert symbols, and relate this to the partition into genera. The results are classical, but our geometrical approach is of independent interest as it may yield new insights into the geometry of Gauss composition, and unify the picture over function fields. Finally we provide a geometric interpretation in the modular orbifold PSL(2, Z) \ H for when two points or two closed geodesics correspond to PSL(2, K)-equivalent quadratic forms, in terms of hyperbolic distances and angles between those modular cycles. These geometric quantities are related to linking numbers of modular knots. Their distribution properties could be studied using the geometry of the quadratic lattice (sl(2, Z), det) but such investigations are not pursued here.
