论文标题
非交通性Arakelov几何形状的点和高度函数的函数
Functor of Points and Height Functions for Noncommutative Arakelov Geometry
论文作者
论文摘要
我们提出了一个非共同空间的点函子的概念，该概念在双模型类别中，并具有由Hermitian结构和高度功能的概念确定的动作函数，以将点的经典函数解释为物理Sigma模型进行建模。我们基于体积和痕迹的不同概念，包括基于Hattori-Stallings等级的不同概念，讨论了此类高度功能的不同选择。我们表明，高度函数决定了与点函子相关的可观察到的代数上的动态时间演变。我们特别关注非交互性算术曲线的情况，其中相关代数是矩阵代数的总和，而不是数字字段上的分区代数，并且在我们的方法中，我们讨论了更高的算术空间的更一般性的概念，在我们的方法中，我们的方法表明了琼斯索引作为高度函数的解释。
We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.
