The Weil Proof and the Geometry of the Adèles Class Space
The Weil Proof and the Geometry of the Adèles Class Space
复制标题
韦尔证明和阿黛尔类空间的几何
DOI：
10.1007/978-0-8176-4745-2_8
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
M. Marcolli
中科院分区：
文献类型：
--
作者：
A. Connes;C. Consani;M. Marcolli
本文探讨了函数场的黎曼假设的Weil证明和Adèles类空间的几何之间的类比，Adèles类空间是Connes的黎曼zeta函数的零点的谱实现的非交换空间。我们考虑了在非交换的Adèles类空间中包含整体域的Idèles类群所定义的“限制映射”的上核（在循环模的阿贝尔范畴中）的循环同调。Weil的显式公式则可以表示为一个Lefschetz迹公式，表示idèles类群在这个上同调上的诱导作用。在这个公式中，黎曼假设等价于相关迹对的正性。这一结果表明，在Weil证明的步骤和涉及Adèles类空间的非交换几何的相应概念之间可能存在一个字典，具有良好的对应性，度和余度等工作概念。特别是，我们构造了函数域曲线的代数点的数域的模拟，在这里体现为经典的观点（低温KMS态）的量子统计力学系统自然地关联到idèles类群的作用的周期轨道，即，到非交换空间上的几何方面的跟踪公式是支持的。
This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.