Weil-etale cohomology of arithmetic schemes

作者为有限域上的变量构造了一个Grothendieck拓扑，这样，模标准猜想，ζ函数的特殊值可以计算为某些动力轮系或轮系复合体的欧拉特征（在适当的意义上）。他现在打算为数环上的方案构造一个类似的Grothendieck拓扑。黎曼ζ函数（由黎曼在19世纪发现）及其推广表达了分析和数论之间的许多深刻关系。这些关系应该是深层底层拓扑结构的结果。这个项目是对该结构必须拥有的属性的调查。

The proposer has constructed a Grothendieck topology for varieties over finite fields such that, modulo standard conjectures, special values of zeta-functions can be computed as Euler characteristics (in a suitable sense) of certain motivic sheaves or complexes of sheaves. He now intends to work on the construction of a similar Grothendieck topology for schemes over number rings. The Riemann zeta-function (discovered by Riemann in the nineteenth century) and its generalizations express many deep relations between analysis and number theory. These relations should be consequences of a deep underlying topological structure. This project is an investigation into the properties which that structure would have to possess.