Motivic cohomology and arithmetic geometry

geisser, Thomas h .摘要题目：动机上同论与算术几何。一方面，他试图将关于ζ函数的weil -etale上同调和特殊值的结果从有限域上的光滑投影变数推广到有限域上的一般变数，以及局部域上的变数。另一个项目是考察de Rham-Witt复和光滑品种在完全离散估值环上的拓扑循环同调的性质。拓扑迹同调具有一个Frobenius算子、一个伽罗瓦作用和一个类似于Fontaine函子的滤波。研究者想利用这种结构来构造虚上同和结晶上同，并将其应用于算术问题。在算术代数几何中，研究多项式方程的解。尽管这个领域有两千多年的历史，但最近发现密码学有各种各样的应用。一种研究多项式解集的方法是将不变量称为上同调和ζ函数联系起来，然后研究这些不变量。由于不变量以非常不同的方式定义，因此找到它们之间的关系可以将一个知识转化为另一个知识。研究者研究了ζ函数与一种新的上同调（称为Weil-etale上同调）和一种新的不变量（称为拓扑循环同调）之间的关系。

DMS-0300133Geisser, Thomas H.AbstractTitle: Motivic cohomology and arithmetic geometryThe investigator is working on two projects.On the one hand, he tries to extend his results onWeil-etale cohomology and special values of zeta functionsfrom smooth projective varieties over finite fieldsto general varieties over finite fields, and to varietiesover local fields. The other project is the examinationof properties of the de Rham-Witt complex and topologicalcyclic homology of smooth varieties over completediscrete valuation rings. Topological trace homology TRhas a Frobenius operator, a Galois action and a filtrationanalog to Fontaine's functor. The investigator wantsto exploit this structure to construct etale and crystalline cohomology, and to apply this to arithmetic problems. In arithmetic algebraic geometry, solutions of polynomial equations are studied. Even though this field is more than two thousand years old,it turned out recently that there is a variety of applications to cryptography. One method to study a solution set of polynomialsis to associate invariants called cohomology and zeta functionsto it, and then study those invariants instead.Since the invariants are defined in very different ways, findingrelationships between them allows to translate knowledge onone into knowledge on the other. The investigator studiesthe relationship between zeta functions and a new cohomology,called Weil-etale cohomology, on the one hand, and a newinvariant called topological cyclic homology on the other hand.