Motivic cohomology and arithmetic geometry

一方面，他试图将他关于Weil-etale上同调和zeta函数特殊值的结果从有限域上的光滑投射簇推广到有限域上的一般簇和局部域上的簇;另一个项目是研究完全离散赋值环上光滑簇的de Rham-Witt复形和拓扑循环同调的性质。拓扑迹同调TR有一个Frobenius算子、一个Galois作用和一个类似于方丹函子的滤子.研究者希望利用这种结构来构造etale和crystalic上同调，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。尽管这个领域已经有两千多年的历史了，但最近发现密码学有各种各样的应用。研究多项式解集的一种方法是将称为上同调和zeta函数的不变量与之联系起来，然后研究这些不变量。由于不变量的定义方式非常不同，找到它们之间的关系可以将一个不变量的知识转化为另一个不变量的知识。研究了zeta函数与Weil-etale上同调和拓扑循环同调的关系。

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.