Motivic cohomology and arithmetic geometry

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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.