Study of algebraic cycles in arithmetic and algebraic geometry

2014 Fiscal Year Final Research Report

• Project/Area Number: 22340003
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Institute of Technology (2011-2014); The University of Tokyo (2010)
• Principal Investigator: SAITO Shuji 東京工業大学, 理工学研究科, 教授 (50153804)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: algebraic cycles / motivic cohomology / Hasse principle / Class field theory

Outline of Final Research Achievements

The research consists of three parts:(I) Higher dimensional class field theory, (II) Cohomological Hasse principle, (III) Finiteness of motivic cohomology.

(I) We generalized higher dimensional class field theory of Kato-Saito by using Chow groups of zero-cycles with modulus and gives a proof of the rank-one case of a conjecture of Deligne-Drinfeld on the existence of smooth l-adic sheaves on varieties over finite fields. (II)We proved rthe prime-to-p part of the Kato conjecture on cohomological Hasse principle and gives several applications on various problems in arithmetic geometry; special values of zeta functions, and a generalization of higher dimensional class field theory in the framework of theory of motivic cohomology, and a geometric application to resolution of singularities. (III) We gave new contributions to the finiteness conjecture of motivic (co)homology of arithmetic schemes. It shows the finiteness of motivic (co)homology with finite coefficient of arithmetic schemes.

Free Research Field

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