2023 Fiscal Year Final Research Report
Arithmetic cohomology over local fields
| Project/Area Number |
18K03258
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Rikkyo University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | Brauer group / Local fields / Motivic cohomology / Birch-Swinnerton-Dyer / Class field theory |
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| Outline of Final Research Achievements |
The research on Weil-etale cohomology for schemes over henselian discrete valuation rings and arithmetic schemes led to 5 publications, three in an international collaboration with B. Morin (France), and two with T. Suzuki.
(1) B.Morin and we proved a result regarding the p- and l-corank of the Brauer group of a smooth and proper scheme over a p-adic local ring, generalizing work of Colliot-Thelene, S.Saito, and Sato. (2) B.Morin and I outlined the definition of a Weil-etale cohomology theory for varieties over local fields which satisfy a Pontrjagin duality theory, and prove a duality result in weight zero. (3) B.Morin and I use the above to prove results on class field theory over local fields, generalizing and improving work of S.Saito and Yoshida.
(4) T.Suzuki and I proved a Weil-etale version of the Birch and Swinnerton-Dyer conjecture for abelian varieties, and (5) generalized the result to one-motives. In particular, we obtain a new proof of the Tamagawa number formula of Oda.
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| Free Research Field |
Motivic cohomology
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| Academic Significance and Societal Importance of the Research Achievements |
Basic research does not have direct application, but contributes to the knowledge of humanity with applications in the future in mind. During the research students were involved and educated. Since my research involved an international collaboration, it also strengthens international understanding.
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