2022 Fiscal Year Research-status Report
Arithmetic cohomology over local fields
| Project/Area Number |
18K03258
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| Research Institution | Rikkyo University |
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Principal Investigator |
ガイサ トーマス 立教大学, 理学部, 教授 (30571963)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | Weil etale cohomology / Local class field theory / Duality / Locally compact groups / One-motives / Birch Swinnerton Dyer |
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| Outline of Annual Research Achievements |
In my ongoing project on Weil-etale cohomology for schemes over henselian
discrete valuation rings, finite fields, and arithmetic schemes, I was able to finalize publication of the following results:
Joint with B.Morin, we outline the definition of a Weil-etale cohomology theory for varieties over local fields which satisfy a Pontrjagin duality theory. The groups are objects of the heart of the t-structure on the derived category of locally compact abelian groups (this work is accepted for publication and published online).
As an application we prove results on class field theory over local fields, generalizing and improving work of S.Saito and Yoshida. We give an integral model for the fundamental group, and some extra information on the kernel of the reciprocity map (a preprint is submitted for publication).
In joint work with T.Suzuki, we generalized our work on the Weil-etale version of the Birch and Swinnerton-Dyer conjecture to one-motives. In particular, our work gives a new proof of the Tamagawa number formula of Oda (this is published).
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The work was slightly delayed due to the corona measures, and the project had to continue into an extra year.
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| Strategy for Future Research Activity |
It is the last year of the project and it only remains to publish the last remaining preprint, to survey the results we obtained, and to attend conferences to present my results.
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| Causes of Carryover |
Research was delayed due to corona and corona measures. It was difficult or impossible to meet my co-workers and to hold conferences.
I am planing to use the remaining amount of my grand to attend conferences and discuss the results.
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