2020 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 – 2023-03-31
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| Keywords | Weil-etale cohomology / Brauer group / Tamagawa number formula / BSD conjecture / Brauer Manin obstruction |
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| Outline of Annual Research Achievements |
The goal of the project is to define Weil-etale cohomology in certain situations, for example for regular flat and proper schemes over henselian discrete valuation rings, or for arithmetic schemes. In the previous year I made progress on several aspects of this project.
In joint work with T.Suzuki we proved a version of the Birch and Swinnerton-Dyer conjecture for one-motives over global function fields. This generalized our joint work for abelian varieties (which was produced in the first year of the project), and also reproves Ono's Tamagawa number formula for tori. We wrote a preprint which is submitted for publication.
I also continued my joint work with B.Morin. Our results on the Brauer group of a smooth and proper scheme over a local field (generalizing work of Colliot-Thelene, S.Saito, and Sato) lead to a preprint, which is submitted for publication. The main result is to show a formula relating the p-corank and the l-corank of the Brauer group of a regular proper model.
Our work on Weil-etale cohomology for varieties over local fields is now in the final stages, and we hope to release a preprint soon. In this preprint, we give an add hoc definition of Weil-etale cohomology groups over local fields which satisfy a Pontrjagin duality theory. Our groups are objects of heart of the t-structure on the derived category of locally compact abelian groups. As an application we are working on a preprint on applications to class field theory over local fields (generalizing and improving work of S.Saito and Yoshida) by giving an integral model for the fundamental group.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We made progress towards defining a Weil-etale cohomology theory for varieties over local fields and to prove results with this theory. In particular, we produced two preprints with related results. The first one is about the Birch and Swinnerton Dyer conjecture for one-motives over global function fields, and the second one about the Brauer group of varieties over local fields. Currently B.Morin and I are working on two more preprints to be released soon. Moreover, I am studying research papers in order to prepare for the next phase of the project (see below).
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| Strategy for Future Research Activity |
After finishing the above mentioned papers, I plan to return to the original project on Weil etale cohomology and related results. In particular, I would like to deepen our understanding of the relationship to syntomic and de Rham cohomology as well as the relationship of our definition with results on the pro-etale site. For the latter I will need to study the work of Bhatt-Scholze on the pro-etale site, and for the former I will need some time to get familiar with results of Bhatt-Morrow-Scholze on "integral p-adic Hodge theory" and Clausen-Matthew on Etale K-theory. These works suggest that it is useful to study motivic cohomology for the syntomic site, which I will try to explore. Another avenue I would like to persue is to undestand the relationshiop with prisms, defined by Bhatt-Scholze.
Finally, our results on class field theory of varieties over p-adic fields are abstract, and it is an interesting project to give an explicit construction of the groups involved and the reciprocity map. I hope to start a joint research project with A.Schmidt in Heidelberg, generalizing our joint work done for varieties over finite and algebraically closed fields.
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| Causes of Carryover |
Due to the corona virus pandemic all travel had to be suspended. In particular, all invitations for my collaborators to come to Japan as well as my trips to conferences and joint research had to be postponed to the next year.
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| Remarks |
These are the ArXiv pages of our preprints
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Research Products
(6 results)
