2021 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 / Local class field theory / Duality / Locally compact groups |
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| Outline of Annual Research Achievements |
I continued in my research Weil-etale cohomology for schemes over henselian
discrete valuation rings, finite fields or arithmetic schemes. Last year, I obtained the following results:
In my joint work with B.Morin, we released a preprint outlining the definition of
Weil-etale cohomology for varieties over local fields. 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. We are also in the final stages of publishing applications of this work 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, and giving some information on the kernel of the reciprocity map.
Finally, I started a quantitative analysis of the Artin-Tate formula for surfaces over finite fields. Using Weil-etale cohomology I obtained a formula comparing terms only involving the Brauer group with terms only involving the Neron-Severi group. In this second half of this work I give estimate for both sides, and study this for abelian surfaces.
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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
Our goals in defining a Weil-etale cohomology theory for varieties over local fields was achieved for weights 0 and d, and we gave an ad-hoc definition of the groups for arbitrary weights. So far this research project has yielded three publications, two preprints, and two more preprints are in preparation.
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| Strategy for Future Research Activity |
In my joint work with B.Morin we are in the process of finishing our third paper on class field theory over local fields. We are still trying to find improvement of our results. I started research on the Artin-Tate formula, this will help to understand Brauer groups and Neron-Severi groups of surfaces over finite fields better. Another paper on the application of this results to K3 surfaces is planned. C.Liedke (Muenchen) and I are also in discussions about a project studying K3-surfaces over finite fields by using the Weil-polynomial.
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| Causes of Carryover |
Due to the corona pandemic, international travel as well as conferences had to be postponed. I will do those international visits in the next academic year. My current plans are the following trips:
May 2022: Oberwolfach(Germany), August 2022: Heidelberg, Wuppertal (Germany).
Also, due to border restrictions, I could not invite foreign researchers for joint research, and those invitations were postponed to 2022. In February 2022, Shane Kelly, Shuji Saito, and I are planing to pool our unused funds to organize an international conference on motives in Tokyo.
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