| Project/Area Number |
23K25768
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| Project/Area Number (Other) |
23H01071 (2023)
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Multi-year Fund (2024)
Single-year Grants (2023) |
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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 |
ガイサ トーマス 立教大学, 理学部, 教授 (30571963)
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| Project Period (FY) |
2024-04-01 – 2028-03-31
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| Project Status |
Granted (Fiscal Year 2024)
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| Budget Amount *help |
¥16,900,000 (Direct Cost: ¥13,000,000、Indirect Cost: ¥3,900,000)
Fiscal Year 2027: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2026: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2025: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2024: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2023: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
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| Keywords | ブラウア群 / 有限体 / Neron-Severi 群 / Weil-etale cohomology / Frobenius / Brauer group / Artin Tate formula / Neron-Severi group / 有限体上の局面 / Tate-Shafarevich group |
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| Outline of Research at the Start |
The first part of the proposal is to apply the a new version of the Artin-Tate formula to calculate Brauer groups and determinants of Neron-Severi groups of special classes of surface. The second part is to find a similar version of the
Birch and Swinnerton-Dyer conjecture for abelian surfaces over global fields of characteristic p.
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| Outline of Annual Research Achievements |
I extended my work of Brauer groups and Neron-Severi groups of surfaces over finite fields to include K3-surfaces. This complements my previous estimates I obtained for abelian surfaces. I also simplified some of the proofs and improved some estimates. This research was presented at a conference on Motivic Homotopy theory in Heidelberg and a conference on Brauer group at the Imperial College in London.
In related work, I came back to Weil-etale cohomology. This plays an important role in the study of special values of zeta-functions for varieties over finite fields and is defined as the derived fixed points of the Frobenius endomorphism on the etale cohomology over the algebraic closure of the finite field. It has been asked by several people if it is possible to replace the algebraic closure of the finite field by an arbitrary algebraically closed field. I have obtained partial results in this direction, mostly for cohomology with torsion coefficients, and presented those results at a conference on algebraic geometry in Zurich.
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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
I obtained results in the amount I was expecting.
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| Strategy for Future Research Activity |
I would like to continue to study the independence of Weil-etale cohomology of the algebraically closed field. A stronger question is the independence of motivic and etale motivic cohomology with finite coefficients of the algebraic closed field. Very little is known in this direction. This generalized the topic of the grant, because the Neron-Severi group is known to be independent of the algebraically closed field, and the Brauer group is not.
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