2020 Fiscal Year Final Research Report
Arithmetic D-modules and its applications
| Project/Area Number |
16H05993
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| Research Category |
Grant-in-Aid for Young Scientists (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Tomoyuki Abe 東京大学, カブリ数物連携宇宙研究機構, 准教授 (70609289)
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 過収束アイソクリスタル / 分岐理論 / 無限圏 |
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| Outline of Final Research Achievements |
There are two main results of the research project. First of all, Deligne conjectured that the p-adic coefficients theory and l-adic coefficients theory for varieties over finite field should be "equivalent" in some sense. I proved one direction of this conjecture, namely the direction to construct l-adic coefficients out of p-adic coefficients, together with Prof. Esnault.
The other result is to show some fundamental results to apply the method of semistable reduction theorem for isocrystals, which could be regarded as a fundamental theorem for p-adic cohomology theory, to the theory of characteristic cycles for l-adic coefficients. The results consist of two parts. One is to show a certain convergence of a sequence needed to define characteristic cycles, and the other is to construct an infinity enhancement of motivic cohomology theory which is expected to be used in the future project.
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| Free Research Field |
数論幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
数論的D加群などのp進コホモロジーはこれまで基礎論が中心で，実際に数論幾何学の問題に応用された例は数少ない．今回の研究で数論幾何学の一つの中心問題であるDeligneの予想（の一部）へ応用ができ，さらにl進コホモロジー論の特性サイクルの理論への広い意味での応用の道が開けた．結果の重要性はもとより，これにより今までp進コホモロジー論でしか顧みられてこなかった数々の知見を応用ができたことは大きな成果に値するものと思う．
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