2023 Fiscal Year Final Research Report
Study of Gross-Zagier type formula via perfectiod spaces
| Project/Area Number |
19K21829
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 11:Algebra, geometry, and related fields
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Mieda Yoichi 東京大学, 大学院数理科学研究科, 准教授 (70526962)
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| Project Period (FY) |
2019-06-28 – 2024-03-31
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| Keywords | パーフェクトイド空間 / Gross-Zagier型公式 / 志村多様体 / Rapoport-Zink空間 / 数論的交叉数 |
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| Outline of Final Research Achievements |
The original goal was to study the conjecture so-called the arithmetic fundamental lemma by using the theory of perfectoid spaces. However, in the first year the conjecture had been solved by Wei Zhang, so I slightly changed the goal. I studied the Tate conjecture for the unitary Shimura varieties and a generalization of the p-adic Gross-Zagier formula due to Darmon-Rotger. I got some important ideas, but I need further research to achieve concrete results. I also investigated the relation between Fargues-Scholze's local Langlands correspondence and the usual one, and obtained some results in the case of the symplectic group Sp(6).
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| Free Research Field |
数論
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| Academic Significance and Societal Importance of the Research Achievements |
ユニタリ型志村多様体のTate予想に関する研究，p進Gross-Zagier公式を一般化する研究は，BSD予想の一般化であるBeilinson-Bloch-加藤予想への貢献に直接結び付くものである．本研究によって得たアイデアにより，研究を進めるべき方向性が明確になったため，近い将来に具体的な成果が得られることが期待できる．また，局所Langlands対応に関する成果は，Fargues-Scholzeの構成が正統的なものであることを保証するとともに，局所志村多様体のエタールコホモロジーの決定という，従来から興味を持たれてきた問題にも応用を持つものである．
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