2022 Fiscal Year Research-status Report
Geometry of Shimura varieties in positive characteristic
| Project/Area Number |
21K13765
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| Research Institution | Saitama University |
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Principal Investigator |
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| Project Period (FY) |
2021-04-01 – 2026-03-31
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| Keywords | 志村多様体 / Gジップのスタック / 保型形式 / 簡約群 |
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| Outline of Annual Research Achievements |
Building up on previous work with my coauthors Naoki Imai (Tokyo University) and Wushi Goldring (Stockholm University), we completed three preprints in 2022. In those papers, we show instances of a general conjecture regarding the weight of automorphic forms in characteristic p. We also prove certain vanishing results for the cohomology of Shimura varieties, which improves on results from previous papers. We have submitted our preprints in peer-reviewed journals.
I presented this research at a conference in Niseko in September 2022. I was also invited to a Oberwokfach workshop on the arithmetic of Shimura varieties, where I discussed my research with several mathematicians, and planned on future research projects.
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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
Together with my coauthors, we made good progress on several problems, resulting in the completion of three different papers. We also found new interesting problems to investigate in our future research.
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| Strategy for Future Research Activity |
We plan to continue working on the "weight conjecture" for Hodge-type Shimura varieties. Recently, we showed several instances of this conjecture for several low-dimensional Shimura varieties. Also, in the case of unitary Shimura varieties of signature (n-1,1), we proved a weak version. In the future, we would like to extand our results to all unitary Shimura varieties, and eventually to all Hodge-type Shimura varieties. I plan to take part in several conferences to connect with researchers and extand my network.
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| Causes of Carryover |
I plan to participate in various international conferences, and visit researchers abroad to pursue my research (the details of these business trips are not yet finalized). I also plan to purchase a Windows computer to use for my research (the software that I use is not compatible with Mac).
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