| Project/Area Number |
23KF0187
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Multi-year Fund |
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| Section | 外国 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | University of Tsukuba |
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Principal Investigator |
山木 壱彦 筑波大学, 数理物質系, 教授 (80402973)
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| Co-Investigator(Kenkyū-buntansha) |
HELMINCK PAUL 筑波大学, 数理物質系, 外国人特別研究員
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| Project Period (FY) |
2023-11-15 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2025: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 2024: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2023: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | Modular curves / Tamagawa numbers / toric ranks / smooth tropicalizations |
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| Outline of Research at the Start |
The tropicalization of a variety is a polyhedral shadow that contains a large amount of arithmetic information on the algebraic variety. This is especially true when the tropicalization is smooth, in the sense that there are no singularities. In practice however, many tropicalizations are not sufficiently smooth. In this project, we will investigate the possibility of transforming a tropical variety into a smooth one, with an eye towards applications in number theory.
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| Outline of Annual Research Achievements |
The researcher first investigated finding sufficiently smooth tropicalizations of modular curves, which have a vast amount of applications in number theory. He used the associated ramified coverings of the projective line to find these, leading to the first general formulas for the tropical invariants of modular curves. These include their toric ranks and Tamagawa numbers (arXiv:2403.09995, 53 pages).
The researcher also investigated the homotopy type of smooth tropicalizations in order to understand their cohomology groups from a number-theoretic perspective. The researcher made important steps towards providing coverings of these spaces by Eilenberg-Maclane polyhedral domains. We expect these to give rise to explicit descriptions of the homotopy type in terms of glued braid groups.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
The results obtained thus far have exceeded the expected results as set out in the project plan. This is best exemplified by the recent preprint arXiv:2403.09995 released by the researcher on finding sufficiently smooth tropicalizations of modular curves, in which he developed a new group-theoretic technique for explicitly finding the pruned skeleton of a curve. This gave explicit formulas for tropical invariants of modular curves such as the toric ranks and Tamagawa numbers.
The researcher also investigated the homotopy type of higher-dimensional smooth tropicalizations, making important steps toward obtaining rigid-analytic descriptions of these. These descriptions will allow us to use combinatorial methods to study the corresponding Galois representations.
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| Strategy for Future Research Activity |
The researcher will continue his work on smooth tropicalizations, and he expects to have a rigid analytic description of their cohomology groups within the foreseeable future. This constitutes a slight change of direction in the research plan, as the researcher has found that in order to understand moduli of smooth tropicalizations and their number-theoretic implications, more foundational results in terms of the homotopy type of a smooth tropicalization are necessary. Indeed, currently only complex-analytic comparisons are known, and these are not sufficient to fully describe the corresponding Galois representations. In keeping with the plan however, the researcher will focus on understanding the key example of smooth tropical K3-surfaces.
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