2019 Fiscal Year Final Research Report
Arithmetic non-linear differential equations and Frobenius structure
| Project/Area Number |
17K14170
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Tokyo Denki University (2019)
Hiroshima University (2017-2018) |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | 数論幾何学 / 超幾何関数 / p-進微分方程式 |
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| Outline of Final Research Achievements |
We studied p-adic differential equations, which have a feature of arithmetics and a feature of differential equations. As a result, we found a new relationship between hypergeometric differential equations, which appears in many areas in mathematics, and p-adic numbers. Concretely speaking, we proved that p-adic hypergeometric equations are overholonomic under a certain condition about p-adic Liouville numbers. (The overholonomicity is a kind of cohomological finiteness condition.)
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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 |
p-進微分方程式は，フロベニウス構造という整数論に由来する構造がある場合にはよい性質を満たすことが多いが，そうでない場合の挙動は難しく，特に overholonomic なp-進D-加群のクラスは具体例がほとんど知られていなかった．本研究では，このような具体例を体系的に構成しただけでなく，それが超幾何微分方程式という微分方程式的にも自然な対象から得られることを発見した点で意義深い．
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