2021 Fiscal Year Final Research Report
New developments in arithmetic invariant theory
| Project/Area Number |
17H02835
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kobe University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | 数論的不変式論 / 概均質ベクトル空間 / ３次体 / 判別式 |
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| Outline of Final Research Achievements |
We study the arithmetic invariant theory. We improved the error term estimate in the counting function for cubic fields. In particular, we significantly improved the uniformity estimate with respect to the local conditions. Also, we showed that there are large numbers of cubic and quartic fields whose discriminants have few prime factors. As a related topic, we obtained explicit formulas of the orbital exponential sums for around ten new cases of prehomogeneous vector spaces. We also obtained some partial results in the arithmetic of coregular spaces.
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| Free Research Field |
整数論
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| Academic Significance and Societal Importance of the Research Achievements |
数論的不変式論は最近非常に活発な研究が行われている分野である。本研究課題において、基盤的な成果をいくつか挙げることができた。この分野のもっとも典型的な問題である３次体の判別式を数える問題で成果を挙げることができ、その方法は、他の場合にも応用できると考えられる。軌道指数和の公式は、概均質ベクトル空間や代数幾何学を超えて、数学の他分野と関連を持つ可能性がある。
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