Study of exponential sums associated with prehomogeneous vector spaces
| Project/Area Number |
16K13747
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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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) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 指数和 / 篩法 / 概均質ベクトル空間 / 代数群 / 篩 / 概素数 / 代数学 / 整数論 |
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| Outline of Final Research Achievements |
We studied exponential sums associated with prehomogeneous vector spaces. We obtained explicit formulas for several prehomogeneous vector spaces.
We also studied applications to number theory. By developing sieve methods, we obtained two major results: (1) We showed there exist "many" quartic fields whose discriminants have at most eight prime factors. (2) We improved the error term estimate in the counting function for cubic fields. We largely improved the uniform estimate with respect to the splitting conditions of the cubic fields.
We further developed a method for evaluating exponential sums, and obtained a further improvement to sieve methods in question.
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| Academic Significance and Societal Importance of the Research Achievements |
指数和は整数論における基本的で重要な研究対象の一つだが、概均質ベクトル空間に伴う指数和は、かなり値が小さくなるという著しい特徴が観察されている。本研究では、具体的に指数和を計算することで、このことをさまざまな場合に実際に確かめることができた。またこの特徴（小ささ）を直接活用する整数論的な応用を与えた。この成果はさまざまな応用を持つことが期待される。指数和を計算する手法を改良できたことも、意義ある成果だと考えられる。
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Report
(5 results)
Research Products
(26 results)
