Study of exponential sums associated with prehomogeneous vector spaces

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K13747
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Taniguchi Takashi 神戸大学, 理学研究科, 准教授 (60422391)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 指数和 / 篩法 / 概均質ベクトル空間 / 代数群

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.

Free Research Field

整数論

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。