New developments in arithmetic invariant theory

• Project/Area Number: 17H02835
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Taniguchi Takashi 神戸大学, 理学研究科, 教授 (60422391)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥17,160,000 (Direct Cost: ¥13,200,000、Indirect Cost: ¥3,960,000); Fiscal Year 2021: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2020: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2019: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2018: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2017: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
• Keywords: 数論的不変式論 / 概均質ベクトル空間 / ３次体 / 判別式 / 指数和 / 余正則空間 / 解析数論 / 解析的整数論

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.

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。

Research Products

• [Int'l Joint Research] Princeton University/University of South Carolina(米国)
• [Int'l Joint Research] Purdue university/University of South Caroloina/ICERM(米国)
• [Int'l Joint Research] Princeton University/University of South Caroloina(米国)
• [Int'l Joint Research] University of Toronto(カナダ)
• [Int'l Joint Research] Princeton University/University of South Carolina(米国)
• [Int'l Joint Research] 西湖大学(中国)
• [Int'l Joint Research] Princeton University/University of South Carolina(米国)
• [Int'l Joint Research] University of Oxford/University of Warwick(英国)
• [Journal Article] Uniform bounds for lattice point counting and partial sums of zeta functions (2021)
  • Author(s): Lowry-Duda David、Taniguchi Takashi、Thorne Frank
  • Journal Title: Mathematische Zeitschrift Volume: 300 Issue: 3 Pages: 2571-2590
  • DOI: 10.1007/s00209-021-02862-z
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves (2020)
  • Author(s): Bhargava M.、Shankar A.、Taniguchi T.、Thorne F.、Tsimerman J.、Zhao Y.
  • Journal Title: Journal of the American Mathematical Society Volume: 33 Issue: 4 Pages: 1087-1099
  • DOI: 10.1090/jams/945
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The local-global property for bitangents of plane quartics (2020)
  • Author(s): Ishitsuka Yasuhiro、Ito Tetsushi、Ohshita Tatsuya、Taniguchi Takashi、Uchida Yukihiro
  • Journal Title: JSIAM Letters Volume: 12 Issue: 0 Pages: 41-44
  • DOI: 10.14495/jsiaml.12.41
  • NAID: 130007879424
  • Peer Reviewed / Open Access
• [Journal Article] Levels of distribution for sieve problems in prehomogeneous vector spaces (2020)
  • Author(s): Taniguchi Takashi、Thorne Frank
  • Journal Title: Mathematische Annalen Volume: 376 Issue: 3-4 Pages: 1537-1559
  • DOI: 10.1007/s00208-019-01933-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Orbital exponential sums for prehomogeneous vector spaces
  • Author(s): Takashi Taniguchi and Frank Thorne
  • Journal Title: Amerian Journal of Mathematics Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Presentation] An introduction to Shintani's zeta function for the space of binary cubic forms (2019)
  • Author(s): Takashi Taniguchi
  • Organizer: The 10th Westlake Number Theory Colloquium
  • Int'l Joint Research / Invited
• [Presentation] Bounds on 2-torsion in class groups of number fields (2018)
  • Author(s): Takashi Taniguchi
  • Organizer: RIMS Workshop 2018 "Analytic Number Theory and Related Topics"
  • Int'l Joint Research / Invited
• [Presentation] 代数体のイデアル類群の2-partについて (2018)
  • Author(s): 谷口隆
  • Organizer: 第17回北陸数論研究集会
  • Invited
• [Presentation] Variants of Ohno-Nakagawa's dual identity (2017)
  • Author(s): 谷口隆
  • Organizer: 神戸整数論集会2017
  • Int'l Joint Research / Invited
• [Presentation] On the numbers of cubic and quartic fields with almost prime discriminants (2017)
  • Author(s): 谷口隆
  • Organizer: 第4回京都保形形式研究集会
  • Int'l Joint Research / Invited
• [Presentation] Uniformity in Landau’s method and applications (2017)
  • Author(s): 谷口隆
  • Organizer: Arithmetic Geometry and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Uniformity in Landau’s method and applications (2017)
  • Author(s): 谷口隆
  • Organizer: 上越整数論セミナー
  • Invited
• [Remarks] Takashi Taniguchi
  • URL: http://www.math.kobe-u.ac.jp/HOME/tani/index.html
• [Remarks] Takashi Taniguchi
  • URL: http://www.math.kobe-u.ac.jp/HOME/tani/
• [Remarks] Takashi Taniguchi
  • URL: http://www.math.kobe-u.ac.jp/~tani/
• [Funded Workshop] Kobe Number Theory Workshop 2017 (2017)