Number theory of prehomogeneous vector spaces

• Project/Area Number: 17K05169
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Yukie Akihiko 京都大学, 理学研究科, 教授 (20312548)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 概均質ベクトル空間 / 代数群 / 不変式論 / 密度定理 / 有理軌道 / 局所密度 / ゼータ関数

Outline of Final Research Achievements

(1) I determined the interpretation of two prehomogeneous vector spaces. (2) I proved a density theorem related to the space of non-split tri-Hermitain forms. For a fixed cubic field and quadratic fields, the composite field is a sextic field. I determined the density of relative hR (class number times the regulator) of this composite field and the quadratic field. (3) I applied the general theory of GIT stratification to 4 concrete cases and determined the stratification and all rational orbits for 3 cases. For the remaining case I determined the parameter set of vectors.

Academic Significance and Societal Importance of the Research Achievements

概均質ベクトル空間は不変式論と解析的整数論が交錯する分野で特にそれから得られる密度定理は解析的整数論の主要な目的である。そのためには有理軌道で問題の意味を解釈することが必要であり，方法としてはベクトル空間の stratication を決定することが不可欠である。これらのことを本研究ではいくつかの場合に実現し，実際に新しい密度定理を証明することに成功したことは意義がある。

Research Products

• [Journal Article] On the density theorem related to the space of non-split tri-{H}ermitian forms II (2020)
  • Author(s): Yukie, Akihiko
  • Journal Title: Manuscripta Math. Volume: 162 Issue: 1-2 Pages: 221-239
  • DOI: 10.1007/s00229-019-01116-x
  • Peer Reviewed
• [Journal Article] RATIONAL ORBITS OF PRIMITIVE TRIVECTORS IN DIMENSION SIX (2019)
  • Author(s): Yukie, Akihiko
  • Journal Title: Tohoku Mathematical Journal, Second Series Volume: 71 Issue: 1 Pages: 35-52
  • DOI: 10.2748/tmj/1552100441
  • ISSN: 0040-8735, 2186-585X
  • Year and Date: 2019-03-30
  • Peer Reviewed
• [Journal Article] On the density theorem related to the space of non-split tri-Hermitian forms I (2019)
  • Author(s): Akihiko Yukie
  • Journal Title: J. Number Theory Volume: 194 Pages: 117-169
  • DOI: 10.1016/j.jnt.2018.07.015
  • Peer Reviewed
• [Journal Article] Rational orbits of the space of pairs of exceptional Jordan algebras (2018)
  • Author(s): Ryo Kato and Akihiko Yukie
  • Journal Title: J. NUmber Theory Volume: 189 Pages: 304-352
  • DOI: 10.1016/j.jnt.2017.12.008
  • Peer Reviewed
• [Journal Article] Rational orbits of the space of pairs of exceptional Jordan algebras (2018)
  • Author(s): R. Kato and A. Yukie
  • Journal Title: J. Number Theory Volume: to appear
  • Peer Reviewed
• [Journal Article] Rational orbits of primitive trivectors in dimension six (2018)
  • Author(s): A. Yukie
  • Journal Title: Tohoku Math. J. Volume: to appear
  • Peer Reviewed
• [Journal Article] Rational orbits of prehomogeneous vector spaces (2018)
  • Author(s): A. Yukie
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: to appear
  • Peer Reviewed
• [Presentation] On the density theorem related to the space of non-split tri-Hermitian forms (2020)
  • Author(s): Yukie, Akihiko
  • Organizer: 保型形式とL関数の解析的, 幾何的, p進的研究
  • Int'l Joint Research / Invited
• [Presentation] On the density theorem related to the space of non-split tri-{H}ermitian forms (2019)
  • Author(s): Yukie, Akihiko
  • Organizer: 第6回京都保型形式研究集会
  • Invited
• [Presentation] On the local density of prehomoegenous vector spaces (2017)
  • Author(s): A. Yukie
  • Organizer: 4th Kyoto conference on automorphic forms
  • Invited