rationality problem on extension fields

• Project/Area Number: 24540019
• 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: YAMASAKI Aiichi 京都大学, 理学(系)研究科(研究院), 准教授 (10283590)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: rationality problem / algebraic torus / conic bundle / Noether's problem / unramified cohomology / relation module / Bravais群 / 有理性問題 / Noether問題 / 有限群

Outline of Final Research Achievements

Let X be an affine surface defined by the equation z^2=P(x)y^2+Q(x). The rationality problem of X has been solved in terms of the polynomials P(x) and Q(x).
The rationality problems of algebraic torus has been solved up to dimension 5. Rationality problems for algebraic tori of dimension 3 was determined by Kunyavskii. But as for algebraic tori of dimension 4 and 5, rationality problems were solved only for special cases.

Research Products

• [Journal Article] Rationality problem for algebraic tori (2015)
  • Author(s): A.Hoshi, A.Yamasaki
  • Journal Title: Memoirs of the American Mathematical Society Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Rationality problem of three-dimensional monomial group asctios. (2012)
  • Author(s): A.Yamasaki
  • Journal Title: Journal of Algebra Volume: 370 Pages: 46-78
  • DOI: 10.1016/j.jalgebra.2012.07.018
  • Peer Reviewed
• [Presentation] Rationality problem of conic bundles (2014)
  • Author(s): A.Yamasaki
  • Organizer: New Developments in Algebraic Geometry
  • Place of Presentation: 台湾大学
  • Year and Date: 2014-09-05
  • Invited
• [Presentation] Rationality problem for algebraic tori (2014)
  • Author(s): A.Hoshi, A.Yamasaki
  • Organizer: SEOUL ICM 2014, Short Communications
  • Place of Presentation: Seoul Coex
  • Year and Date: 2014-08-19
• [Presentation] Rationality problem of conic bundles (2014)
  • Author(s): A.Yamasaki
  • Organizer: SEOUL ICM 2014, Short Communications
  • Place of Presentation: Seoul Coex
  • Year and Date: 2014-08-16
• [Presentation] Class numbers and algebraic tori (2014)
  • Author(s): 山崎愛一,星明考,Ming-chang Kang
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学
• [Presentation] Rationality of generalized Chatelet surfaces (2013)
  • Author(s): 山崎愛一
  • Organizer: One-day workshop of algebra
  • Place of Presentation: 台湾大學
  • Invited
• [Presentation] 代数的トーラスとシャトレー曲面の有理性問題 (2013)
  • Author(s): 山崎愛一
  • Organizer: 西早稲田数論セミナー
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] z^2 = P(x) y^2 + Q(x) の有理性問題 (2013)
  • Author(s): 山崎愛一
  • Organizer: 日本数学会年会
  • Place of Presentation: 愛媛大学
• [Presentation] Krull-Schmidt theorem fails for dimension 5 (2013)
  • Author(s): 星明考,山崎愛一
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学
• [Presentation] 位数256の群のisoclinism family (2013)
  • Author(s): 山崎愛一
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学
• [Presentation] GL(5;Z), GL(6;Z) の有限部分群の分類表の整備と楕円曲線の計算につ いて (2012)
  • Author(s): 山崎愛一
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] Rationality problem for algebraic tori (2012)
  • Author(s): 星明考,山崎愛一
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学
• [Presentation] Rationality problem of algebraic tori (II) (2012)
  • Author(s): 山崎愛一
  • Organizer: One-day Workshop of Algebra
  • Place of Presentation: 台湾大學
  • Invited
• [Presentation] GL(n, Z) の有限部分群と楕円曲線 (2012)
  • Author(s): 山崎愛一
  • Organizer: Magma で開く数学の世界
  • Place of Presentation: 高知大学
  • Invited