Analysis of the structures of Iwasawa modules by arithmetic special elements

• Project/Area Number: 25400013
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokushima
• Principal Investigator: TAKAHASHI HIROKI 徳島大学, 大学院社会産業理工学研究部(理工学域), 教授 (90291476)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: Greenberg予想 / 一般Greenberg予想 / 岩澤加群 / 円単数 / 岩澤不変量 / Kummer-Vandiver予想 / イデアル類群 / 円分体 / 特殊元 / Vandiver予想

Outline of Final Research Achievements

The purpose of this research was to investigate concrete structures of Iwasawa modules associated to various p-adic Galois representations by using special elements, and to clarify detailed reasons of Greenberg's conjecture for totally real number fields and Greenberg's generalized conjecture for general algebraic number fields. Concerning the former conjecture, we computed structures of Iwasawa modules for cyclotomic Z_p-extensions of composite fields of p-cyclotomic fields and quadratic fields with the discriminant D<10 (resp.D<200) in the range 6,000,000<p<13,000,000 (resp. 300,000<p<600,000), and checked that the actual numbers are close to the expected numbers. Concerning the latter conjecture, we computed paring of p-unit groups of 4p-cyclotomic fields for Milnor K_2-groups in the range p<32768, and checked that the actual numbers of nontrivial zeros are close to the expected numbers.

Research Products

• [Journal Article] Normal integral basis of an unramified quadratic extension over a cyclotomic Z2-extension (2016)
  • Author(s): Humio Ichimura and Hiroki Sumida-Takahashi
  • Journal Title: Journal de Theorie des Nombres de Bordeaux Volume: 28 Issue: 2 Pages: 325-345
  • DOI: 10.5802/jtnb.942
  • NAID: 120006884834
  • URL: https://tokushima-u.repo.nii.ac.jp/records/2008108
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the 2-adic Iwasawa lambda invariants of the p-cyclotomic fields and their quadratic twists (2014)
  • Author(s): Humio Ichimura, Shoichi Nakajima and Hiroki Sumida-Takahashi
  • Journal Title: International Journal of Number Theory Volume: 10 Issue: 02 Pages: 283-296
  • DOI: 10.1142/s1793042113500929
  • Peer Reviewed
• [Presentation] 円分体の一般Greenberg予想とK群の特殊元 (2017)
  • Author(s): 高橋浩樹
  • Organizer: 九州代数的整数論2017
  • Place of Presentation: 九州大学（福岡県福岡市）
• [Presentation] 円分体の特殊元と岩澤不変量 (2015)
  • Author(s): 高橋浩樹
  • Organizer: 北陸数論研究集会
  • Place of Presentation: 金沢大学（石川県，金沢市）
  • Year and Date: 2015-12-26
• [Presentation] 代数体の岩澤不変量について (2014)
  • Author(s): 高橋浩樹
  • Organizer: 愛媛大学代数セミナー
  • Place of Presentation: 愛媛大学（愛媛県，松山市）
  • Year and Date: 2014-05-26
• [Presentation] 円分体の一般Greenberg予想に関するある計算法 (2014)
  • Author(s): 高橋浩樹
  • Organizer: 大阪大学整数論・保型形式セミナー
  • Place of Presentation: 大阪大学（大阪府，豊中市）
  • Year and Date: 2014-05-23
• [Remarks] Exploring the Galois Universe
  • URL: http://hiro2.pm.tokushima-u.ac.jp/~hiroki/major/galois1.html
• [Remarks] Exploring the Galois Universe
  • URL: http://hiro2.pm.tokushima-u.ac.jp/~hiroki/major/galois1.html