Iwasawa theoretic approach to the absolute Galois groups of number fields

• Project/Area Number: 16K05080
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Waseda University
• Principal Investigator: Ozaki Manabu 早稲田大学, 理工学術院, 教授 (80287961)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 岩澤理論 / ガロワ群 / 代数的整数論 / Galois群 / 絶対Galois群 / 算術的同値 / Dedekindゼータ函数 / Neukirch-内田の定理 / 代数体

Outline of Final Research Achievements

By introducing Iwasawa theoretic method to the study of the absolute Galois groups of number fields, I have succeeded in (1) generalizing the Neukirch-Uchida theorem to number fields of infinite degrees, (2) characterizing Dedekind zeta functions from small quotients of the absolute Galois groups, (3) establishing freeness conjecture for the imaginary quadratic fields.

Academic Significance and Societal Importance of the Research Achievements

本研究によって代数的整数論における岩澤理論的な手法の有効性をさらに高めることができ，今後の研究に対して新たなる方向付けを与えることができた．

Research Products

• [Journal Article] Non-abelian Iwasawa theory of Z_p-extensions---overview and outlook. (Japanese) (2017)
  • Author(s): Manabu Ozaki
  • Journal Title: RIMS Kokyuroku Bessatsu, Volume: B64 Pages: 313-330
  • Peer Reviewed
• [Journal Article] A GCD and LCM-like inequality for multiplicative lattices. (2016)
  • Author(s): Daniel D. Anderson, Takashi Aoki, Shuzo Izumi, Yasuo Ohno, and Manabu Ozaki
  • Journal Title: Tamkang J. Math. Volume: 47 Issue: 3 Pages: 261-270
  • DOI: 10.5556/j.tkjm.47.2016.1822
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] ある種の無限次代数体の絶対Galois群について (2016)
  • Author(s): 尾崎 学
  • Organizer: 北陸数論研究集会2016
  • Place of Presentation: 金沢大学サテライトプラザ
  • Invited