Analysis of multivariable Iwasawa modules by using special elements of K-groups

• Project/Area Number: 17K05176
• 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: SUMIDA-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): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: Greenberg予想 / Kummer-Vandiver予想 / 一般Greenberg予想 / K-群 / イデアル類群 / 単数 / 特殊元 / 単数群 / K群 / 岩澤加群 / Vandiver予想 / 円単数 / カップ積 / 円分体

Outline of Final Research Achievements

Concernig some conjecture (generalized Greenberg's conjecture, Greenberg's conjecture for totally real fields, the Kummer-Vandiver conjecture), we computed special elements of K-groups of cyclotomic fields, and investigated reasons of validity or invalidity of these conjectures. Particularly, concernig Greenberg's generalized conjecture, we computed parings of p-units, and check the conjecture for 4p-cyclotomic fields and all p<65536 such that p is congruent to 3 modulo 4 except for 3 primes. The distributions of the number of nontrivial zeros are close to the predicted distributions, which can be one of reasons of the validity of Greenberg's generalized conjecture.

Academic Significance and Societal Importance of the Research Achievements

円分体のイデアル類群や単数群は古典的には不定方程式の解法において有効であり，その重要性は200年ほど前から数論研究者に認識されている．最近では耐量子計算機暗号の候補として円分イデアル格子暗号や同種写像による楕円曲線暗号などが候補に挙がっているが，その理由としては定義の簡明さと構造の複雑さという点と多数の研究者が調査対象としている点が挙げられる．本研究は，その重要な対象の基本的な現象に関する予想が成立するか否かについて調査している．

Research Products

• [Journal Article] On the class groups of certain imaginary cyclic fields of 2-power degree (2022)
  • Author(s): ICHIMURA Humio、SUMIDA-TAKAHASHI Hiroki
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 74 Issue: 3 Pages: 945-972
  • DOI: 10.2969/jmsj/86438643
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access
• [Journal Article] A Generalized Problem Associated to the Kummer-Vandiver Conjecture (2022)
  • Author(s): Sumida-Takahashi Hiroki
  • Journal Title: Arnold Mathematical Journal Volume: - Issue: 3 Pages: 381-391
  • DOI: 10.1007/s40598-022-00220-3
  • Peer Reviewed
• [Journal Article] On the l-part of the Class Groups of Imaginary Cyclic Fields of Conductor p and Degree 2l^n (2022)
  • Author(s): Hiroki Sumida-Takahashi, Naoki Furuya and Kodai Kitano
  • Journal Title: Journal of Mathematics, Tokushima University Volume: 56 Pages: 1-10
  • Open Access
• [Journal Article] On the Class Group of an Imaginary Cyclic Field of Conductor 8p and 2-power Degree (2021)
  • Author(s): ICHIMURA Humio、SUMIDA-TAKAHASHI Hiroki
  • Journal Title: Tokyo Journal of Mathematics Volume: 44 Issue: -1
  • DOI: 10.3836/tjm/1502179326
  • Peer Reviewed
• [Presentation] ある種の 2 ベキ次巡回拡大体のイデアル類群について (2021)
  • Author(s): 高橋浩樹
  • Organizer: 早稲田整数論セミナー
• [Presentation] ゼータ値と円分体 (2018)
  • Author(s): 高橋浩樹
  • Organizer: 岐阜数理科学セミナー
• [Remarks] Exploring the Galois Universe
  • URL: https://math0.pm.tokushima-u.ac.jp/~hiroki/major/galois1-e.html
• [Remarks] Exploring the Galois Universe
  • URL: https://math0.pm.tokushima-u.ac.jp/~hiroki/major/galois1.html