New Developments in Non-Abelian Iwasawa Theory

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03432
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Waseda University
• Principal Investigator: Ozaki Manabu 早稲田大学, 理工学術院, 教授 (80287961)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ガロワ群 / 類数 / K-群 / ゼータ函数

Outline of Final Research Achievements

The summary of the results obtained in this research is as follows:
1. For a totally imaginary finite extension k of the rational number field Q　and the cyclotomic Z^－extension Ω of Q, when their intersection is Q, it was　shown that the structure of the Galois group X(kΩ)=Gal(L/kΩ) of the maximal　unramified Abelian extension L/kΩ characterizes the Galois closure of k. 2. Let k be a totally real number field. It was shown that X(k(μ)) completely determines the Dedekind zeta function of k. 3. For a finitely generated pro-p extension of an algebraic number field, it was proven that the class numbers of the intermediate fields converges p-adically, and a simple relationship was discovered between the p-adic limits of the class numbers and that of the order of the K_2-groups using the p-adic L-function.

Free Research Field

数論

Academic Significance and Societal Importance of the Research Achievements

Ｄedekindゼータ函数と種々のＧalois群や類群などの数論的対象物との関係性を追究することは数論における重要なテーマの一つである．本研究に於いて従来知られていなかった興味深いそれらの関係性を発見することができた．
また，研究の過程で新たな研究課題を見出すことができたので，今後の研究の一つの指針を与えることもできた．