On the triviality of the unramified Iwasawa modules of the maximal multiple ℤ_p-extensions

2024/06/04 by Keiji Okano, Okano, Keiji
Mathematics · #11R23 #11R37 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2406.01892

openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a number field k and an odd prime number p, we consider the maximal multiple ℤ_p-extension k of k and the unramified Iwasawa module X(k), which is the Galois group of the maximal unramified abelian p-extension of k. In this article, we classify the CM-fields k in which p splits completely and for which X(k) = 0. In addition, we provide an alternative proof of the sufficient condition for X(k)=0, based on the ideas of Minardi, Itoh, and Fujii in the study of the generalized Greenberg conjecture.

Related

• On Greenberg's generalized conjecture
• On Greenberg's generalized conjecture for families of number fields
• Galois coinvariants of the unramified Iwasawa modules of multiple ℤ_p-extensions
• On generators of unramified Iwasawa module over an imaginary quadratic field
• Weak Greenberg's generalized conjecture for imaginary quadratic fields