Galois coinvariants of the unramified Iwasawa modules of multiple ℤ_p-extensions

2019/03/30 by Takashi Miura, Miura, Takashi, Kazuaki Murakami +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1904.00163

openalex publication_date 2019/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a CM-field K and an odd prime number p, let \widetilde K' be a certain multiple ℤ_p-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module X_{\widetilde K'} of \widetilde K' as a ℤ_p[[\rm Gal(\widetilde K'/K)]]-module. Our first main result is a description of the order of a Galois coinvariant of X_{\widetilde K'} in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic ℤ_p-extension of K under a certain assumption on the splitting of primes above p. Second one is that if K is an imaginary quadratic field and p does not split in K, we give a necessary and sufficient condition for which X_{\widetilde K} is ℤ_p[[\rm Gal(\widetilde K/K)]]-cyclic under several assumptions on the Iwasawa λ-invariant and the ideal class group of K, where \widetilde K is the ℤ_p²-extension of K.

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