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- Title
Minimal resolutions of Iwasawa modules.
- Authors
Kataoka, Takenori; Kurihara, Masato
- Abstract
In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic Z p -extension K ∞ / K , we consider X K ∞ , S = Gal (M K ∞ , S / K ∞) where S is a finite set of places of k containing all ramifying places in K ∞ and archimedean places, and M K ∞ , S is the maximal abelian pro-p-extension of K ∞ unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of X K ∞ , S as a Z p [ [ Gal (K ∞ / k) ] ] -module, using the p-rank of Gal (K / k) . This result explains the complexity of X K ∞ , S as a Z p [ [ Gal (K ∞ / k) ] ] -module when the p-rank of Gal (K / k) is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of X K ∞ , S , and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of X K ∞ , S .
- Subjects
CYCLOTOMIC fields; NONABELIAN groups
- Publication
Research in Number Theory, 2024, Vol 10, Issue 3, p1
- ISSN
2522-0160
- Publication type
Article
- DOI
10.1007/s40993-024-00549-y