\(\Gamma\)-transforms of rational function measures on \(\mathbb Z_ S\)
type: Item
instance of: scholarly article
title: \(\Gamma\)-transforms of rational function measures on \(\mathbb Z_ S\)
zbMATH Open document ID: 0637.12004
DOI: 10.1007/BF01404675
author: Warren Sinnott
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143475
review text: Let \(k\) be a finite abelian extension of \(\mathbb Q\), let \(p_ 1,\dots,p_ s\) be distinct primes, and let \(K/k\) be the cyclotomic \(\mathbb Z_{p_ 1}\times\dots\times\mathbb Z_{p_ s}\)-extension of \(k\). \textit{E. C. Friedman} [Invent. Math. 65, 425--440 (1982; Zbl 0495.12007)] generalizing previous results of \textit{B. Ferrero} and the reviewer [Ann. Math. (2) 109, 377--395 (1979; Zbl 0443.12001)] and of the reviewer [Invent. Math. 49, 87--97 (1978; Zbl 0403.12007)] showed that if \(\ell\) is distinct from \(p_ 1,\dots,p_ s\) then the \(\ell\)-part of the class number of \(k'\) is bounded, where \(k'\) ranges over the finite subextensions of \(K\), and he also gave a simple formula for the behavior of the \(\ell\)-part of the class number when \(\ell =p_ i\) for some \(i\). The original proofs of these results all relied on the theory of normal \(p\)-adic numbers. More recently, the present author has given new proofs of these results. In [Invent. Math. 75, 273--282 (1984; Zbl 0531.12004)] he proved the result of Ferrero and the reviewer (namely, the vanishing of the Iwasawa \(\mu\)-invariant), in [Journées arithmétiques, Besançon/Fr. 1985, Astérisque 147--148, 209--224 (1987; Zbl 0616.12004)] he proved the result of the reviewer (the case \(s=1\) and \(\ell \neq p_ 1)\), and in the present paper he proves Friedman's result (for the minus part of the class number). In all three cases, the proofs, which are simpler than the original ones, rely on a study of the gamma transforms of \(p\)-adic measures whose Fourier transforms can be described by rational functions. [Remark: The result of Ferrero and the reviewer was not proved in 1976, as stated in the history at the beginning of the article, but rather in 1978.]
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 4037948
zbMATH Keywords: behavior of the \(\ell \)-part of the class number
zbMATH Keywords: normal p-adic numbers
zbMATH Keywords: gamma transforms of p-adic measures
MaRDI profile type: Publication
cites work: The Iwasawa invariant \(\mu_p\) vanishes for abelian number fields
cites work: Ideal class groups in basic \(\mathbb Z_{p_1}\times\dots\times\mathbb Z_{p_s}\)-extensions of abelian number fields
cites work: On Γ-extensions of algebraic number fields
cites work: On the \(\mu\)-invariant of the \(\Gamma\)-transform of a rational function
cites work: Q3335870
cites work: The non-p-part of the class number in a cyclotomic \(\mathbb{Z}_p\)-extension
cites work: Q3943047
description: scientific article; zbMATH DE number 4037948