\(\Gamma\)-transforms of rational function measures on \(\mathbb Z_ S\)
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.]
behavior of the \(\ell \)-part of the class number
scientific article; zbMATH DE number 4037948