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On a theorem of L. Washington. (English) Zbl 0616.12004
Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 209-224 (1987).
[For the entire collection see Zbl 0605.00004.]
Let \(k\) be an abelian number field, let \(p\) be a prime number, and let \(k_ n\) be the \(n\)-th layer of the cyclotomic \({\mathbb Z}_ p\)-extension of \(k\). For a prime \(\ell\), let \(\ell^{e_ n}\) be the power of \(\ell\) dividing the class number of \(k_ n\). When \(\ell =p\), Iwasawa showed that there exist integers \(\lambda\), \(\mu\), \(\nu\) such that \(e_ n=\lambda n+\mu p^ n+\nu\) for all sufficiently large \(n\).
B. Ferrero and the reviewer [Ann. Math. (2) 109, 377–395 (1979; Zbl 0443.12001)] used the theory of normal numbers to prove that \(\mu =0\). When \(\ell \neq p\), the reviewer [Invent. Math. 49, 87–97 (1978; Zbl 0403.12007)] used similar techniques to show that \(e_ n\) is constant for large \(n\). In [ibid. 75, 273–282 (1984; Zbl 0531.12004)], the present author used the fact that \(p\)-adic \(L\)-functions are \(\Gamma\)–transforms of rational functions to eliminate the use of normal numbers in the proof in the case \(\ell =p\). In the present paper, he uses similar techniques to give a simplification of the proof in the case \(\ell \neq p\).

MSC:
11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions