an:03997933
Zbl 0616.12004
Sinnott, W.
On a theorem of L. Washington
EN
Journ??es arithm??tiques, Besan??on/France 1985, Ast??risque 147/148, 209-224 (1987).
1987
a
11R23 11R42 11S40
cyclotomic fields; p-adic measures; cyclotomic Zp-extension; class number; p-adic L-functions
[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 \(n\).
\textit{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 \(\ell \neq p\).
Lawrence C. Washington (College Park)
Zbl 0605.00004; Zbl 0443.12001; Zbl 0403.12007; Zbl 0531.12004