# zbMATH — the first resource for mathematics

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