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On the $$\mu$$-invariant of the $$\Gamma$$-transform of a rational function. (English) Zbl 0531.12004
The author gives a new and substantially simpler proof of the theorem of B. Ferrero and the reviewer [Ann. Math. (2) 109, 377–395 (1979; Zbl 0443.12001)] that the Iwasawa $$\mu$$-invariant vanishes for cyclotomic $${\mathbb Z}_ p$$-extensions of Abelian number fields. In fact, he deduces this from a more general result, namely the $$\mu$$-invariant of the $$\Gamma$$-transform of the $$p$$-adic measure associated to a rational function is essentially the same as the $$\mu$$-invariant of the rational function (regarded as a power series). Since the generating function of the generalized Bernoulli numbers is closely related to a rational function with a coefficient prime to $$p$$, and the $$\Gamma$$-transform of this rational function yields the $$p$$-adic $$L$$-function, the vanishing of Iwasawa’s $$\mu$$-invariant follows. The main step in the proof is the following proposition: Let $${\mathbb F}$$ be a field of characteristic $$p$$. Suppose that for each $$p$$-adic $$(p-1)$$st root of unity $$\eta$$ we are given a rational function $$r_{\eta}(Z)\in {\mathbb F}(Z)$$ and suppose that in $${\mathbb F}((T-1))$$ we have $$\sum r_{\eta}(T^{\eta})=0$$. Then $$r_{\eta}(Z)+r_{\eta}(Z^{-1})\in {\mathbb F}$$ for each $$\eta$$. This proposition replaces the highly combinatorial arguments involving normal numbers used in the original proof that $$\mu =0.$$
In a forthcoming paper the author uses similar techniques to prove the theorem of the reviewer [Invent. Math. 49, 87–97 (1978; Zbl 0403.12007)] describing the behavior of the non-$$p$$-part of the class number in a cyclotomic $${\mathbb Z}_ p$$-extension and also its extension by E. C. Friedman [Invent. Math. 65, 425–440 (1982; Zbl 0495.12007)] to cyclotomic $${\mathbb Z}_{p_ 1}\times...\times {\mathbb Z}_{p_ s}$$ extensions. He also obtains much better bounds for the $$\lambda$$-invariant than those obtained by B. Ferrero and the reviewer.
R. Gillard has recently been able to apply the author’s techniques to prove that $$\mu =0$$ for certain non-cyclotomic $${\mathbb Z}_ p$$-extensions of imaginary quadratic fields.

MSC:
 11R23 Iwasawa theory 11S40 Zeta functions and $$L$$-functions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 11R58 Arithmetic theory of algebraic function fields 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:
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