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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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[2] Ferrero, B., Washington, L.: The Iwasawa invariant ? p vanishes for abelian number fields. Ann. of Math.109, 377-395 (1979) · Zbl 0443.12001 · doi:10.2307/1971116
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