# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Cassou-Noguès, P.: Valeurs aux entiers négatifs des fonctions zêta et fonctions zêtap-adiques. Invent. math.51, 29-59 (1979) · Zbl 0408.12015 · doi:10.1007/BF01389911  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  Katz, N.:p-adicL-functions via moduli of elliptic curves. In: Proceedings A.M.S. Summer Institute of Alg. Geom. at Arcata, Calif., 1974  Katz, N.: Another look atp-adicL-functions for totally real fields. Math. Ann.255, 33-43 (1981) · Zbl 0497.14006 · doi:10.1007/BF01450554  Iwasawa, K.: Onp-adicL-functions, Ann. of Math.89, 198-205 (1969) · Zbl 0186.09201 · doi:10.2307/1970817  Iwasawa, K.: Lectures onp-adicL-functions, Ann. of Math. Studies 74. Princeton University Press 1972  Lang, S.: Cyclotomic fields. Berlin-Heidelberg-New York: Springer 1978 · Zbl 0395.12005  Lang, S.: Cyclotomic fields II. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0435.12001  Mazur, B.: Analysep-adique. Secret Bourbaki redaction. 1973  Washington, L.: Introduction to Cyclotomic fields. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0484.12001  Washington, L.: The non-p-part of the class number in a cyclotomicZ p -extension. Invent. Math.49, 87-97 (1978) · Zbl 0403.12007 · doi:10.1007/BF01399512  Lang, S.: Introduction to Modular Forms. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0344.10011  Friedman, E.: Ideal class groups in basic $$Z_{p_1 } \times \ldots \times Z_{p_s }$$ -extensions of abelian number fields. Invent. math.65, 425-440 (1982) · Zbl 0495.12007 · doi:10.1007/BF01396627
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.