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Relations among Iwasawa invariants. (English) Zbl 0608.12005
For a prime $$p$$, let $$\mathbb Q_{\infty}$$ denote the cyclotomic $$\mathbb Z_ p$$-extension of the rational number field. Consider finite extensions of $$\mathbb Q_{\infty}$$ (in the complex number field), called $$\mathbb Z_ p$$-fields by the authors. Let $$K/F$$ be a finite Galois extension of $$\mathbb Z_ p$$-fields, let $$E_ 1,\ldots,E_ t$$ be intermediate fields of $$K/F$$, and put $$H_ i=\text{Gal}(K/E_ i)$$. The authors prove that, for the norm idempotents $$\varepsilon_ i=\sum h_ i | H_ i|^{-1}$$ $$(h_ i\in H_ i)$$, any relation $$\sum^{t}_{i=1}r_ i \varepsilon_ i=0$$ with rational coefficients $$r_ i$$ implies the relation $$\sum^{t}_{i=1}r_ i \lambda_ i=0$$ between the Iwasawa $$\lambda$$-invariants $$\lambda_ i$$ of $$E_ i$$. The proof makes use of certain characters of the groups $$H_ i$$.
From this result, analogs of two theorems of R. D. M. Accola [Proc. Am. Math. Soc. 25, 598–602 (1970; Zbl 0212.42502)] on Riemann surfaces are deduced. Moreover, it is shown how a similar result (with $$\lambda^-_ i$$ in place of $$\lambda_ i)$$ can be obtained from a theorem of K. Iwasawa [Tôhoku Math. J., II. Ser. 33, 263–288 (1981; Zbl 0468.12004)], provided that $$p>2$$ and $$K$$ and $$F$$ are $$\mathbb Z_ p$$-fields of CM-type.

##### MSC:
 11R18 Cyclotomic extensions 11R23 Iwasawa theory
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##### References:
 [1] Accola, R., Two theorems on Riemann surfaces with non-cyclic automorphism groups, (), 598-602 · Zbl 0212.42502 [2] Iwasawa, K., Riemann-Hurwitz formula and p-adic Galois representations for number fields, Tokoku math. J., 33, 263-288, (1981) · Zbl 0468.12004 [3] Kani, E., Relations between the genera and between the Hasse-Witt invariants of Galois coverings of curves, Canad. math. bull., 28, 321-327, (1985) · Zbl 0557.14017
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