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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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[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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