Osaka Journal of Mathematics

On tamely ramified Iwasawa modules for the cyclotomic $\mathbb {Z}_{p}$-extension of abelian fields

Tsuyoshi Itoh

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Let $p$ be an odd prime, and $k_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of an abelian field $k$. For a finite set $S$ of rational primes which does not include $p$, we will consider the maximal $S$-ramified abelian pro-$p$ extension $M_{S} (k_{\infty})$ over $k_{\infty}$. We shall give a formula of the $\mathbb{Z}_{p}$-rank of $\mathrm{Gal}(M_{S} (k_{\infty})/k_{\infty})$. In the proof of this formula, we also show that $M_{\{q\}} (k_{\infty})/L(k_{\infty})$ is a finite extension for every real abelian field $k$ and every rational prime $q$ distinct from $p$, where $L(k_{\infty})$ is the maximal unramified abelian pro-$p$ extension over $k_{\infty}$.

Article information

Osaka J. Math., Volume 51, Number 2 (2014), 513-537.

First available in Project Euclid: 8 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11R18: Cyclotomic extensions


Itoh, Tsuyoshi. On tamely ramified Iwasawa modules for the cyclotomic $\mathbb {Z}_{p}$-extension of abelian fields. Osaka J. Math. 51 (2014), no. 2, 513--537.

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  • A. Brumer: On the units of algebraic number fields, Mathematika 14 (1967), 121–124.
  • B. Ferrero and L.C. Washington: The Iwasawa invariant $\mu_{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), 377–395.
  • R. Greenberg: On a certain $l$-adic representation, Invent. Math. 21 (1973), 117–124.
  • R. Greenberg: On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139–158.
  • C. Greither: Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), 449–499.
  • T. Itoh, Y. Mizusawa and M. Ozaki: On the $\mathbb{Z}_{p}$-ranks of tamely ramified Iwasawa modules, Int. J. Number Theory 9 (2013), 1491–1503.
  • \begingroup K. Iwasawa: On $\mathbf{Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326. \endgroup
  • K. Iwasawa: Riemann–Hurwitz formula and $p$-adic Galois representations for number fields, Tôhoku Math. J. (2) 33 (1981), 263–288.
  • \begingroup C. Khare and J.-P. Wintenberger: Ramification in Iwasawa modules, arXiv:1011.6393, (2010). \endgroup
  • Y. Mizusawa and M. Ozaki: On tame pro-$p$ Galois groups over basic $\mathbb{Z}_{p}$-extensions, Math. Z. 273 (2013), 1161–1173.
  • J. Neukirch, A. Schmidt and K. Wingberg: Cohomology of Number Fields, second edition, Grundlehren der Mathematischen Wissenschaften 323, Springer, Berlin, 2008.
  • L. Salle: On maximal tamely ramified pro-2-extensions over the cyclotomic $\mathbb{Z}_{2}$-extension of an imaginary quadratic field, Osaka J. Math. 47 (2010), 921–942.
  • T. Tsuji: Semi-local units modulo cyclotomic units, J. Number Theory 78 (1999), 1–26.
  • T. Tsuji: On the Iwasawa $\lambda$-invariants of real abelian fields, Trans. Amer. Math. Soc. 355 (2003), 3699–3714.
  • L.C. Washington: Introduction to Cyclotomic Fields, second edition, Graduate Texts in Mathematics 83, Springer, New York, 1997.