Functiones et Approximatio Commentarii Mathematici

On $\lambda$-invariants of $\mathbb{Z}_\ell$-extensions over real abelian number fields of prime power conductors

Takashi Fukuda, Keiichi Komatsu, and Takayuki Morisawa

Full-text: Open access


For each prime number $\ell$ less than $10^4$, we construct an infinite family of abelian number fields for which Iwasawa $\lambda_{\ell}$-invariants vanish.

Article information

Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 95-104.

First available in Project Euclid: 25 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory
Secondary: 11Y40: Algebraic number theory computations

Iwasawa invariant computation.


Fukuda, Takashi; Komatsu, Keiichi; Morisawa, Takayuki. On $\lambda$-invariants of $\mathbb{Z}_\ell$-extensions over real abelian number fields of prime power conductors. Funct. Approx. Comment. Math. 47 (2012), no. 1, 95--104. doi:10.7169/facm/2012.47.1.8.

Export citation


  • L. Carlitz, Some arithmetic properties of generalized Bernoulli numbers, Bull. Amer. Math. Soc. 65 (1959), 68–69.
  • J. Coates, $p$-adic $L$-functions and Iwasawa theory, Algebraic Number Fields (Durham Symposium, 1975), ed. by A. Fröhlich, 269–353, Academic Press, London, 1977.
  • B. Ferrero and L. Washington, The Iwasawa invariant $\mu_p$ vanishes for abelian number fields, Ann. Math. 109 (1979), 377–395.
  • E. Friedman, Ideal Class Groups in Basic $\ACOHZ_{p_1}\times\ldots\times\ACOHZ_{p_s}$-Extensions of Abelian Number Fields Invent. Math. 65 (1982), 425–440.
  • E. Friedman and J. W. Sands, On the $\ell$-adic Iwasawa $\lambda$-invariant in a $p$-extension, Math. Comp. 64 (1995), 1659–1674.
  • T. Fukuda and K. Komatsu, Weber's class number problem in the cyclotomic $\ACOHZ_2$-extension of $\ACOHQ$, Experiment. Math. 18-2 (2009), 213–222.
  • R. Greenberg, On the Iwasawa invariants of totally real number fields. Amer. J. Math. 98(1976), 263–284.
  • K. Horie, Certain primary components of the ideal class group of the $\ACOHZ_p$-extension over the rationals, Tohoku Math. J. 59 (2007), 259–291.
  • H. Ichimura and H. Sumida, On the Iwasawa Invariants of certain real abelian fields II, Inter. J. Math. 7 (1996), 721–744.
  • K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258.
  • T. Morisawa, A Class Number Problem in the Cyclotomic $\ACOHZ_{3}$-extension of $\ACOHQ$, Tokyo J. Math. 32 (2009), 549–558.
  • J. Nakagawa and K. Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), 20–24.
  • K. Ono, Indivisibility of class numbers of real quadratic fields, Compositio Math. 119 (1999), 1–11.
  • M. Ozaki and H. Taya, On the Iwasawa $\lambda_2$-invariants of certain families of real quadratic fields, Manuscripta Math. 94 (1997), no. 4, 437- 444.
  • L. Washington, Introduction to Cyclotomic Fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, Berlin/New York, 1997