Osaka Journal of Mathematics

On the Isomorphism classes of Iwasawa modules with $\lambda = 3$ and $\mu = 0$

Kazuaki Murakami

Full-text: Open access


For an odd prime number $p$, we classify the isomorphism classes of finitely generated torsion $\Lambda=\mathbb{Z}_{p}[[T]]$-modules with $\lambda=3$ and $\mu=0$, which are free over $\mathbb{Z}_{p}$. We apply this classification to the Iwasawa module associated to the cyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic field.

Article information

Osaka J. Math., Volume 51, Number 4 (2014), 829-867.

First available in Project Euclid: 31 October 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants 13C05: Structure, classification theorems


Murakami, Kazuaki. On the Isomorphism classes of Iwasawa modules with $\lambda = 3$ and $\mu = 0$. Osaka J. Math. 51 (2014), no. 4, 829--867.

Export citation


  • R. Ernvall and T. Metsänkylä: Computation of the zeros of $p$-adic $L$-functions, Math. Comp. 58 (1992), 815–830.
  • C. Franks: Classifying $\Lambda$-modules up to isomorphism and applications to Iwasawa theory, PhD Dissertation, Arizona State University May (2011).
  • T. Fukuda: Iwasawa $\lambda$-invariants of imaginary quadratic fields, J. College Industrial Technology Nihon Univ. 27 (1994), 35–88.
  • H. Ichimura and H. Sumida: On the Iwasawa invariants of certain real abelian fields II, Internat. J. Math. 7 (1996), 721–744.
  • K. Iwasawa: On $\Gamma$-extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183–226.
  • K. Murakami: On the isomorphism classes of Iwasawa modules, Master's thesis Tokyo University of Science (2010), in Japanese.
  • M. Koike: On the isomorphism classes of Iwasawa modules associated to imaginary quadratic fields with $\lambda=2$, J. Math. Sci. Univ. Tokyo 6 (1999), 371–396.
  • Y. Mizusawa:
  • D.G. Northcott: Finite Free Resolutions, Cambridge Univ. Press, Cambridge, 1976.
  • M. Saito and H. Wada: A table of ideal class groups of imaginary quadratic fields, Sophia Kokyuroku in Math. 28, (1988).
  • H. Sumida: Greenberg's conjecture and the Iwasawa polynomial, J. Math. Soc. Japan 49 (1997), 689–711.
  • H. Sumida: Isomorphism classes and adjoints of certain Iwasawa modules, Abh. Math. Sem. Univ. Hamburg 70 (2000), 113–117.
  • L.C. Washington: Introduction to Cyclotomic Fields, second edition, Graduate Texts in Mathematics 83, Springer, New York, 1997.