On a new invariant determining the isomorphism classes of \(\Lambda \)-modules with \(\lambda =3\)


For a prime number p and the ring of power series \(\Lambda ={\mathbb {Z}}_p[[T]]\), we define a new invariant which determines the isomorphism classes of finitely generated torsion \(\Lambda \)-modules with \(\lambda =3\). We apply this result to the classical Selmer group attached to an elliptic curve over \({\mathbb {Q}}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Cremona, J.E.: Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  2. 2.

    Franks, C.: Classifying \(\Lambda \)-modules up to Isomorphism and Applications to Iwasawa Theory. Ph.D. dissertation, Arizona State University (2011)

  3. 3.

    Greenberg, R.: Iwasawa theory for elliptic curves in Arithmetic theory of elliptic curves (Cetraro, 1997). Springer Lect. Notes Math. 1716(1999), 51–144 (1999)

    Article  Google Scholar 

  4. 4.

    Koike, M.: On the isomorphism classes of Iwasawa modules associated to imaginary quadratic fields with \(\lambda = 2\). J. Math. Sci. Univ. Tokyo 6, 371–396 (1999)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Kato, K.: \(p\)-adic Hodge theory and values of zeta functions of modular forms, cohomologies \(p\)-adiques et applications arithmetiques, III. Asterisque 295, 117–290 (2004)

    MATH  Google Scholar 

  6. 6.

    Kurihara, M.: Iwasawa theory and Fitting ideals. J. Reine Angew. Math. 561, 39–86 (2003)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Kurihara, M.: Refined Iwasawa theory and Kolyvagin systems of Gauss sum type. Proc. Lond. Math. Soc. 104, 728–769 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kurihara, M.: Refined Iwasawa theory for \(p\)-adic representations and the structure of Selmer groups. Munster J. Math. 7, 149–223 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Mazur, B., Rubin, K.: Organizing the arithmetic of elliptic curves. Adv. Math. 198, 504–546 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Murakami, K.: On the isomorphism classes of Iwasawa modules with \(\lambda =3\) and \(\mu =0\). Osaka J. Math. 51, 829–865 (2014)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Murakami, K.: Isomorphism classes of modules over Iwasawa algebra with \(\lambda =4\). Tokyo J. Math. 37, 101–132 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Northcott, D.G.: Finite Free Resolutions. Cambridge University Press, Cambridge-New York (1976)

    Google Scholar 

  13. 13.

    Pollack, R.: http://math.bu.edu/people/rpollack/

  14. 14.

    Skinner, C., Urban, E.: The Iwasawa main conjectures for \({{\rm GL}}_{2}\). Invent. Math. 195, 1–277 (2014)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sumida, H.: Greenberg’s conjecture and the Iwasawa polynomial. J. Math. Soc. Jpn. 49, 689–711 (1997)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Sumida, H.: Isomorphism classes and adjoints of certain Iwasawa modules. Abh. Math. Sem. Univ. Hamburg 70, 113–117 (2000)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Washington, L.C.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn. Springer, New York (1997)

    Google Scholar 

Download references


.The author would like to thank Professor Masato Kurihara for his valuable suggestions, which led me to get the results in this paper. The author also would like to express his gratitude to the referee for reading this article carefully and giving many helpful suggestions. The author is partially supported by JSPS Core-to-core program, Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory.

Author information



Corresponding author

Correspondence to Kazuaki Murakami.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Murakami, K. On a new invariant determining the isomorphism classes of \(\Lambda \)-modules with \(\lambda =3\). manuscripta math. 164, 409–430 (2021). https://doi.org/10.1007/s00229-020-01190-6

Download citation

Mathematics Subject Classification

  • Primary 11R23
  • Secondary 11G05