Abstract
Given an odd prime number p and an imaginary quadratic field K, we establish a relationship between the p-rank of the class group of K, and the classical \(\lambda \)-invariant of the cyclotomic \(\mathbb {Z}_p\)-extension of K. Exploiting this relationship, we prove statistical results for the distribution of \(\lambda \)-invariants for imaginary quadratic fields ordered according to their discriminant. Some of our results are conditional since they rely on the original Cohen–Lenstra heuristics for the distribution of the p-parts of class groups of imaginary quadratic fields. Some results are unconditional results ad are obtained by leveraging theorems of Byeon, Craig and others.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No data was generated or analyzed in this paper.
References
Andrews, G.E.: The theory of partitions. Number 2. Cambridge university press (1998)
Byeon, D.: Imaginary quadratic fields with noncyclic ideal class groups. Ramanujan J. 11(2), 159–163 (2006)
Chattopadhyay, J., Saikia, A.: On the \( p \)-ranks of the ideal class groups of imaginary quadratic fields (2021). arXiv preprint arXiv:2112.00472
Cohen, H., Lenstra, H.W: Heuristics on class groups of number fields. In: Number Theory Noordwijkerhout 1983, pages 33–62. Springer (1984)
Craig, Maurice: A construction for irregular discriminants. Osaka J. Math. 14(2), 365–402 (1977)
Ellenberg, J.S., Jain, S., Venkatesh, A.: Modeling \(\lambda \)-invariants by p-adic random matrices. Commun. Pure Appl. Math. 64(9), 1243–1262 (2011)
Ferrero, B., Washington, L.C: The Iwasawa invariant \(\mu \)p vanishes for abelian number fields. Ann. Math. 377–395 (1979)
Horie, K.: A note on basic Iwasawa \(\lambda \)-invariants of imaginary quadratic fields. Invent. Math. 88(1), 31–38 (1987)
Iwasawa, K.: On \({\mathbb{Z}}_\ell \)-extensions of algebraic number fields. Ann. Math. 246–326 (1973)
Jochnowitz, N.: A p-adic conjecture about derivatives of L-series attached to modular forms. Contemp. Math. 165, 239–239 (1994)
Kulkarni, K., Levin, A.: Hilbert’s irreducibility theorem and ideal class groups of quadratic fields (2021). arXiv preprint arXiv:2111.15582
Levin, A., Shengkuan, Y., Wiljanen, L.: Quadratic fields with a class group of large 3-rank (2019). arXiv preprint arXiv:1910.12276
Luca, F., Pacelli, A.M.: Class groups of quadratic fields of 3-rank at least 2: effective bounds. J. Number Theory 128(4), 796–804 (2008)
Sands, J.W.: On the non-triviality of the basic Iwasawa \(\lambda \)-invariant for an infinitude of imaginary quadratic fields. Acta Arithmetica 65(3), 243–248 (1993)
Washington, L.C.: Introduction to cyclotomic fields, volume 83. Springer Science & Business Media (1997)
Washington, L.C.: Some remarks on Cohen–Lenstra heuristics. Math. Comput. 47(176), 741–747 (1986)
Yu, G.: Imaginary quadratic fields with class groups of 3-rank at least 2. Manusc. Math. 163(3), 569–574 (2020)
Acknowledgements
The author’s research is supported by the CRM Simons postdoctoral fellowship. He thanks the referee for helpful suggestions.
Funding
The funding has been received from Simons Foundation.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author is employed at the Centre de recherches mathematiques Montreal, advised by Matilde Lalin and Antonio Lei.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Ray, A. A Note on the Distribution of Iwasawa Invariants of Imaginary Quadratic Fields. Bull Braz Math Soc, New Series 54, 36 (2023). https://doi.org/10.1007/s00574-023-00353-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00574-023-00353-9