Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 2

On the $\mu$ equals zero conjecture for fine Selmer groups in Iwasawa theory

Pages: 641 – 680

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n2.a8


Shaunak V. Deo (Department of Mathematics, Indian Institute of Science, Bangalore, India)

Anwesh Ray (Centre de recherches mathématiques, Université de Montréal, Quebec, Canada)

R. Sujatha (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)


We study the Iwasawa theory of the fine Selmer groups associated to Galois representations arising from modular forms. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $\mu = 0$ conjecture is shown to hold for various Galois representations of interest.


Iwasawa $\mu$-invariant, fine Selmer groups, adjoint representations, deformations of Galois representations

2010 Mathematics Subject Classification

Primary 11R23. Secondary 11F11, 11F80, 11G05.

The first-named author was supported by the CRM-Simons bridge postdoctoral fellowship.

The third-named author gratefully acknowledges support from NSERC Discovery grant 2019-03987

Received 3 October 2022

Received revised 8 December 2022

Accepted 11 January 2023

Published 7 April 2023