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Asian Journal of Mathematics
Volume 27 (2023)
Number 3
Ordinary deformations are unobstructed in the cyclotomic limit
Pages: 405 – 422
DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n3.a4
Authors
Abstract
The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $\mu$-invariants vanish (as is often expected), then $R_\infty$ is free over the ring of Witt vectors of $k$.
Keywords
Galois deformation theory, ordinary deformations, Iwasawa theory
2010 Mathematics Subject Classification
11F80, 11R23, 11S25
Received 8 April 2021
Accepted 15 March 2023
Published 7 November 2023