Asian Journal of Mathematics

Volume 27 (2023)

Number 3

Ordinary deformations are unobstructed in the cyclotomic limit

Pages: 405 – 422



Ashay Burungale (California Institute of Technology, Pasadena Calif., U.S.A.; and University of Texas at Austin, Tx., U.S.A.)

Laurent Clozel (Mathématiques Université Paris-Sud, Orsay, France)


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 $k$.


Galois deformation theory, ordinary deformations, Iwasawa theory

2010 Mathematics Subject Classification

11F80, 11R23, 11S25

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Received 8 April 2021

Accepted 15 March 2023

Published 7 November 2023