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arXiv · 1904.09522

Ordinary deformations are unobstructed in the cyclotomic limit

Ashay Burungale·Laurent Clozel
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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 $k$.

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Ashay Burungale, Laurent Clozel. 2019-04-21. Ordinary deformations are unobstructed in the cyclotomic limit. https://arxiv.org/abs/1904.09522

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