Let
$n$
be either
$2$
or an odd integer greater than
$1$
, and fix a prime
$p>2(n+1)$
. Under standard ‘adequate image’ assumptions, we show that the set of components of
$n$
-dimensional
$p$
-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on
$n$
) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2) 179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.
$G$-valued Galois deformation rings when $\ell \neq p$
