We show that the -adic eigencurve is smooth at classical weight points which are regular at and give a precise criterion for étaleness over the weight space at those points. Our approach uses deformations of Galois representations.
"On the eigencurve at classical weight points." Duke Math. J. 165 (2) 245 - 266, 1 February 2016. https://doi.org/10.1215/00127094-3165755