We prove in this article that, for any prime and tame level , the projection from the eigencurve to the weight space satisfies a rigid analytic version of the valuative criterion for properness introduced by Buzzard and Calegari. This gives a negative answer to a question of Coleman and Mazur.
"The eigencurve is proper." Duke Math. J. 165 (7) 1381 - 1395, 15 May 2016. https://doi.org/10.1215/00127094-3450536