Failure of the Local to Global Principle in the Eigencurve

Abstract

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away from p) under one of two conditions: on a generically principal series component where monodromy vanishes; or on a generically special component where the ratio of the Satake parameters degenerates. We prove, under mild restrictive hypotheses, that such points are the intersection of generically principal series and special components. This is a geometric analogue of Ribet's level raising and lowering theorems.