Class groups and local indecomposability for non-CM forms

In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those p -ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at p . It is expected that such p -ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman–Greenberg’s question using Galois deformation theory. In particular, for p -ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the p -indivisibility of a certain class group.