Class groups and local indecomposability for non-CM forms

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