  In the late 1990's, 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.
