Abstract: 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.
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