In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those -ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at . It is expected that such -ordinary eigenforms are precisely those with complex multiplication.
In this paper, we study Coleman–Greenberg’s question using Galois deformation theory. In particular, for -ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the -indivisibility of a certain class group.
Cite this article
Francesc Castella, Carl Wang-Erickson, Haruzo Hida, Class groups and local indecomposability for non-CM forms. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1103–1160DOI 10.4171/JEMS/1107