JournalsjemsVol. 24, No. 4pp. 1103–1160

Class groups and local indecomposability for non-CM forms

  • Francesc Castella

    University of California Santa Barbara, USA
  • Carl Wang-Erickson

    University of Pittsburgh, UK
  • Haruzo Hida

    UCLA, Los Angeles, USA
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Abstract

In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those pp-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at pp. It is expected that such pp-ordinary eigenforms are precisely those with complex multiplication.

In this paper, we study Coleman–Greenberg’s question using Galois deformation theory. In particular, for pp-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the pp-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. 1103–1160

DOI 10.4171/JEMS/1107