Journal of the European Mathematical Society

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Published online first: 2021-06-08
DOI: 10.4171/JEMS/1107

Class groups and local indecomposability for non-CM forms

Francesc Castella[1], Carl Wang-Erickson[2] and Haruzo Hida[3]

(1) University of California Santa Barbara, USA
(2) University of Pittsburgh, UK
(3) UCLA, Los Angeles, USA

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.

Keywords: Ordinary modular forms, complex multiplication, Galois representations, anti-cyclotomic Iwasawa theory

Castella Francesc, Wang-Erickson Carl, Hida Haruzo: Class groups and local indecomposability for non-CM forms. J. Eur. Math. Soc. Electronically published on June 8, 2021. doi: 10.4171/JEMS/1107 (to appear in print)