Class groups and local indecomposability for non-CM forms

Abstract

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.

40 pages, with a 11-page appendix by Haruzo Hida. v3: improvements to exposition, minor corrections

References in corpus (3)

• Derived Galois deformation rings
• Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings
• Pseudo-modularity and Iwasawa theory

Cited by in corpus (2)

• Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism
• Critical Lambda-adic modular forms and bi-ordinary complexes