Class groups and local indecomposability for non-CM forms

TL;DR

This paper investigates the Galois representations associated with p-ordinary eigenforms, exploring their decomposability and connection to complex multiplication, using Galois deformation theory and class group properties.

Contribution

It provides a new criterion linking class group divisibility to the local indecomposability of Galois representations for p-ordinary eigenforms with CM.

Findings

Decomposability relates to class group p-indivisibility.

Proves the conjecture for eigenforms congruent to CM forms.

Connects local Galois representation properties with global class group data.

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.