Let f be a classical p-ordinary cuspidal eigenform of weight at least two. Ralph Greenberg asked for a characterization of those with complex multiplication.

The following characterization has been proposed, with supporting evidence provided by Ghate and Vatsal: the associated 2-dimensional p-adic representation of the absolute Galois group of is split (that is, isomorphic to a sum of two characters) after restriction to a decomposition group at p. In joint work with Francesc Castellà, we give additional evidence supporting this answer to Greenberg’s question. The strength of our conclusions is proportional to the smallness of certain anti-cyclotomic Iwasawa-theoretic ideal class groups.