Speaker: Carl Wang-Erickson (Imperial College)

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 Castell√†, 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.

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