Number Theory Seminar

It is well-understood that p-ordinary Hecke eigenforms give rise to global 2-dimensional Galois representations which become reducible with an unramified quotient after restriction to a decomposition group at p. Coleman and Greenberg independently asked for a characterization of those p-ordinary forms whose associated Galois representation is also semi-simple after restriction to this decomposition group. It has been suspected that all such p-ordinary forms have complex multiplication, a restrictive global property. We will discuss joint work with Francesc Castella, in which we give a criterion for this suspicion to be true, and give a construction of "bi-ordinary" modular forms as a tool to explore the case that the criterion fails.