A non-abelian analogue of Herbrand-Ribet

Following the natural instinct that when a group operates on a number field k, every term in the class number formula factorizes “compatibly” according to the representation theory (both complex and modular) of the group, we are led to some questions about the p-part of the class group of k.  The case when k is the cyclotomic extension Q(\mu_p) is the famous Herbrand-Ribet theorem.  We generalize these questions to k = Q(E[p]), where E[p] is the group of p-torsion points on an elliptic curve E over Q.  We answer these questions in a special case.