Ideal Class Groups of Number Fields and Bloch-Kato?s Tate-Shafarevich Groups for Symmetric Powers of Elliptic Curves

For an elliptic curve E over Q, putting K = Q(E[p]) which is the p-th division field of E for an odd prime p, we study the ideal class group ClK of K as a Gal(K/Q)-module. More precisely, for any j with 1 5 j 5 p - 2, we give a condition that ClK circle times Fp has the symmetric power Symj E[p] of E[p] as its quotient Gal(K/Q)-module, in terms of Bloch-Kato's Tate-Shafarevich group of Symj VpE. Here VpE denotes the rational p-adic Tate module of E. This is a partial generalization of a result of Prasad and Shekhar for the case j = 1.