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 for odd prime p, we study ideal class group ClK K as a Gal(K/ℚ)-module. More precisely, any j with 1⩽j⩽p-2, give condition that ClK⊗Fp has symmetric power SymjE[p] E[p] its quotient Gal(K/ℚ)-module, in terms Bloch-Kato’s Tate-Shafarevich SymjVpE. Here VpE denotes rational p-adic Tate module E. This partial generalization result Prasad and Shekhar case j=1.