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 Cl K of K as a Gal( K/ Q ) -module. More precisely, for any j with 1 6 j 6 p − 2 , we give a condition that Cl K ⊗ F p has the symmetric power Sym j E [ p ] of E [ p ] as its quotient Gal( K/ Q ) -module, in terms of Bloch-Kato’s Tate-Shafarevich group of Sym j V p E . Here V p E 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 .