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

Abstract

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⩽j⩽p-2, we give a condition that ClK⊗Fp has the symmetric power SymjE[p] of E[p] as its quotient Gal(K/ℚ)-module, in terms of Bloch-Kato’s Tate-Shafarevich group of SymjVpE. 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.