Duality of Selmer groups of an abelian variety over a number field

Abstract

Let $A$ be an abelian variety defined over a number field $K$ and let $A^{\vee}$ be the dual abelian variety. For an odd prime $p$, we consider two Selmer groups attached to $A[p]$ and relate the orders of these groups along with those of their corresponding duals to the order of the component groups of $A^{\vee}$ at primes $v$.