Pontryagin duality for Iwasawa modules and abelian varieties

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\mathfrak{a}_n\}$ and $\{\mathfrak{b}_n\}$, endowed with an action of $\mathbb{Z}_p^d$ such that $\mathfrak{a}_n$ can be identified with the Pontryagin dual of $\mathfrak{b}_n$ for all $n$.

Let $K$ be a global field. Let $L$ be a $\mathbb{Z}_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.