Algebraic functional equations and completely faithful Selmer groups

Let $E$ be an elliptic curve—defined over a number field $K$—without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p\ge 5$. We construct a pairing on the dual $p^{\mathrm{\infty }}$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_{\mathrm{\infty }}\mathrm{/}K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\mathrm{Gal}(K_{\mathrm{\infty }}\mathrm{/}K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$.