Pairings and functional equations over the $GL_2$-extension

In this talk we are going to construct a pairing on the dual Selmer group over the $GL_2$-extension $Q(E[p^{\infty}])$ of an elliptic curve without complex multiplication and with good ordinary reduction at $p$ whenever the dual Selmer satisfies certain--conjectured--torsion properties. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the $p$-adic $L$-function. As an application we reduce the parity conjecture for the $p$-Selmer rank and the analytic root number for the twists of elliptic curves with self-dual Artin representations to the case when the Artin representation factors through the (finite) quotient of $Gal(Q(E[p^{\infty}])/Q)$ by its maximal pro-$p$ normal subgroup. This gives a new proof of the parity conjecture whenever the elliptic curve has a $p$-isogeny over the rationals.