Pairings and functional equations over the GL2 -extension†

Abstract

We construct a pairing on the dual Selmer group over the GL 2-extension ℚ(E[p∞]) of an elliptic curve without complex multiplication and with good ordinary reduction at a prime p ⩾ 5 whenever it 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 representation to the case when the Artin representation factors through the quotient of Gal (ℚ(E[p∞])/ℚ) by its maximal pro-p normal subgroup. This gives a new proof of the parity conjecture whenever the elliptic curve E has a p-isogeny over the rationals.