Tuesday 22 April 2008, MR13, 2:30pm
Pairings and functional equations over the GL2-extension
We construct a pairing on the dual Selmer group over the
GL2-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 - conjectural
- 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
Q(E[p∞])/Q by its maximal pro-p normal subgroup.
This gives a proof of the parity
conjecture whenever the curve E has a p-isogeny over the rationals.