Tuesday 22 April 2008, MR13, 2:30pm
Pairings and functional equations over the GL_{2}extension
Gergely Zábrádi
(Cambridge)
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  conjectural
 torsion properties. This gives a functional equation of the
characteristic element which is compatible with the conjectural
functional equation of the padic Lfunction. As an
application we reduce the parity conjecture for the pSelmer
rank and the analytic root number for the twists of elliptic
curves with selfdual Artin representation to the case when the
Artin representation factors through the quotient of
Q(E[p^{∞}])/Q by its maximal prop normal subgroup.
This gives a proof of the parity
conjecture whenever the curve E has a pisogeny over the rationals.



