Number Theory Seminar

Tuesday 22 April 2008, MR13, 2:30pm

Pairings and functional equations over the GL2-extension
Gergely Zábrádi (Cambridge)


     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.

Last modified on 10/4/2008.

Comments/corrections to Tim Dokchitser