Pairings and functional equations over the GL_2-extension

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• Gergely Zabradi (Cambridge)
• Tuesday 22 April 2008, 14:30-15:30
• MR13.

If you have a question about this talk, please contact Tim Dokchitser.

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.

This talk is part of the Number Theory Seminar series.

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