Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Self-duality of Selmer groups

TIM DOKCHITSERa1 and VLADIMIR DOKCHITSERa2

a1 Robinson College, Cambridge CB39AN. e-mail: t.dokchitser@dpmms.cam.ac.uk

a2 Gonville & Caius College, Cambridge CB21TA. e-mail: v.dokchitser@dpmms.cam.ac.uk

Abstract

The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the $\Q_p$G-representation naturally associated to the p-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

(Received May 11 2008)

Footnotes

† Supported by a Royal Society University Research Fellowship.