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Proceedings of the London Mathematical Society Advance Access originally published online on November 19, 2008
Proceedings of the London Mathematical Society 2009 98(3):775-796; doi:10.1112/plms/pdn049
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© 2008 London Mathematical Society

Generalised Euler characteristics of Selmer groups

Sarah Livia Zerbes

Department of Mathematics
Huxely Building
Imperial College London
London
SW7 2AZ
United Kingdom
Current address:
Department of Mathematics
Harrison Building
University of Exeter
Exeter
EX4 4QF
United Kingdom
s.zerbes@exeter.ac.uk

Received 5 September 2007. Revision received 6 May 2008.

Let E be an elliptic curve defined over a number field F, and let p ≥ 5 be a prime. In this paper, we study the structure of the Selmer group of E, over p-adic Lie extensions F{infty}/F. In particular, under certain global and local conditions on F{infty} we relate the generalised Gal(F{infty}/F)-Euler characteristic of Sel(E/F{infty}) to the generalised Euler characteristic of the Selmer group over the cyclotomic Zp-extension of F. This invariant generalises the classical Euler characteristic to the case when rankZE(F) > 0. Moreover, we show that the global and local conditions on F{infty} are satisfied for a large class of p-adic Lie extensions of F.

2000 Mathematics Subject Classification 11R23 (primary), 11R34 (secondary).


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