Generalised Euler characteristics of Selmer groups

Let E be an elliptic curve defined over a number field F, and let p be a prime >= 5. In this paper we study the structure of the Selmer group of E over p-adic Lie extensions $F_{\mathrm{\infty }}$ of F. In particular, under certain global and local conditions on $F_{\mathrm{\infty }}$ we relate the generalised $Gal(F_\infty / F)$-Eulercharacteristicof$Sel(E / F_\infty)$ 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 the rank of E(F) is positive. Moreover, we show that the global and local conditions on $F_{\mathrm{\infty }}$ are satisfied for a large class of p-adic Lie extensions of F .