Generalised Euler characteristics of Selmer groups

Abstract

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 $F_{\infty }$ of 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 the rank of E(F) is positive. 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 .

23 pages, much updated and reorganised

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