Generalised Euler characteristics of Selmer groups

Abstract

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∞/F. In particular, under certain global and local conditions on F∞ we relate the generalised Gal(F∞/F)-Euler characteristic of Sel(E/F∞) 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∞ are satisfied for a large class of p-adic Lie extensions of F.