On a local to global principle in étale K-groups of curves

Abstract

Let X be a smooth, proper and geometrically irreducible curve X defined over a number field F and let X be a regular and proper model of X over OF;Sl : In this paper we address the problem of detecting the linear dependence over Zl of elements in the etale K-theory of X : To be more specific, let P 2Ket 2n.X / and let O ƒ K 2n.X / be a Zl -submodule. Let rv W K 2n.X /! K 2n.Xv/ be the reduction map for v ... Sl . We prove, under some conditions on X; that if rv. O P / 2 rv. O ƒ/ for almost all v of OF;Sl then O P 2 O ƒCKet 2n.X /tor :