On reduction map for étale $K$-theory of curves

Abstract

In this paper we investigate reduction of nontorsion elements in theétale K-theory of a curve X over a global field F. We show that the reduction map can be well understood in terms of Galois cohomology of l-adic representations.Dedicated to Victor Snaith on the occasion of his 60-th birthday. IntroductionAssume that X is a smooth, proper and geometrically irreducible curve of genus g, defined over a global field F. Let l be an odd rational prime different from the characteristic of the field F. In the function field case we fix the set of places at infinity. Let n > 0 and let S l be the set of places of F which consists of places of bad reduction of X, places at infinity, and in the number field case, primes lying over l. Denote by X a smooth and proper model of X over the ring of S l -integers of[DF]. For places v ∈S l , we consider the reduction map:induced onétale K-theory by the injection X v → X of the special fiber at v. Theorem . Let J be the Jacobian variety of X and assume that End

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