Adams Operations on Higher Arithmetic $K$-theory

Abstract

We construct Adams operations on the rational higher arithmetic $K$-groups of a proper arithmetic variety. The definition applies to the higher arithmetic $K$-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic $K$-theory. The definition relies on the chain morphism representing Adams operations in higher algebraic $K$-theory given previously by the author. It is shown that this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang.