# Publications of the Research Institute for Mathematical Sciences

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**Volume 46, Issue 1, 2010, pp. 115–169**

**DOI: 10.2977/PRIMS/3**

Adams Operations on Higher Arithmetic *K*-theory

^{[1]}(1) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

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.

*Keywords: *Higher Bott–Chern form, higher arithmetic *K*-group, arithmetic variety, Adams operation

Feliu Elisenda: Adams Operations on Higher Arithmetic *K*-theory. *Publ. Res. Inst. Math. Sci.* 46 (2010), 115-169. doi: 10.2977/PRIMS/3