Adams Operations on Higher Arithmetic K-theory

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. In this paper it is shown that a slight modification of this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang. Introduction This paper contributes to the development of a higher arithmetic intersection theory following the steps of the higher algebraic intersection theory but suitable for arithmetic varieties. In [BF08], the author, together with Burgos, defined the higher arithmetic Chow ring for any arithmetic variety over a field, extending the construction given by Goncharov in [Gon05] which was valid only for proper arithmetic varieties. The question that arises is whether these groups are related to the higher arithmetic K-groups as given by Takeda or as suggested by Deligne and Soulé (see below). To this end, and inspired by the algebraic analogue, in this paper we endow the higher arithmetic K-groups of an arithmetic variety (tensored by Q) with a (pre)-λ-ring structure. Let X be an arithmetic variety over the ring of integers Z. In order to define the arithmetic Chern character on hermitian vector bundles, Gillet and Soulé have introduced in [GS90b] the arithmetic K0-group, denoted by K̂0(X). They endowed K̂0(X) with a pre-λ-ring structure, which was shown to be a λ-ring structure by Rössler in [Roe01]. This group fits in an exact sequence (*) K1(X) ρ −→ ⊕ p≥0 D2p−1(X, p)/ imdD → K̂0(X) → K0(X) → 0, with ρ the Beilinson regulator (up to a constant factor) and D∗(X, p) the Deligne complex of differential forms with p-twist computing Deligne-Beilinson cohomology with R coefficients and twisted by p, H∗ D(X,R(p)). Two different definitions for higher arithmetic K-theory have been proposed. Initially, it was suggested by Deligne and Soulé (see [Sou92, §III.2.3.4] and [Del87, Remark 5.4]) that these groups should fit in a long exact sequence · · · → Kn+1(X) ρ −→ H2p−n−1 D (X,R(p)) → K̂n(X) → Kn(X) → . . . , 2000 Mathematics Subject Classification. 14G40 (primary), 19E08 (secondary). Supported partially by the DGICYT BFM2003-02914. 1