Concentration theorem and relative fixed point formula of Lefschetz type in Arakelov geometry

In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. K\"{o}hler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.