The theory of Coleman power series for 𝐾₂

The purpose of this paper is to define “Coleman power series” associated to norm compatible systems in K 2 K_2 groups of complete discrete valuation fields of mixed characteristic ( 0 , p ) (0,p) with imperfect residue fields k {\mathsf {k}} such that [ k : k p ] = p [{\mathsf {k}}:{\mathsf {k}}^p]=p . These “Coleman power series” are elements of K 2 K_2 groups of certain power series rings. We use our “Coleman power series” to obtain some results on modular forms, and we also study properties of our “Coleman power series”.