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The theory of Coleman power series for $$K_2$$. (English) Zbl 1053.11088
Let $$H$$ be a complete discrete valuation field of mixed characteristic $$(0,p)$$ with imperfect residue field $$k$$ such that $$[k: k^p]=p$$, $$p$$ being a prime number. The author establishes the $$K_2$$-analogue of such a local field $$H$$ of the Coleman isomorphism associating a power series to a norm compatible system of units in the tower of usual $$p$$-adic local fields. Namely, she associates an element of the $$K_2$$-group of a certain power series ring to a norm compatible system of elements of the $$K_2$$-groups in a certain field-tower over $$H$$. As an important generalization of the theories of Iwasawa (cyclotomic case) and Coates-Wiles (elliptic case), she studies the $$K_2$$-power series associated to a norm compatible system constructed by K. Kato using the Beilinson elements when $$H$$ is the $$p$$-adic completion of the function field of a modular curve. The author also gives the relation between her $$K_2$$-Coleman isomorphism and Parschin-Kato’s two-dimensional local class field theory.

##### MSC:
 11S31 Class field theory; $$p$$-adic formal groups 11G55 Polylogarithms and relations with $$K$$-theory 11F11 Holomorphic modular forms of integral weight 19C99 Steinberg groups and $$K_2$$
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