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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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