The Syntomic Regulator for K–theory of Fields

We define complexes analogous to Goncharov’s complexes for the K–theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K–theory, there is a map from the cohomology of those complexes to the K–theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K–theory with the syntomic regulator. The result can be described in terms of a p–adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson’s cyclotomic elements. The result is again given by the p–adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.