Coleman Power Series for K 2 and p - Adic Zeta Functions of Modular Forms dedicated to Professor

For a usual local field of mixed characteristic (0, p), we have the theory of Coleman power series [Co]. By applying this theory to the norm compatible system of cyclotomic elements, we obtain the p-adic Riemann zeta function of Kubota-Leopoldt [KL]. This application is very important in cyclotomic Iwasawa theory. In [Fu1], the author defined and studied Coleman power series for K2 for certain class of local fields. The aim of this paper is following the analogy with the above classical case, to obtain p-adic zeta functions of various cusp forms (both in one variable attached to cusp forms, and in two variables attached to ordinary families of cusp forms) by Amice-Vélu, Vishik, Greenberg-Stevens, and Kitagawa,... by applying the K2 Coleman power series to the norm compatible system of Beilinson elements defined by Kato [Ka2] in the projective limit of K2 of modular curves. 2000 Mathematics Subject Classification: Primary 11F85; Secondary 11G55, 19F27