Coleman power series for $K_2$ and $p$-adic zeta functions of modular forms

Abstract

For a usual local field of mixed characteristic $(0,p)$, we have the theory of Coleman power series [R. Coleman, Invent. Math. 53, 91--116 (1979; Zbl 0429.12010)]. By applying this theory to the norm compatible system of cyclotomic elements, we obtain the $p$-adic Riemann zeta function of T. Kubota and H. W. Leopoldt [J. Reine Angew. Math. 214--215, 328--339 (1964; Zbl 0186.09103)]. This application is very important in cyclotomic Iwasawa theory. In [RIMS Kokyuroku 1200, 48--59 (2001; Zbl 0985.11506)], the author defined and studied Coleman power series for $K_2$ 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 $K_2$ Coleman power series to the norm compatible system of Beilinson elements defined by K. Kato [in: Arithmetic algebraic geometry. Lect. Notes Math. 1553, 50--163 (1993; Zbl 0815.11051)] in the projective limit of $K_2$ of modular curves.