Coleman power series for and -adic zeta functions of modular forms
For a usual local field of mixed characteristic , 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 -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 for certain class of local fields. The aim of this paper is following the analogy with the above classical case, to obtain -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 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 of modular curves.