DOCUMENTA MATHEMATICA, Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 387-442

Takako Fukaya

Coleman Power Series for $K_2$ and $p$-Adic Zeta Functions of Modular Forms

For a usual local field of mixed characteristic $(0,p)$, we have the theory of Coleman power series \cite{Co}. By applying this theory to the norm compatible system of cyclotomic elements, we obtain the $p$-adic Riemann zeta function of Kubota-Leopoldt \cite{KL}. This application is very important in cyclotomic Iwasawa theory.\par In \cite{Fu}, 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 Kato \cite{Ka7} in the projective limit of $K_2$ of modular curves.

2000 Mathematics Subject Classification: Primary 11F85; Secondary 11G55, 19F27

Keywords and Phrases: $p$-adic zeta function, modular form, algebraic $K$-theory, Euler system.

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