#### 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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