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Zeta function of representations of compact $p$-adic analytic groups

Author(s): A. Jaikin-Zapirain
Journal: J. Amer. Math. Soc. 19 (2006), 91-118.
MSC (2000): Primary 20E18; Secondary 20C15, 20G25, 22E35
Posted: September 7, 2005
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Abstract: Let $G$ be an FAb compact $p$-adic analytic group and suppose that $p>2$ or $p=2$ and $G$ is uniform. We prove that there are natural numbers $n_1, \ldots, n_k$ and functions $f_1(p^{-s}),\ldots, f_k(p^{-s})$ rational in $p^{-s}$ such that

\begin{displaymath}\zeta^G(s)=\sum_{\lambda\in \operatorname{Irr}(G)} \lambda(1) ^{-s}=\sum_{i=1}^kn_i^{-s}f_i(p^{-s}).\end{displaymath}

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Additional Information:

A. Jaikin-Zapirain
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: andrei.jaikin@uam.es

DOI: 10.1090/S0894-0347-05-00501-1
PII: S 0894-0347(05)00501-1
Keywords: Profinite groups, zeta functions, representations
Received by editor(s): June 2, 2004
Posted: September 7, 2005
Additional Notes: This work has been supported by the FEDER, MEC Grant MTM2004-04665, and the Ramón y Cajal Program
Copyright of article: Copyright 2005, American Mathematical Society

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