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Stability results for local zeta functions of groups algebras, and modules

Published online by Cambridge University Press:  01 August 2017

TOBIAS ROSSMANN
TOBIAS ROSSMANN*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Posttach 100131, D- 33501 Bielefeld, Germany. e-mail: [email protected] Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
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Abstract

Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 435 - 444
Copyright
Copyright © Cambridge Philosophical Society 2017 

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