Functional equations for zeta functions of groups and rings
Functional equations for zeta functions of groups and rings
We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively.
subgroup growth, representation growth, nilpotent groups, Igusa’s local zeta function, p-adic integration, local functional equations, Kirillov theory
1181-1218
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79
2010
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79
Abstract
We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively.
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Published date: 2010
Keywords:
subgroup growth, representation growth, nilpotent groups, Igusa’s local zeta function, p-adic integration, local functional equations, Kirillov theory
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Pure Mathematics
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Local EPrints ID: 56879
URI: http://eprints.soton.ac.uk/id/eprint/56879
PURE UUID: 62c6b17e-9c75-4276-8951-1bff1e4ca9ee
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Date deposited: 12 Aug 2008
Last modified: 15 Mar 2024 11:04
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Author:
Christopher Voll
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