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Functional equations for local normal zeta functions of nilpotent groups

Functional equations for local normal zeta functions of nilpotent groups
Functional equations for local normal zeta functions of nilpotent groups
We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.
zeta functions of groups, local functional equations, pfaffians
274-295
Voll, C.
df3a6745-2a8e-4e8f-82e2-f921e019f4ee
Beauville, A.
b9bfcd6b-a7c1-4f95-8137-dc66d0c99fad
Voll, C.
df3a6745-2a8e-4e8f-82e2-f921e019f4ee
Beauville, A.
b9bfcd6b-a7c1-4f95-8137-dc66d0c99fad

Voll, C. and Beauville, A. (2005) Functional equations for local normal zeta functions of nilpotent groups. Geometric And Functional Analysis, 15 (1), 274-295. (doi:10.1007/s00039-005-0506-y).

Record type: Article

Abstract

We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.

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More information

Published date: February 2005
Keywords: zeta functions of groups, local functional equations, pfaffians

Identifiers

Local EPrints ID: 42198
URI: http://eprints.soton.ac.uk/id/eprint/42198
PURE UUID: f18f6c00-55a4-41b3-b2cc-87c5e58a6e64

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Date deposited: 23 Nov 2006
Last modified: 15 Mar 2024 08:46

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Contributors

Author: C. Voll
Author: A. Beauville

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