Functional equations for local normal zeta functions of nilpotent groups

2003/05/26 by Christopher Voll, Voll, Christopher
Mathematics · #11M41 #20E07 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT #msc:11M41 #msc:20E07

paper · pdf · doi:10.48550/arxiv.math/0305362

17 pages. Appendix by Arnaud Beauville. Incorporated referee's suggestions

openalex publication_date 2003/05/26 · arxiv created 2004/02/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

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