Functional equations for zeta functions of groups and rings

• 成果类型: Article
• 署名作者: Voll, Christopher
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 1181-1218
• 关键词: nilpotent groups; counting subgroups; REPRESENTATIONS; RESOLUTION; index

摘要

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 J-) groups, and the normal zeta functions of J-groups of class 2. 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 J-groups, respectively.