Functional equations for zeta functions of groups and rings

By Christopher Voll. 2006

97 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2bdf49597a73740f6c6d79b78552126bb62cc213

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In Discrete Mathematics and Combinatorics

1. Nir Avni
2. B. Klopsch
3. Uri Onn
4. Alexander Stasinski
5. Christopher Voll

In Geometry and Topology

1. Nir Avni
2. B. Klopsch
3. Uri Onn
4. Alexander Stasinski
5. Christopher Voll

In Mathematical Physics

1. Nir Avni
2. B. Klopsch
3. Uri Onn
4. Alexander Stasinski
5. Christopher Voll

Citations in the corpus

• Representation zeta functions of compact p-adic analytic groups and arithmetic groups (2010, methodology): “1 of [51] provide explicit formulae for the p-adic integrals ZW(o),I(s) under additional invariance and regularity conditions – specified just before [51, Theorem 2.”
• Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B (2011, background): “The expression of the local zeta function of ζ Zn (s) in terms of a sum, like the one defining the function Bn, illustrates a general approach to the study of local (subgroup and representation) zeta functions of T -groups developed in [43].”