Zeta functions of groups and rings: Uniformity
Consider the following zeta functions (firstly studied by F. J. Grunewald, D. Segal and G. C. Smith) for finitely generated abstract or profinite groups \(G\): \(\zeta^*_G(s)=\sum_{H\in\chi^*(G)}|G:H|^{-s}\) where \(*\in\{\leq,\triangleleft,\wedge\}\) and \(\chi^\leq(G)=\{H\mid H\) is a subgroup of finite index in \(G\}\), \(\chi^\triangleleft(G)=\{H\mid H\) is a normal subgroup of finite index in \(G\}\), \(\chi^\wedge(G)=\{H\mid H\) is a subgroup of finite index in \(G\) and \(\widehat H\cong\widehat G\}\), here \(\widehat G\) denotes the profinite completion of \(G\). The ``local factors'' associated with these zeta functions for each prime \(p\) are given by: \(\zeta^*_{G,p}(s)=\sum_{H\in\chi^*_p(G)}|G:H|^{-s}\) where \(\chi^*_p(G)=\{H\in\chi^*(G)\mid H\) has \(p\)-power index in \(G\}\). For the class of finitely generated torsion-free nilpotent groups these zeta-functions can be decomposed as an Euler product of the local factors: \(\zeta^*_G(s)=\prod_p\zeta^*_{G,p}(s)\). In the paper under review contributions to the following questions are made: Question 1: For a fixed group \(G\), how does \(\zeta^*_{G,p}(s)\) behave as \(p\) varies over all primes \(p\)? Question 2: For a fixed prime \(p\), how does \(\zeta^*_{G,p}(s)\) behave as \(G\) varies over \(p\)-adic analytic groups of fixed dimension \(d\) (or nilpotent groups of a fixed Hirsch length \(d\))? Question 3: For a fixed finitely generated nilpotent group \(G\) do there exist finitely many rational functions \(W^*_1(Y,X),\dots,W^*_n(Y,X)\) of two variables over \(\mathbb{Q}\) such that for each prime \(p\) there is an \(i\) for which \(\zeta^*_p(s)=W^*_i(p,p^{-s})\)? Partial answers to questions 1 and 2 are given in the reviewed article. The proof uses the method of Denef and van den Dries for the calculation of \(p\)-adic integrals over semi-algebraic, respectively over sub-analytic sets and results of Macintyre, which allows the calculation of this \(p\)-adic integrals uniformly for all primes \(p\).
scientific article; zbMATH DE number 590933
scientific article