We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class- @math -nilpotent Lie rings, introduced in [8], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.
The paper @cite_10 , which introduced the groups @math , computes their @math , enumerating the subgroups of finite index in @math whose profinite completions are isomorphic to that of @math . By general principles, these zeta functions also satisfy Euler product decompositions indexed by the rational primes, whose factors are rational functions in @math . In the notation of the current paper, [Theorem 1.4] BermanKlopschOnn 18 establishes that the Euler factor @math of @math at a rational prime @math , enumerating the relevant subgroups of @math of @math -power index, is of the form @math for explicitly given numerical data'' @math , for integers @math , @math , comparable to (but different from) those given in .
@cite_10
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The pro-isomorphic zeta function ( ^ _ (s) ) of a finitely generated nilpotent group ( ) is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of ( ). Such zeta functions can be expressed as Euler products of p-adic integrals over the ( Q _p )-points of an algebraic automorphism group associated to ( ). In this way they are closely related to classical zeta functions of algebraic groups over local fields.