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.
For a nonzero prime ideal @math in a number ring @math , the @math -ideal zeta functions @math may not be confused with the @math -ideal zeta functions of @math , considered as @math -Lie algebras. For @math , i.e. in the case of the Heisenberg Lie ring @math ; cf. Example ), the latter have been computed for primes @math which are unramified in @math in @cite_6 and for primes @math which are nonsplit in @math in @cite_1 . In the forthcoming paper @cite_7 we generalize these computations to cover the ideal zeta functions of algebras arising from a large class of Lie rings, including the Grenham Lie rings @math , via base extensions with various compact discrete valuation rings. The paper gives a survey of applications of Igusa functions in the area of zeta functions of groups and rings, and is built on a generalization of this class of functions.
@cite_1 @cite_7 @cite_6
2054570374 1555308356
We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sln(Qp). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions. We report on progress and problems concerning the analytical behaviour of the zeta functions of groups and rings. We also describe how these generating functions are special cases of adelic cone integrals for which our results hold.