Enumerating finite class-2-nilpotent groups on 2 generators

We compute the numbers g(n,2,2) of nilpotent groups of order n, of class atmost 2 generated by at most 2 generators, by giving an explicit formula for theDirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}

Ideal zeta functions associated to a family of class-2-nilpotent Lie rings

We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in [1], 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.Comment: 16 pages, minor revisions, including referee's suggestions. To appear in the Quarterly Journal of Mathematic

Igusa-type functions associated to finite formed spaces and their functional equations

We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations. Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type $A$. While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.Comment: Slightly revised version, to appear in Trans. Amer. Math. Soc

Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.Comment: 19 pages; to appear in Israel J. Mat