Geometric structure of class two nilpotent groups and subgroup growth

TL;DR

This paper derives explicit formulas for the normal zeta functions of certain class two nilpotent groups, linking their subgroup growth to counting points on algebraic varieties and analyzing their analytic properties.

Contribution

It provides a new explicit expression for the zeta functions of these groups and explores their geometric and analytic properties, including functional equations.

Findings

Explicit formulas for local zeta functions

Dependence on counting rational points on varieties

Results on polynomial subgroup growth and poles

Abstract

In this paper we derive an explicit expression for the normal zeta function of class two nilpotent groups whose associated Pfaffian hypersurface is smooth. In particular, we show how the local zeta function depends on counting mod p rational points on related varieties, and we describe the varieties that can appear in such a decomposition. As a corollary, we also establish explicit results on the degree of polynomial subgroup growth in these groups, and we study the behaviour of poles of this zeta function. Under certain geometric conditions, we also confirm that these functions satisfy a functional equation.