Associating curves of low genus to infinite nilpotent groups via the zeta function

Abstract

It is known from work of du Sautoy and Grunewald in [duSG1] that the zeta functions counting subgroups of finite index in infinite nilpotent groups depend upon the behaviour of some associated system of algebraic varieties on reduction modp. Further to this, in [duS1, duS2] du Sautoy constructed a group whose local zeta function was determined by the number of points on the elliptic curveE:Y ²=X ³−X. In this work we generalise du Sautoy’s construction to define a class of groups whose local zeta functions are dependent upon the number of points on the reduction of a given elliptic curve with a rational point. We also construct a class of groups that behave the same way in relation to any curve of genus 2 with a rational point. We end with a discussion of problems arising from this work.