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

Cornelius Griffin
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Abstract

It is known from work of du Sautoy and Grunewald in \cite{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 \cite{duS3, duS4} du Sautoy constructed a group whose local zeta function was determined by the number of points on the elliptic curve $E:Y^2=X^3-X.$ In this work we generalise du Sautoy\rq 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.

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Cornelius Griffin. 2002-09-20. Associating curves of low genus to infinite nilpotent groups via the zeta function. https://arxiv.org/abs/math/0209271

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