A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups

A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points modp on the elliptic curvey 2=x 3−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform asp varies. A proof is given that counting points on the elliptic curvey 2=x 3−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms.