# Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

### Ehud Hrushovski

Hebrew University of Jerusalem, Israel and University of Oxford, UK### Ben Martin

University of Aberdeen, UK### Silvain Rideau

Université Paris Diderot, France

## Abstract

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context.

As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.

## Cite this article

Ehud Hrushovski, Ben Martin, Silvain Rideau, Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers). J. Eur. Math. Soc. 20 (2018), no. 10, pp. 2467–2537

DOI 10.4171/JEMS/817