JournalsjemsVol. 20, No. 10pp. 2467–2537

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

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Abstract

We prove that the theory of the pp-adics Qp{\mathbb Q}_p admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp){\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each nn. We also prove that the elimination of imaginaries is uniform in pp. Using pp-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 pp) 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