Definable equivalence relations and zeta functions of groups

Ehud Hrushovski, Ben Martin, Silvain Rideau, Raf Cluckers

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13 Citations (Scopus)
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We prove that the theory of the p-adics Qp admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp) 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.
Original languageEnglish
Pages (from-to)2467-2537
Number of pages71
JournalJournal of the European Mathematical Society
Issue number10
Publication statusPublished - 18 Jul 2018


  • Elimination of imaginaries
  • invariant extensions of types
  • cell decompositions
  • rational zeta functions
  • subgroup zeta functions
  • representation zeta functions


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