Uniform rationality of the Poincaré series of definable, analytic equivalence relations on local fields

Abstract

Poincaré series of p p -adic, definable equivalence relations have been studied in various cases since Igusa’s and Denef’s work related to counting solutions of polynomial equations modulo p n p^n for prime p p . General semi-algebraic equivalence relations on local fields have been studied uniformly in p p recently by Hrushovski, Martin and Rideau (2014). In this paper we generalize their rationality result to the analytic case, uniformly in p p , we build further on their appendix given by Cluckers as well as on work by van den Dries (1992), on work by Cluckers, Lipshitz and Robinson (2006). In particular, the results hold for large positive characteristic local fields. We also introduce rational motivic constructible functions and their motivic integrals, as a tool to prove our main results.