Uniform rationality of the Poincar\'e series of definable, analytic equivalence relations on local fields

TL;DR

This paper proves the uniform rationality of Poincaré series for definable, analytic equivalence relations on local fields, extending previous results from semi-algebraic to analytic cases and introducing rational motivic constructible functions.

Contribution

It generalizes the rationality of Poincaré series to the analytic case uniformly in p, and introduces rational motivic constructible functions as a new tool.

Findings

Rationality of Poincaré series established for analytic equivalence relations.

Results hold uniformly for large positive characteristic local fields.

Introduction of rational motivic constructible functions for motivic integration.

Abstract

Poincar\'e series of $p$-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 $p^n$ for prime $p$. General semi-algebraic equivalence relations on local fields have been studied uniformly in $p$ recently in \cite{16}. Here we generalize the rationality result of \cite{16} to the analytic case, unifomly in $p$, building further on the appendix of \cite{16} and on \cite{13b}, \cite{03}. 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.