Submission date: 27 October 2016

Abstract:

Poincaré 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 a power of prime p. General semi-algebraic equivalence relations on local fields have been studied uniformly in p recently, we generalize the rationality result to the analytic case, unifomly in p. 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.

Mathematics Subject Classification: 03C60; 03C10 03C98 11M41 20E07 20C15.

Keywords and phrases: Rationality of Poincaré series, motivic integration, uniform p-adic integration, constructible motivic functions, non-archimedean geometry, subanalytic sets, analytic structure, definable equivalence relations, zeta functions of groups.

Full text arXiv 1610.07952: pdf, ps.