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

2016/10/25 by Kien Huu Nguyen, Nguyen, Kien Huu
Mathematics · #03C10 #03C60 #03C98 #11M41 #20C15 #20E07 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #math.LO #msc:03C10 #msc:03C60 #msc:03C98 #msc:11M41 #msc:20C15 #msc:20E07

paper · pdf · doi:10.48550/arxiv.1610.07952

arxiv created 2016/10/25 · openalex publication_date 2016/10/25 · arxiv updated 2016/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 pⁿ for prime p. General semi-algebraic equivalence relations on local fields have been studied uniformly in p recently in \cite16. Here we generalize the rationality result of \cite16 to the analytic case, unifomly in p, building further on the appendix of \cite16 and on \cite13b, \cite03. 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.

Related

• Uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
• Definable equivalence relations and zeta functions of groups
• Rationalite des series de Poincare et des fonctions Zeta motiviques
• The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational
• Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero