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Transactions of the American Mathematical Society Series B

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


Author: Kien Huu Nguyen
Journal: Trans. Amer. Math. Soc. Ser. B 6 (2019), 274-296
MSC (2010): Primary 03C60; Secondary 03C10, 03C98, 11M41, 20E07, 20C15
DOI: https://doi.org/10.1090/btran/23
Published electronically: October 24, 2019
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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 $ p^n$ for prime $ p$. General semi-algebraic equivalence relations on local fields have been studied uniformly in $ p$ recently by Hrushovski, Martin and Rideau (2014). In this paper we generalize their rationality result to the analytic case, uniformly in $ 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.


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Kien Huu Nguyen
Affiliation: Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France; and Department of Mathematics, Hanoi National University of Education, 136 XuanThuy street, Cau Giay, Hanoi, Vietnam
Address at time of publication: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Email: hkiensp@gmail.com; kien.nguyenhuu@kuleuven.be

DOI: https://doi.org/10.1090/btran/23
Keywords: Rationality of Poincar\'e series, motivic integration, uniform $p$-adic integration, constructible motivic functions, non-Archimedean geometry, subanalytic sets, analytic structure, definable equivalence relations, zeta functions of groups
Received by editor(s): October 25, 2016
Received by editor(s) in revised form: July 19, 2017
Published electronically: October 24, 2019
Additional Notes: The author was supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 61572 2 MOTMELSUM. He also acknowledges the support of the Labex CEMPI (ANR-11-LABX-0007-01).
Article copyright: © Copyright 2019 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)