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Definable sets, motives and $p$-adic integrals


Authors: Jan Denef and François Loeser
Journal: J. Amer. Math. Soc. 14 (2001), 429-469
MSC (2000): Primary 03C10, 03C98, 12E30, 12L12, 14G15, 14G20, 14G27; Secondary 11G25, 11S40, 12L10, 14F20, 14G05, 14G10, 14J20
DOI: https://doi.org/10.1090/S0894-0347-00-00360-X
Published electronically: December 8, 2000
MathSciNet review: 1815218
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Abstract: We associate a canonical virtual motive to definable sets over a field of characteristic zero. We use this construction to show that very general $p$-adic integrals are canonically interpolated by motivic ones.


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  • 1. J. Ax, The elementary theory of finite fields, Ann. of Math., 88 (1968), 239-271. MR 37:5187
  • 2. S. del Baño Rollin, V. Navarro Aznar, On the motive of a quotient variety, Collect. Math., 49 (1998), 203-226. MR 2000d:14009
  • 3. D. Bollaerts, On the Poincaré series associated to the $p$-adic points on a curve, Acta Arith., 51 (1988), 9-30. MR 89i:11130
  • 4. Z. Chatzidakis, L. van den Dries, A. Macintyre, Definable sets over finite fields, J. Reine Angew. Math., 427 (1992), 107-135. MR 94c:03049
  • 5. R. Cluckers, D. Haskell, The Grothendieck ring of the $p$-adic numbers, preprint (5 pages).
  • 6. P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 273-307. MR 49:5013
  • 7. J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math., 77 (1984), 1-23. MR 86c:11043
  • 8. J. Denef, $p$-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math., 369 (1986), 154-166. MR 88d:11030
  • 9. J. Denef, On the evaluation of certain $p$-adic integrals, Séminaire de théorie des nombres, Paris 1983-84, 25-47, Progr. Math., 59, Birkhäuser Boston, Boston, Mass., 1985. MR 88j:11031
  • 10. J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math., 109 (1987), 991-1008. MR 89d:11108
  • 11. J. Denef, F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom., 7 (1998), 505-537. MR 99j:14021
  • 12. J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., 135 (1999), 201-232. MR 99k:14002
  • 13. J. Denef, F. Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani Theorem, Duke Math. J., 99 (1999), 285-309. MR 2000k:14006
  • 14. J. Denef, F. Loeser, Motivic integration, quotient singularities and the McKay correspondence, preprint February 1999.
  • 15. H. Enderton, A mathematical introduction to logic, Academic Press (1972), New York-London. MR 49:2239
  • 16. M. Fried, D. Haran, M. Jarden, Galois stratifications over Frobenius fields, Adv. in Math., 51 (1984), 1-35. MR 86c:12007
  • 17. M. Fried, D. Haran, M. Jarden, Effective counting of the points of definable sets over finite fields, Israel J. Math., 85 (1994), 103-133. MR 95k:12016
  • 18. M. Fried, M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge Band 11 (1986), Springer-Verlag, Berlin, Heidelberg. MR 89b:12010
  • 19. M. Fried, G. Sacerdote, Solving diophantine problems over all residue class fields of a number field and all finite fields, Ann. of Math., 100 (1976), 203-233. MR 58:10722
  • 20. H. Gillet, C. Soulé, Descent, motives and $K$-theory, J. Reine Angew. Math., 478 (1996), 127-176. MR 98d:14012
  • 21. F. Guillén, V. Navarro Aznar, Un critère d'extension d'un foncteur défini sur les schémas lisses, preprint (1995), revised (1996).
  • 22. C. Kiefe, Sets definable over finite fields, their zeta-functions, Trans. Amer. Math. Soc., 223 (1976), 45-59. MR 54:10272
  • 23. S. Lang, A. Weil, Number of points of varieties in finite fields, Amer. J. Math., 76 (1954), 819-827. MR 16:398d
  • 24. A. Macintyre, Rationality of $p$-adic Poincaré series: Uniformity in $p$, Ann. Pure Appl. Logic, 49 (1990), 31-74. MR 92b:11085
  • 25. J. Oesterlé, Réduction modulo $p^{n}$ des sous-ensembles analytiques fermés de ${{\mathbf Z}}^{N}_{p}$, Invent. Math., 66 (1982), 325-341. MR 83j:12014
  • 26. J. Pas, Uniform $p$-adic cell decomposition and local zeta functions, J. Reine Angew. Math., 399 (1989), 137-172. MR 91g:11142
  • 27. M. Presburger, Uber die Vollständigkeit eines gewissen Systems des Arithmetik ..., Comptes-rendus du I Congrès des Mathématiciens des Pays Slaves, Warsaw (1929), 92-101.
  • 28. A. Scholl, Classical motives, in Motives, U. Jannsen, S. Kleiman, J.-P. Serre, Eds., Proceedings of Symposia in Pure Mathematics, Volume 55 Part 1 (1994), 163-187. MR 95b:11060
  • 29. J.-P. Serre, Zeta and $L$ functions, Arithmetical Algebraic Geometry, Proc. Conf. Purdue Univ., Harper & Row, New York (1965), 82-92. MR 33:2606
  • 30. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math., 54 (1981), 323-401. MR 83k:12011
  • 31. W. Veys, Reduction modulo $p^{n}$ of $p$-adic subanalytic sets, Math. Proc. Cambridge Philos. Soc., 112 (1992), 483-486. MR 93i:11142

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Additional Information

Jan Denef
Affiliation: Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Email: Jan.Denef@wis.kuleuven.ac.be

François Loeser
Affiliation: Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France (UMR 8553 du CNRS)
Email: Francois.Loeser@ens.fr

DOI: https://doi.org/10.1090/S0894-0347-00-00360-X
Received by editor(s): October 6, 1999
Received by editor(s) in revised form: October 20, 2000
Published electronically: December 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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