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Geometric Methods in Algebra and Number Theory pp 109–137Cite as

Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration

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Part of the book series: Progress in Mathematics ((PM,volume 235))

Summary

We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs.

(UMR 8553 du CNRS)

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References

  1. R. Cluckers and F. Loeser —Constructible motivic functions and motivic integration”, in preparation.

    Google Scholar 

  2. —,Fonctions constructibles et intégration motivique I”, math. AG/0403349, to appear in C. R. Acad. Sci. Paris Sér. I Math.

    Google Scholar 

  3. —,Fonctions constructibles et intégration motivique II”, math. AG/0403350, to appear in C. R. Acad. Sci. Paris Sér. I Math.

    Google Scholar 

  4. R. Cluckers —Classification of semi-algebraic p-adic sets up to semi-algebraic bijection”, J. Reine Angew. Math.540 (2001), p. 105–114.

    MATH  MathSciNet  Google Scholar 

  5. P. J. Cohen —Decision procedures for real and p-adic fields”, Comm. Pure Appl. Math. 22 (1969), p. 131–151.

    MATH  MathSciNet  Google Scholar 

  6. C. Cunningham and T. Hales —Good orbital integrals”, math. RT/0311353.

    Google Scholar 

  7. F. Delon —Some p-adic model theory”, European women in mathematics (Trieste, 1997), Hindawi Publ. Corp., Stony Brook, NY, 1999, p. 63–76.

    Google Scholar 

  8. J. Denef —On the evaluation of certain p-adic integrals”, Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, p. 25–47.

    Google Scholar 

  9. —,p-adic semi-algebraic sets and cell decomposition”, J. Reine Angew. Math. 369 (1986), p. 154–166.

    Google Scholar 

  10. —,Arithmetic and geometric applications of quantifier elimination for valued fields”, Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge Univ. Press, Cambridge, 2000, p. 173–198.

    Google Scholar 

  11. J. Denef and F. Loeser —On some rational generating series occuring in arithmetic geometry”, math. NT/0212202.

    Google Scholar 

  12. —,Germs of arcs on singular algebraic varieties and motivic integration”, Invent. Math. 135 (1999), no. 1, p. 201–232.

    Google Scholar 

  13. —,Definable sets, motives and p-adic integrals”, J. Amer. Math. Soc. 14 (2001), no. 2, p. 429–469 (electronic).

    Google Scholar 

  14. —,Motivic integration and the Grothendieck group of pseudo-finite fields”, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, p. 13–23.

    Google Scholar 

  15. L. van den Dries —Dimension of definable sets, algebraic boundedness and Henselian fields”, Ann. Pure Appl. Logic45 (1989), no. 2, p. 189–209, Stability in model theory, II (Trento, 1987).

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Gillet and C. Soulé —Descent, motives and K-theory”, J. Reine Angew. Math.478 (1996), p. 127–176.

    MathSciNet  MATH  Google Scholar 

  17. F. Guillén and V. Navarro Aznar —Un critère d’extension des foncteurs définis sur les schémas lisses”, Publ. Math. Inst. Hautes Études Sci. (2002), no. 95, p. 1–91.

    Article  MATH  Google Scholar 

  18. G. Laumon and B. C. Ngô —Le lemme fondamental pour les groupes unitaires”, math. AG/0404454.

    Google Scholar 

  19. J. Nicaise —Relative motives and the theory of pseudo-finite fields”, math. AG/0403160.

    Google Scholar 

  20. J. Oesterlé —Réduction modulo pn des sous-ensembles analytiques fermés de ZpN”, Invent. Math.66 (1982), no. 2, p. 325–341.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Pas —Uniform p-adic cell decomposition and local zeta functions”, J. Reine Angew. Math.399 (1989), p. 137–172.

    MATH  MathSciNet  Google Scholar 

  22. P. Scowcroft and L. van den Dries —On the structure of semialgebraic sets over p-adic fields”, J. Symbolic Logic53 (1988), no. 4, p. 1138–1164.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.-P. Serre —Quelques applications du théorème de densité de Chebotarev”, Inst. Hautes Études Sci. Publ. Math. (1981), no. 54, p. 323–401.

    MathSciNet  Google Scholar 

  24. G. Terjanian —Un contre-exemple à une conjecture d’Artin”, C. R. Acad. Sci. Paris Sér. A–B262 (1966), p. A612.

    MathSciNet  Google Scholar 

  25. W. Veys —Reduction modulo pn of p-adic subanalytic sets”, Math. Proc. Cambridge Philos. Soc.112 (1992), no. 3, p. 483–486.

    Article  MATH  MathSciNet  Google Scholar 

  26. J.-L. Waldspurger —Endoscopie et changement de caractéristiques”, 2004, preprint.

    Google Scholar 

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Author information

Author notes

  1. Raf Cluckers

    Present address: Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

Authors and Affiliations

  1. Departement wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium

    Raf Cluckers

  2. Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    François Loeser

Authors
  1. Raf Cluckers
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  2. François Loeser
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Editor information

Editors and Affiliations

  1. Department of Mathematics Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Fedor Bogomolov

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Yuri Tschinkel

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© 2005 Birkhäuser Boston

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Cluckers, R., Loeser, F. (2005). Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_5

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