Definable sets, motives and 𝑝-adic integrals

Denef, Jan; Loeser, François

doi:10.1090/S0894-0347-00-00360-X

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6834(online) ISSN 0894-0347(print)

Definable sets, motives and -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
Posted: December 8, 2000
MathSciNet review: 1815218
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 -adic integrals are canonically interpolated by motivic ones.

1. James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 0229613 (37 #5187)
2. Sebastian del Baño Rollin and Vicente Navarro Aznar, On the motive of a quotient variety, Collect. Math. 49 (1998), no. 2-3, 203–226. Dedicated to the memory of Fernando Serrano. MR 1677089 (2000d:14009)
3. Dirk Bollaerts, On the Poincaré series associated to the 𝑝-adic points on a curve, Acta Arith. 51 (1988), no. 1, 9–30. MR 959783 (89i:11130)
4. Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107–135. MR 1162433 (94c:03049)
5. R. Cluckers, D. Haskell, The Grothendieck ring of the -adic numbers, preprint (5 pages).
6. Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258 (49 #5013)
7. J. Denef, The rationality of the Poincaré series associated to the 𝑝-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1–23. MR 751129 (86c:11043), http://dx.doi.org/10.1007/BF01389133
8. Jan Denef, 𝑝-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1986), 154–166. MR 850632 (88d:11030), http://dx.doi.org/10.1515/crll.1986.369.154
9. 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, pp. 25–47. MR 902824 (88j:11031)
10. J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), no. 6, 991–1008. MR 919001 (89d:11108), http://dx.doi.org/10.2307/2374583
11. Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505–537. MR 1618144 (99j:14021)
12. Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700 (99k:14002), http://dx.doi.org/10.1007/s002220050284
13. Jan Denef and François Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), no. 2, 285–309. MR 1708026 (2000k:14006), http://dx.doi.org/10.1215/S0012-7094-99-09910-6
14. J. Denef, F. Loeser, Motivic integration, quotient singularities and the McKay correspondence, preprint February 1999.
15. Herbert B. Enderton, A mathematical introduction to logic, Academic Press, New York, 1972. MR 0337470 (49 #2239)
16. Michael Fried, Dan Haran, and Moshe Jarden, Galois stratification over Frobenius fields, Adv. in Math. 51 (1984), no. 1, 1–35. MR 728998 (86c:12007), http://dx.doi.org/10.1016/0001-8708(84)90002-1
17. Michael D. Fried, Dan Haran, and Moshe Jarden, Effective counting of the points of definable sets over finite fields, Israel J. Math. 85 (1994), no. 1-3, 103–133. MR 1264342 (95k:12016), http://dx.doi.org/10.1007/BF02758639
18. Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860 (89b:12010)
19. M. Fried and G. Sacerdote, Solving Diophantine problems over all residue class fields of a number field and all finite fields, Ann. of Math. (2) 104 (1976), no. 2, 203–233. MR 0491477 (58 #10722)
20. H. Gillet and C. Soulé, Descent, motives and 𝐾-theory, J. Reine Angew. Math. 478 (1996), 127–176. MR 1409056 (98d:14012), http://dx.doi.org/10.1515/crll.1996.478.127
21. F. Guillén, V. Navarro Aznar, Un critère d'extension d'un foncteur défini sur les schémas lisses, preprint (1995), revised (1996).
22. Catarina Kiefe, Sets definable over finite fields: their zeta-functions, Trans. Amer. Math. Soc. 223 (1976), 45–59. MR 0422281 (54 #10272), http://dx.doi.org/10.1090/S0002-9947-1976-0422281-1
23. Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 0065218 (16,398d)
24. Angus Macintyre, Rationality of 𝑝-adic Poincaré series: uniformity in 𝑝, Ann. Pure Appl. Logic 49 (1990), no. 1, 31–74. MR 1076249 (92b:11085), http://dx.doi.org/10.1016/0168-0072(90)90050-C
25. Joseph Oesterlé, Réduction modulo 𝑝ⁿ des sous-ensembles analytiques fermés de 𝑍^{𝑁}_{𝑝}, Invent. Math. 66 (1982), no. 2, 325–341 (French). MR 656627 (83j:12014), http://dx.doi.org/10.1007/BF01389398
26. Johan Pas, Uniform 𝑝-adic cell decomposition and local zeta functions, J. Reine Angew. Math. 399 (1989), 137–172. MR 1004136 (91g:11142), http://dx.doi.org/10.1515/crll.1989.399.137
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. J. Scholl, Classical motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163–187. MR 1265529 (95b:11060)
29. Jean-Pierre Serre, Zeta and 𝐿 functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 82–92. MR 0194396 (33 #2606)
30. Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559 (83k:12011)
31. Willem Veys, Reduction modulo 𝑝ⁿ of 𝑝-adic subanalytic sets, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 3, 483–486. MR 1177996 (93i:11142), http://dx.doi.org/10.1017/S0305004100071152

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|