Definable sets, motives and 𝑝-adic integrals

Denef, Jan¹ ; Loeser, François²

¹ Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
² Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France (UMR 8553 du CNRS)

Journal of the American Mathematical Society, Tome 14 (2001) no. 2, pp. 429-469

Résumé

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.

Détail

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

Bibliographie

[1] Ax, James The elementary theory of finite fields Ann. of Math. (2) 1968 239 271

[2] Del Baã±O Rollin, Sebastian, Navarro Aznar, Vicente On the motive of a quotient variety Collect. Math. 1998 203 226

[3] Bollaerts, Dirk On the Poincaré series associated to the 𝑝-adic points on a curve Acta Arith. 1988 9 30

[4] Chatzidakis, Zoã©, Van Den Dries, Lou, Macintyre, Angus Definable sets over finite fields J. Reine Angew. Math. 1992 107 135

[5] Deligne, Pierre La conjecture de Weil. I Inst. Hautes Études Sci. Publ. Math. 1974 273 307

[6] Denef, J. The rationality of the Poincaré series associated to the 𝑝-adic points on a variety Invent. Math. 1984 1 23

[7] Denef, Jan 𝑝-adic semi-algebraic sets and cell decomposition J. Reine Angew. Math. 1986 154 166

[8] Denef, J. On the evaluation of certain 𝑝-adic integrals 1985 25 47

[9] Denef, J. On the degree of Igusa’s local zeta function Amer. J. Math. 1987 991 1008

[10] Denef, Jan, Loeser, Franã§Ois Motivic Igusa zeta functions J. Algebraic Geom. 1998 505 537

[11] Denef, Jan, Loeser, Franã§Ois Germs of arcs on singular algebraic varieties and motivic integration Invent. Math. 1999 201 232

[12] Denef, Jan, Loeser, Franã§Ois Motivic exponential integrals and a motivic Thom-Sebastiani theorem Duke Math. J. 1999 285 309

[13] Enderton, Herbert B. A mathematical introduction to logic 1972

[14] Fried, Michael, Haran, Dan, Jarden, Moshe Galois stratification over Frobenius fields Adv. in Math. 1984 1 35

[15] Fried, Michael D., Haran, Dan, Jarden, Moshe Effective counting of the points of definable sets over finite fields Israel J. Math. 1994 103 133

[16] Fried, Michael D., Jarden, Moshe Field arithmetic 1986

[17] Fried, M., Sacerdote, G. Solving Diophantine problems over all residue class fields of a number field and all finite fields Ann. of Math. (2) 1976 203 233

[18] Gillet, H., Soulã©, C. Descent, motives and 𝐾-theory J. Reine Angew. Math. 1996 127 176

[19] Kiefe, Catarina Sets definable over finite fields: their zeta-functions Trans. Amer. Math. Soc. 1976 45 59

[20] Nakayama, Tadasi On Frobeniusean algebras. I Ann. of Math. (2) 1939 611 633

[21] Macintyre, Angus Rationality of 𝑝-adic Poincaré series: uniformity in 𝑝 Ann. Pure Appl. Logic 1990 31 74

[22] Oesterlã©, Joseph Réduction modulo 𝑝ⁿ des sous-ensembles analytiques fermés de 𝑍^{𝑁}_{𝑝} Invent. Math. 1982 325 341

[23] Pas, Johan Uniform 𝑝-adic cell decomposition and local zeta functions J. Reine Angew. Math. 1989 137 172

[24] Scholl, A. J. Classical motives 1994 163 187

[25] Serre, Jean-Pierre Zeta and 𝐿 functions 1965 82 92

[26] Serre, Jean-Pierre Quelques applications du théorème de densité de Chebotarev Inst. Hautes Études Sci. Publ. Math. 1981 323 401

[27] Veys, Willem Reduction modulo 𝑝ⁿ of 𝑝-adic subanalytic sets Math. Proc. Cambridge Philos. Soc. 1992 483 486