## Definable sets, motives and $$p$$-adic integrals.(English)Zbl 1040.14010

In the paper under review, the authors considerably generalize several of their previously established methods and results in arithmetic geometry, and that in various directions. Namely, in a previous paper [J. Denef and F. Loeser, Invent. Math. 135, 201–232 (1999; Zbl 0928.14004)], they had developed a general theory of integration – called motivic integration – on the space of arcs of an algebraic variety $$X$$ over a field $$k$$ of characteristic zero. This theory, which they now refer to as “geometric” motivic integration, is here replaced by a suitable arithmetic analogue which appears to be much better adapted to the study of certain rationality questions concerning Poincaré series of arithmetic schemes. More precisely, the authors develop a different kind of motivic integration theory, the so-called “arithmetic” motivic integration, which is largely based on methods from general field arithmetic and first-order model theory in mathematical logic. On the other hand, the Poincaré series $$P_p(T)$$ associated to any pair $$(X,p)$$,where $$X$$ is a reduced and separated scheme of finite type over $$\mathbb{Z}$$ and $$p$$ is a prime number, is known to be a rational function. This fact was proved by the first author [J. Denef, Invent. Math. 77, 1–23 (1984; Zbl 0537.12011)]. Then, just a little later, J. Denef, J. Pas, and A. Macintyre proved, independently, that the degrees of the nominator and denominator of the rational function $$P_p(T)$$ are bounded independently of the prime number $$p$$. In view of these deep results, the basic task of the paper under review was to derive a much stronger uniformity result by constructing a canonical rational function $$P_{\text{ar}}(T)$$ which specializes to $$P_p(T)$$ for almost all prime numbers $$p$$. In fact, it is precisely to this end that the authors develop their new “arithmetic” motivic integration theory, in the first part of the paper, which then turns out to be the right tool for constructing the universal rational function $$P_{\text{ar}}(T)$$ by means of the Grothendieck ring of certain Chow motives. The paper consists of ten sections whose contents are as follows:
After a first section devoted to preliminaries on Grothendieck rings of Chow motives, the authors develop in section 2 what is needed, in the sequel, from the theory of Galois stratifications in field arithmetic. This and the construction of virtual motives from Galois stratifications associated to first-order logic formulas, as carried out in section 3, relies on the fundamental work of M. D. Fried and M. Jarden [“Field arithmetic”, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 11, Berlin (1986; Zbl 0625.12001)]. Here the authors also introduce a new class of objects, the so-called “definable subassignments”, which are more geometric in nature than first-order Galois formulas. Section 4 discusses definable subassignments for rings, and section 5 does the same for power series rings. These constructions are then used to define a measure on the set of definable subassignments of the space of germs of arcs of an arithmetic scheme $$X$$ and this leads to the basic theory of arithmetic motivic integration developed in section 6. Applications to general rationality results for power series associated to algebraic varieties, in the spirit of the earlier works of J. Denef and F. Loeser mentioned above, are presented in section 7. The general results obtained here strikingly demonstrate the enormous power of the authors’ new arithmetic motivic integration theory, and this insight is even strengthened by the results exhibited in the remaining three sections. Namely, in section 8, the authors show that arithmetic motivic integration indeed specializes to $$p$$-adic integration.
Finally, the arithmetical Poincaré series $$P_{\text{ar}}(T)$$ for an algebraic variety $$X$$ over a field of characteristic zero is established in section 9 where it is also proved that $$P_{\text{ar}}(T)$$ factually specializes to the usual $$p$$-adic Poincaré series when $$k$$ is a number field.
As the authors point out, the arithmetical Poincaré series seems to contain much more information about the underlying variety $$X$$ than the authors’ formerly studied geometric counterpart $$P_{\text{geom}}(T)$$ defined by geometric motivic integration. This observation is demonstrated by means of a concrete example discussed in the concluding section 10. Actually, the authors compute the two different Poincaré series $$P_{\text{ar}}(T)$$ and $$P_{\text{geom}}(T)$$ for branches of plane algebraic curves, thereby showing that the poles of $$P_{\text{ar}}(T)$$ completely determine the Puiseux pairs of the branch, whereas the series $$P_{\text{geom}}(T)$$ only encodes the multiplicity at the origin.
All in all, the significance of the methods and results exhibited in this important paper can barely be overestimated, since these achievements unquestionably signify a major step forward in the theory of motivic integration within arithmetic geometry. Moreover, the various contributions of the two authors in this realm, over many years and in their entirety, must be seen as being pioneering and epoch-making in the field.

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G20 Local ground fields in algebraic geometry 03C10 Quantifier elimination, model completeness, and related topics 03C98 Applications of model theory 03C68 Other classical first-order model theory 03C95 Abstract model theory 12E30 Field arithmetic 12L12 Model theory of fields 14G15 Finite ground fields in algebraic geometry 14G27 Other nonalgebraically closed ground fields in algebraic geometry 11G25 Varieties over finite and local fields 11S40 Zeta functions and $$L$$-functions 12L10 Ultraproducts and field theory 14F20 Étale and other Grothendieck topologies and (co)homologies 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 14F42 Motivic cohomology; motivic homotopy theory

### Citations:

Zbl 0928.14004; Zbl 0537.12011; Zbl 0625.12001
Full Text:

### References:

  James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239 – 271. · Zbl 0195.05701  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. · Zbl 0929.14033  Dirk Bollaerts, On the Poincaré series associated to the \?-adic points on a curve, Acta Arith. 51 (1988), no. 1, 9 – 30. · Zbl 0608.12021  Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107 – 135. · Zbl 0759.11045  R. Cluckers, D. Haskell, The Grothendieck ring of the $$p$$-adic numbers, preprint (5 pages). · Zbl 0988.03058  Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273 – 307 (French). · Zbl 0287.14001  J. Denef, The rationality of the Poincaré series associated to the \?-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1 – 23. · Zbl 0537.12011  Jan Denef, \?-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1986), 154 – 166. · Zbl 0584.12015  J. Denef, On the evaluation of certain \?-adic integrals, Séminaire de théorie des nombres, Paris 1983 – 84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, pp. 25 – 47.  J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), no. 6, 991 – 1008. · Zbl 0659.14017  Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505 – 537. · Zbl 0943.14010  Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201 – 232. · Zbl 0928.14004  Jan Denef and François Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), no. 2, 285 – 309. · Zbl 0966.14015  J. Denef, F. Loeser, Motivic integration, quotient singularities and the McKay correspondence, preprint February 1999. · Zbl 1080.14001  Herbert B. Enderton, A mathematical introduction to logic, Academic Press, New York-London, 1972. · Zbl 0298.02002  Michael Fried, Dan Haran, and Moshe Jarden, Galois stratification over Frobenius fields, Adv. in Math. 51 (1984), no. 1, 1 – 35. · Zbl 0554.12016  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. · Zbl 0826.11027  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. · Zbl 0625.12001  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. · Zbl 0376.02042  H. Gillet and C. Soulé, Descent, motives and \?-theory, J. Reine Angew. Math. 478 (1996), 127 – 176. · Zbl 0863.19002  F. Guillén, V. Navarro Aznar, Un critère d’extension d’un foncteur défini sur les schémas lisses, preprint (1995), revised (1996).  Catarina Kiefe, Sets definable over finite fields: their zeta-functions, Trans. Amer. Math. Soc. 223 (1976), 45 – 59. · Zbl 0372.02032  S. Lang, A. Weil, Number of points of varieties in finite fields, Amer. J. Math., 76 (1954), 819-827. · Zbl 0058.27202  Angus Macintyre, Rationality of \?-adic Poincaré series: uniformity in \?, Ann. Pure Appl. Logic 49 (1990), no. 1, 31 – 74. · Zbl 0731.12015  Joseph Oesterlé, Réduction modulo \?$$^{n}$$ des sous-ensembles analytiques fermés de \?^{\?}_{\?}, Invent. Math. 66 (1982), no. 2, 325 – 341 (French). · Zbl 0473.12015  Johan Pas, Uniform \?-adic cell decomposition and local zeta functions, J. Reine Angew. Math. 399 (1989), 137 – 172. · Zbl 0666.12014  M. Presburger, Uber die Vollständigkeit eines gewissen Systems des Arithmetik ..., Comptes-rendus du I Congrès des Mathématiciens des Pays Slaves, Warsaw (1929), 92-101.  A. J. Scholl, Classical motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163 – 187. · Zbl 0814.14001  Jean-Pierre Serre, Zeta and \? functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 82 – 92. · Zbl 0171.19602  Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323 – 401 (French). · Zbl 0496.12011  Willem Veys, Reduction modulo \?$$^{n}$$ of \?-adic subanalytic sets, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 3, 483 – 486. · Zbl 0821.32004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.