International Congress of Mathematicians

22 J. Denef F. Loeseri > 1. One verifies that the operator N p can be applied to any element of Mi oc ,for p >• 0, yielding a rational number. The same holds for the Hodge-Delignepolynomial which now belongs to Q(«,w). By the method of section 7, we provedin [10] the followingTheorem 8.1. Let X be an algebraic variety over a field k of characteristiczero, let h be a definable subassignement of hc(x)tan d h n a definable family ofdefinable subassignements of hc(x) •(1) The motivic volume v(h) is contained in Mi oc -(2) The power series ^v(h n )T n £ -Mi 0C [[T]] is rational, with denominator a prodnuct of factors of the form 1 — L _a T 6 , with a, b £ N, 6 ^ 0.Let X be a reduced separable scheme of finite type over Z, and let A = (.4 P ) P>0be a definable family of subsets of X(Z P ), meaning that on each affine open, of asuitable finite affine covering of X, A p can be described by a DVR-formula overZ. (Here p runs over all large enough primes.) To A we associate in a canonicalway, its motivic volume V(ìIA) £ Mioc, in the following way: Let h f \ be a definablesubassignement of /i£(x®Q)> given by DVR-formulas that define A. Because theseformulas are not canonical, the subassignement h f \ is not canonical. But by theAx-Kochen-Ersov Principle (see 4.2), the set ìIA(K) is canonical for each pseudofinitefield K containing Q. Hence V(ìIA) £ Mioc is canonical, by Theorem 6.2.(1).By the method of section 7, we proved in [10] the following comparison result:Theorem 8.2. With the above notation, for all large enough primes p,N P (V(ìIA)) equals the measure of A p with respect to the canonical measure on X(Z P ).When X ® Q is nonsingular and of dimension d, the canonical measure onX(Z P ) is defined by requiring that each fiber of the map X(Z P ) —t X(Z p /p m ) hasmeasure p^md whenever m >• 0. For the definition of the canonical measure in thegeneral case, we refer to [25].The above theorem easily generalizes to integrals instead of measures, but thisyields little more because quite general p-adic integrals (such as the orbital integralsappearing in the Langlands program) can be written as measures of the definablesets we consider. For example the p-adic integral J \f(x)\dx on Z p equals the p-adicmeasure of {(x, t) £ Z d+1 : oid p (f(xj) < ord p (t)}.References[1] J. Ax, The elementary theory of finite fields, Ann. of Math, 88 (1968), 239-271.[2] S. del Bario Rollin, V. Navarro Aznar, On the motive of a quotient variety,Collect. Math., 49 (1998), 203-226.[3] V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonicalsingularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997),1-32, World Sci. Publishing, River Edge, NJ, 1998.[4] V, Batyrev, Non-Archimedean integrals and stringy Euler numbers of logterminalpairs, J. Eur. Math. Soc. (JEMS), 1 (1999), 5-33.[5] Z. Chatzidakis, L. van den Dries, A. Macintyre, Definable sets over finite fields,J. Reine Angew. Math., 427 (1992), 107-135.