Raf Cluckers

Constructible motivic functions and motivic integration

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration. These results have been announced in math.AG/0403349 and math.AG/0403350. Main results and statements unchanged. Many minor slips corrected and some details added.

Fields with analytic structure

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principle, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b-minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, p-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988) 79–138], of [Cohen, Paul J. Decision procedures for real and p-adic fields. Comm. Pure Appl. Math. 22 (1969) 131–151], and of [Fresnel, van der Put, Rigid analytic geometry and its applications. Progress in Mathematics, 218 Birkhauser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.

Analytic p-adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized p-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions). the pieces being geometrically simple sets. called cells. We also classify silbanalytic sets up to subanalytic bijection.

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

Abstract We give a definition, in the ring language, of Z p inside Q p and of F p [ [ t ] ] inside F p ( ( t ) ) , which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z p inside Q p uniformly for all p. For any fixed finite extension of Q p , we give an existential formula and a universal formula in the ring language which define the valuation ring.

Fonctions constructibles et intégration motivique II

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration.

Integrability of oscillatory functions on local fields: transfer principles

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over Q n p implies integrability over F p ((t)) n for large p , and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.

Strictly convergent analytic structures

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on

On the computability of some positive-depth supercuspidal characters near the identity

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Stability under integration of sums of products of real globally subanalytic functions and their logarithms

-adic subanalytic sets, and we continue the study of non-archimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language

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

L_K