Raf Cluckers | Université des Sciences et Technologies de Lille (Lille-1) (original) (raw)

Papers by Raf Cluckers

We extend the formalism and results on motivic integration from ["Constructible motivic func... more We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using

decreases as r goes to 0, pointing out, long before this concept has been formalized, the tame be... more decreases as r goes to 0, pointing out, long before this concept has been formalized, the tame behaviour of the local normalized volume of analytic sets. Ten years after Lelong's pioneering paper, Thie proved in (36) that the local density of a complex analytic subset X at a point a is a positive integer by expressing it as a sum

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a ppp-adic f... more We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a ppp-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index

The purpose of this paper is to explain how the identities of various fundamental lemmas fall wit... more The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one collection of fields to others. In particular, once the fundamental lemma has been established for one collection of fields (for example, fields

Progress in Mathematics, 2005

We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex al... more We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs.

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infin... more We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with ... more The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416].

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with ... more The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416].

We present a unifying theory of fields with certain classes of analytic functions, called fields ... more 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,

We present a unifying theory of fields with certain classes of analytic functions, called fields ... more 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,

We consider the ordered field which is the completion of the Puiseux series field over \bR equipp... more We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language.

We consider the ordered field which is the completion of the Puiseux series field over \bR equipp... more We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language.

The Journal of Symbolic Logic, 2012

We introduce a very weak language ℒ<sub>M</sub> on p-... more We introduce a very weak language ℒ<sub>M</sub> on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language ℒ<sub>M</sub> are trivial functions. We also give

The Journal of Symbolic Logic, 2012

We introduce a very weak language ℒ<sub>M</sub> on p-... more We introduce a very weak language ℒ<sub>M</sub> on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language ℒ<sub>M</sub> are trivial functions. We also give

Bulletin of Symbolic Logic, 2001

Journal of Pure and Applied Algebra, 2007

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivi... more Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula. * Supported by a postdoctorial fellowship of the Fund for Scientific Research -Flanders (Belgium) (F. W. O) and the European Commission -Marie Curie European Individual Fellowship HPMF CT 2005-007121. * * With partial support from the FCT (Fundação para a Ciência e Tecnologia) program POCTI (Portugal/FEDER-EU) and Fundação Calouste Gulbenkian.

Journal of Mathematical Logic, 2007

We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. ... more We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef -Pas [25] . Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of P. J. Cohen [11], J. Denef , and the link between cell decomposition and integration was first made by Denef .

Journal of the London Mathematical Society, 2008

We consider the ordered field which is the completion of the Puiseux series field over R equipped... more We consider the ordered field which is the completion of the Puiseux series field over R equipped with a ring of analytic functions on [−1, 1] n which contains the standard subanalytic functions as well as functions given by tadically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields Rn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [HP] of a sentence which is not true in any o-minimal expansion of R (shown in [LR3] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in Rn, but not true in any o-minimal expansion of any of the fields R, R 1 , . . . , R n−1 .

Journal of Number Theory - J NUMBER THEOR, 2008

In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine A... more In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114], it is shown that a p-adic semi-algebraic set can be partitioned in such a way that each part is semi-algebraically isomorphic to a Cartesian product ∏i=1lR(k) where the sets R(k) are very basic subsets of Qp. It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114] that this result can be adapted to become useful to p-adic integration theory, by controlling the Jacobians of the occurring isomorphisms. In this paper we show that the isomorphisms can be chosen in such a way that the valuations of their Jacobians equal the valuations of products of coordinate functions, hence obtaining a kind of explicit p-adic resolution of singularities for semi-algebraic p-adic functions. We do this by restricting the used isomorphisms to a few specific types of functions, and by contr...

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Z d (s) = G(q −ds , q dλ 1 , · · · , q dλt ) H(q −ds , q dλ 1 , · · · , q dλt ) .

We extend the formalism and results on motivic integration from ["Constructible motivic func... more We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using

decreases as r goes to 0, pointing out, long before this concept has been formalized, the tame be... more decreases as r goes to 0, pointing out, long before this concept has been formalized, the tame behaviour of the local normalized volume of analytic sets. Ten years after Lelong's pioneering paper, Thie proved in (36) that the local density of a complex analytic subset X at a point a is a positive integer by expressing it as a sum

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a ppp-adic f... more We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a ppp-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index

The purpose of this paper is to explain how the identities of various fundamental lemmas fall wit... more The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one collection of fields to others. In particular, once the fundamental lemma has been established for one collection of fields (for example, fields

Progress in Mathematics, 2005

We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex al... more We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs.

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infin... more We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with ... more The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416].

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with ... more The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416].

We present a unifying theory of fields with certain classes of analytic functions, called fields ... more 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,

We present a unifying theory of fields with certain classes of analytic functions, called fields ... more 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,

We consider the ordered field which is the completion of the Puiseux series field over \bR equipp... more We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language.

We consider the ordered field which is the completion of the Puiseux series field over \bR equipp... more We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language.

The Journal of Symbolic Logic, 2012

We introduce a very weak language ℒ<sub>M</sub> on p-... more We introduce a very weak language ℒ<sub>M</sub> on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language ℒ<sub>M</sub> are trivial functions. We also give

The Journal of Symbolic Logic, 2012

We introduce a very weak language ℒ<sub>M</sub> on p-... more We introduce a very weak language ℒ<sub>M</sub> on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language ℒ<sub>M</sub> are trivial functions. We also give

Bulletin of Symbolic Logic, 2001

Journal of Pure and Applied Algebra, 2007

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivi... more Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, Comptes rendus de l'Académie des Sciences, 339 (2004) 411 -416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula. * Supported by a postdoctorial fellowship of the Fund for Scientific Research -Flanders (Belgium) (F. W. O) and the European Commission -Marie Curie European Individual Fellowship HPMF CT 2005-007121. * * With partial support from the FCT (Fundação para a Ciência e Tecnologia) program POCTI (Portugal/FEDER-EU) and Fundação Calouste Gulbenkian.

Journal of Mathematical Logic, 2007

We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. ... more We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef -Pas [25] . Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of P. J. Cohen [11], J. Denef , and the link between cell decomposition and integration was first made by Denef .

Journal of the London Mathematical Society, 2008

We consider the ordered field which is the completion of the Puiseux series field over R equipped... more We consider the ordered field which is the completion of the Puiseux series field over R equipped with a ring of analytic functions on [−1, 1] n which contains the standard subanalytic functions as well as functions given by tadically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields Rn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [HP] of a sentence which is not true in any o-minimal expansion of R (shown in [LR3] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in Rn, but not true in any o-minimal expansion of any of the fields R, R 1 , . . . , R n−1 .

Journal of Number Theory - J NUMBER THEOR, 2008

In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine A... more In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114], it is shown that a p-adic semi-algebraic set can be partitioned in such a way that each part is semi-algebraically isomorphic to a Cartesian product ∏i=1lR(k) where the sets R(k) are very basic subsets of Qp. It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105–114] that this result can be adapted to become useful to p-adic integration theory, by controlling the Jacobians of the occurring isomorphisms. In this paper we show that the isomorphisms can be chosen in such a way that the valuations of their Jacobians equal the valuations of products of coordinate functions, hence obtaining a kind of explicit p-adic resolution of singularities for semi-algebraic p-adic functions. We do this by restricting the used isomorphisms to a few specific types of functions, and by contr...

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Z d (s) = G(q −ds , q dλ 1 , · · · , q dλt ) H(q −ds , q dλ 1 , · · · , q dλt ) .