arXiv subjects

R. Cluckers

Publications and source records attributed to R. Cluckers.

At least 19 recordsLinked to original sources

arXiv2015

Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila. Along the way we prove, for definable functions in a general context of non-archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

math.AG↗

arXiv2013

Chai's Conjecture and Fubini properties of dimensional motivic integration

We prove that a conjecture of Chai on the additivity of the base change conductor for semi-abelian varieties over a discretely valued field is equivalent to a Fubini property for the dimensions of certain motivic integrals. We prove this Fubini property when the valued field has characteristic zero.

math.AG↗

arXiv2012

Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

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 richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to previously known versions of motivic integration by e.g. the second author and J. Sebag and to classical p-adic integrals.

math.AG↗

arXiv2009

Local metric properties and regular stratifications of p-adic definable sets

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-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 in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone.We also prove that these distinguished tangent cones stabilize for small enough subgroups. We finally obtain the $p$-adic counterpart of the Cauchy-Crofton formula for the density. To prove these results we use the Lipschitz decomposition of definable $p$-adic sets of arXiv:0904.3853v1 and prove here the genericity of the regularity conditions for stratification such as $(w_f)$, $(w)$, $(a_f)$, $(b)$ and $(a)$ conditions.

math.LO↗

arXiv2009

Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions

We prove that a (globally) subanalytic p-adic function which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken subanalytic. We also prove the analogous result for a subanalytic family of functions depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in finite extensions of the field of p-adic numbers. These results are p-adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p-adic fields, we need to introduce new methods adapted to the p-adic situation.

math.AG↗

arXiv2008

An introduction to b-minimality

We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, J. Math. Log., 7 no. 2 (2007) 195--227, math.LO/0610183]. We compare this notion with other notions like o-minimality, C-minimality, p-minimality, and so on.

math.LO↗

arXiv2008

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.

math.AG↗

arXiv2008

$b$-minimality

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. 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, J. Denef, and the link between cell decomposition and integration was first made by Denef.

math.LO↗

arXiv2007

Transfer Principle for the Fundamental Lemma

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 of positive characteristic), it is also valid for others (fields of characteristic zero).

math.RT↗

arXiv2007

Exponential sums: questions by Denef, Sperber, and Igusa

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local exponential sums modulo $p^m$. We generalize Igusa's conjecture of the introduction of [Igusa, J., \textit{Lectures on forms of higher degree}, Lect. math. phys., Springer-Verlag, {\bf{59}} (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $p$ case. We generalize some results by Katz of [Katz, N. M., \textit{Estimates for "singular" exponential sums}, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.

math.NT↗

arXiv2007

Igusa and Denef-Sperber Conjectures on nondegenerate p-adic exponential sums

We prove the intersection of Igusa's Conjecture of [Igusa, J., "Lectures on forms of higher degree", Lect. math. phys., Springer-Verlag, 59 (1978)] and the Denef - Sperber Conjecture of [Denef, J. and Sperber, S., "Exponential sums mod p^n and Newton polyhedra", Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate exponential sums modulo p^m.

math.NT↗

arXiv2007

Orbital integrals for linear groups

For a linear group $G$ acting on an absolutely irreducible variety $X$ over the rationals $\QQ$, we describe the orbits of $X(\QQ_p)$ under $G(\QQ_p)$ and of $X(\FF_p((t)))$ under $G(\FF_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\QQ_p$ or $\FF_p((t))$ is $\leq 0$ for $p$ big enough, and similarly for all finite field extensions of $\QQ_p$ and $\FF_p((t))$.

math.AG↗

arXiv2007

Igusa's conjecture on exponential sums modulo $p$ and $p^2$ and the motivic oscillation index

We prove the modulo $p$ and modulo $p^2$ cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo $p^m$ when $p$ and $m$ vary. We introduce the motivic oscillation index of a polynomial $f$ and prove the stronger, analogue bounds for $m=1,2$ using this index instead of the original bounds. The modulo $p^2$ case of our bounds holds for all polynomials; the modulo $p$ case holds for homogeneous polynomials and under extra conditions also for nonhomogeneous polynomials. We obtain natural lower bounds for the motivic oscillation index by using results of Segers. We also show that, for $p$ big enough, Igusa's local zeta function has a nontrivial pole when there are $\FF_p$-rational singular points on $f=0$. We introduce a new invariant of $f$, the flaw of $f$.

math.NT↗

arXiv2006

Integration of positive constructible functions against Euler characteristic and dimension

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et integration motivic I, C. R. Math. Acad. Sci. Paris 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.

math.LO↗

arXiv2005

Constructible exponential functions, motivic Fourier transform and transfer principle

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We define also motivic Schwartz-Bruhat spaces on which motivic Fourier transformation induces an isomorphism. Our motivic integrals specialize to non archimedian integrals. We give a general transfer principle comparing identities between functions defined by integrals over local fields of characteristic zero, resp. positive, having the same residue field. We also prove new results about p-adic integrals of exponential functions.

math.AG↗

arXiv2005

Fonctions constructibles exponentielles, transformation de Fourier motivique et principe de transfert

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We define also motivic Schwartz-Bruhat spaces on which motivic Fourier transformation induces an isomorphism. Our motivic integrals specialize to non archimedian integrals. We give a general transfer principle comparing identities between functions defined by integrals over local fields of characteristic zero, resp. positive, having the same residue field. Details of constructions and proofs will be given elsewhere.

math.NT↗

arXiv2005

Analytic cell decomposition and analytic motivic integration

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 $\FF_q((t))$ 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 \emph{analytic} motivic integration and \emph{analytic} motivic constructible functions in the line of R. Cluckers and F. Loeser [\emph{Fonctions constructible et intégration motivic I}, Comptes rendus de l'Académie des Sciences, {\bf 339} (2004) 411 - 416].

math.AG↗

arXiv2004

Ax-Kochen-Er{š}ov Theorems for $p$-adic integrals and motivic integration

This survey paper, to appear in he proceedings of the Miami Winter School ``Geometric Methods in Algebra and Number Theory'', is concerned with extending classical results à la Ax-Kochen-Er{š}ov to $p$-adic integrals in a motivic setting, using the framework of constructible motivic functions we introduced in math.AG/0410203. We conclude the paper by discussing briefly the relevance of our results to the study of orbital integrals and the Fundamental Lemma.

math.AG↗