arXiv subjects

Raf Cluckers

Publications and source records attributed to Raf Cluckers.

At least 55 records · Page 3Linked to original sources

arXiv2014

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 $p$-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 $L_K$ consisting of the language of valued fields together with all strictly convergent power series over a complete, rank one valued field $K$ can be expanded, in a definitial way, to a larger language corresponding to an analytic structure (with separated power series) from [CL1], hence inheriting all properties from loc. cit., including geometric properties for the definable sets like certain forms of quantifier elimination. Our expansion comes from adding specific, existentially definable functions, which are solutions of certain henselian systems of equations. Moreover, we show that, even when $K$ is algebraically closed, one does not have quantifier elimination in $L_K$ itself, and hence, passing to expansions is unavoidable in general. We pursue this study in the wider generality of extending non-separated power series rings to separated ones, and give new examples, in particular of the analytic structure over $\mathbb{Z}[[t]]$ that can be interpreted and understood now in all complete valued fields. In a separate direction, we show in rather large generality that Weierstrass preparation implies Weierstrass division.

math.LO↗

arXiv2013

Local integrability results in harmonic analysis on reductive groups in large positive characteristic

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer principles for integrability and boundedness conditions for motivic exponential functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic.

math.RT↗

arXiv2013

Motivic functions, integrability, and uniform in p bounds for orbital integrals

This is a short announcement and summary of the results of arxiv:1111.7057, arxiv.org:1111.4405, and Appendix B to arxiv:1208.1945. In particular, we emphasize the exposition of the ideas related to model theory and motivic integration, and simplify the definitions, to make these results easily accessible to non-model theorists.

math.RT↗

arXiv2013

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

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.

math.LO↗

arXiv2013

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 $Q_p^n$ 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.

math.AG↗

arXiv2013

Bounding the decay of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function

We call a function "constructible" if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we also show that any continuous, integrable, constructible function of a single variable has an integrable Fourier transform.

math.CA↗

arXiv2012

Lebesgue classes and preparation of real constructible functions

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and constructible functions $f$ and $μ$ on $E\times\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E \times (0,\infty]$ for which $y \mapsto f(x,y)$ is in $L^p(|μ|_{x}^{q})$, where $|μ|_{x}^{q}$ is the positive measure on $\RR^n$ whose Radon-Nikodym derivative with respect to the Lebesgue measure is $y\mapsto |μ(x,y)|^q$. We also prove a closely related preparation theorem for $f$ and $μ$. These results relate analysis (the study of $L^p$-spaces) with geometry (the study of zero loci).

math.AG↗

arXiv2011

Quantifier elimination in ordered abelian groups

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates. As a corollary, we find that all definable functions in ordered abelian groups are piecewise affine linear on finitely many definable pieces.

math.LO↗

arXiv2010

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

It was already known that a p-adic, locally Lipschitz continuous semi-algebraic function is piecewise Lipschitz continuous, where the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1. We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parametrized versions and in the subanalytic setting.

math.LO↗

arXiv2010

The Howe-Moore property for real and p-adic groups

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 direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.

math.FA↗

arXiv2010

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

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depends on the group and a real number $r>0$) for the set of equivalence classes of the representations of minimal depth $r$ satisfying some additional assumptions. This parameter space is essentially a geometric object defined over $\Q$. Given a non-Archimedean local field $\K$ with sufficiently large residual characteristic, the part of the character table near the identity element for $G(\K)$ that comes from our class of representations is parameterized by the residue-field points of $B$. The character values themselves can be recovered by specialization from a constructible motivic exponential function. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group $G(\K)$ is computable.

math.RT↗

arXiv2010

Analytic van der Corput Lemma for p-adic and F_q((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems

We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma, where the real condition of sufficient smoothness for the phase is replaced by the condition that the phase is a convergent power series. This van der Corput style result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (analytic) manifolds and opens the way for further applications. As one such further application we give the restriction theorem for Fourier transforms of L^p functions to suitably curved analytic manifolds over non-archimedean local fields, similar to the real restriction result by E. Stein and C. Fefferman.

math.FA↗

arXiv2009

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

We study Lebesgue integration of sums of products of globally subanalytic functions and their logarithms, called constructible functions. Our first theorem states that the class of constructible functions is stable under integration. The second theorem treats integrability conditions in Fubini-type settings, and the third result gives decay rates at infinity for constructible functions. Further, we give preparation results for constructible functions related to integrability conditions.

math.AG↗

arXiv2009

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 principal, 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.

math.LO↗

arXiv2006

Real closed fields with nonstandard and standard analytic structure

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 extend these constructions and results to rank n ordered fields \bR_n (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 \bR (shown in [LR3] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σ_n, true in \bR_n, but not true in any o-minimal expansion of any of the fields \bR,\bR_1,...,\bR_{n-1}.

math.LO↗

arXiv2005

The Fundamental Theorem of prehomogeneous vector spaces modulo $p^m$. With an appendix "L-functions of prehomogeneous vector spaces" by Fumihiro Sato

For a number field $K$ with ring of integers ${\mathcal O}_K$, we prove an analogue over finite rings of the form ${\mathcal O}_K/{\mathcal P}^m$ of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where ${\mathcal P}$ is a big enough prime ideal of ${\mathcal O}_K$ and $m>1$. In the appendix, F. Sato gives an application of the Theorems A, B and the Theorems A, B, C in J. Denef and A. Gyoja [\emph{Character sums associated to prehomogeneous vector spaces}, Compos. Math., \textbf{113} (1998) 237--346] to the functional equation of $L$-functions of Dirichlet type associated with prehomogeneous vector spaces.

math.NT↗

arXiv2004

Multivariate Igusa theory: Decay rates of exponential sums

We obtain general estimates for exponential integrals of the form \[ E_f(y)=\int_{\mathbb{Z}_{p}^{n}}ψ(\sum_{j=1}^r y_j f_j(x))|dx|, \] where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $ψ$ a nontrivial additive character on $\mathbb{Q}_p$. We prove that if $(f_1,...,f_r)$ is a dominant map, then $|E_f(y)| < c|y|^α$ for some $c>0$ and $α<0$, uniform in $y$, where $|y|=\max(|y_i|)_i$. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of $p$-adic subanalytic sets and $p$-adic integration techniques based on $p$-adic cell decomposition. We compare our results to some elementarily obtained explicit bounds for $E_f$ with $f_j$ polynomials.

math.NT↗

arXiv2003

Model theory of valued fields

We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.

math.LO↗