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We propose to grok Lipschitz stratifications from a non-archimedean point of view and thereby show that they exist for closed definable sets in any power-bounded o-minimal structure on a real closed field. Unlike the previous approaches in the literature, our method bypasses resolution of singularities and Weierstrass preparation altogether; it transfers the situation to a non-archimedean model, where the quantitative estimates appearing in Lipschitz stratifications are sharpened into valuation-theoretic inequalities. Applied to a uniform family of sets, this approach automatically yields a family of stratifications which satisfy the Lipschitz conditions in a uniform way., Comment: 44 pages, 5 figures

We prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

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., Comment: Compared to v2/v3: some proofs simplified, the main statement generalized; slightly reorganized. Regarding the automatically generated text overlap note: it overlaps with the Appendix B (which is part of arXiv:1208.1945) written by us; the appendix and this article cross-reference each other, and since the set-up is very similar, some overlap is unavoidable

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective., Comment: 15 pages, to appear in JSL; minor corrections and one proof replaced by a sketch of proof and a reference

© Les auteurs, 2018. Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from [10], [11] and [5]. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field. ispartof: Journal de l'Ecole Polytechnique - Mathematiques vol:5 pages:45-78 status: published

In the context of the local Langlands philosopy for R \mathbb {R} , Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group G ( R ) G(\mathbb {R}) and the space of “complete geometric parameters”—a space of equivariant local systems on a variety on which the Langlands-dual of G ( R ) G(\mathbb {R}) acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of categories. In this article, that conjecture is proven in the case where G ( R ) G(\mathbb {R}) is a generalized Lorentz group SO n , 1 ⁡ ( R ) \operatorname {SO}_{n,1}(\mathbb {R}) .

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., Comment: 44 pages

We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also contain non-local information about the stratified sets. For example, they do not only see the singularities in the valued field, but also those in the residue field. Like Whitney stratifications, t-stratifications exist for different classes of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic subsets. The general framework are definable sets (in the sense of model theory) in a language that satisfies certain hypotheses. We give two applications. First, we show that t-stratifications in suitable valued fields $K$ induce classical Whitney stratifications in $\Bbb R$ or $\Bbb C$; in particular, the existence of t-stratifications implies the existence of Whitney stratifications. This uses methods of non-standard analysis. Second, we show how, using t-stratifications, one can determine the ultra-metric isometry type of definable subsets of $\Bbb Z_p^n$ for $p$ sufficiently big. For those $p$, this proves a conjecture stated in a previous article. In particular, this yields a new, geometric proof of the rationality of Poincar\'e series., Comment: Fixed typos; enhanced the presentation