145 research outputs found

    Definable equivalence relations and zeta functions of groups

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    We prove that the theory of the pp-adics Qp{\mathbb Q}_p admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp){\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p) for each nn. We also prove that the elimination of imaginaries is uniform in pp. Using pp-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed pp) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math. So

    Integration and Cell Decomposition in PP-minimal Structures

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    We show that the class of L\mathcal{L}-constructible functions is closed under integration for any PP-minimal expansion of a pp-adic field (K,L)(K,\mathcal{L}). This generalizes results previously known for semi-algebraic and sub-analytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for PP-minimal structures, a result which is independent of the existence of Skolem functions. %The result is obtained from weak versions of cell decomposition and function preparation which we prove for general PP-minimal structures. A direct corollary is that Denef's results on the rationality of Poincar\'e series hold in any PP-minimal expansion of a pp-adic field (K,L)(K,\mathcal{L}).Comment: 22 page

    Uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums

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    This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sumsComment: Doctoral thesis, University of Lill

    Dimension, matroids, and dense pairs of first-order structures

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    A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of omega and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding a field and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We extend the above result to dense tuples of structures.Comment: Version 2.8. 61 page

    Panorama of p-adic model theory

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    ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud par une revue de la bibliographie
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