English
Lang-Weil type bounds in finite difference fields
Multi angle
Auteurs : Hils, Martin (Auteur de la Conférence)
CIRM (Editeur )
03C13 - Finite structures [See also 68Q15, 68Q19]
03C20 - Ultraproducts and related constructions
03C60 - Model-theoretic algebra
11G25 - Varieties over finite and local fields
11U09 - Model theory, See also {03Cxx}
12L12 - Model theory
CIRM (Editeur )
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Résumé :
(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou)
We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.
Keywords : model theory; finite difference fields; Lang-Weil estimates; Frobenius specializations
Codes MSC : We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.
Keywords : model theory; finite difference fields; Lang-Weil estimates; Frobenius specializations
03C13 - Finite structures [See also 68Q15, 68Q19]
03C20 - Ultraproducts and related constructions
03C60 - Model-theoretic algebra
11G25 - Varieties over finite and local fields
11U09 - Model theory, See also {03Cxx}
12L12 - Model theory
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Informations sur la Vidéo
Réalisateur : Petit, JeanLangue : Anglais Date de publication : 23/06/2023 Date de captation : 30/05/2023 Sous collection : Research talks arXiv category : Logic ; Commutative Algebra Domaine : Algebra ; Logic and Foundations Format : MP4 (.mp4) - HD Durée : 01:07:10 Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023_05-30_Hils_1.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Model theory of valued fields / Théorie des modèles des corps valuésOrganisateurs de la rencontre : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain Dates : 29/05/2023 - 02/06/2023 Année de la rencontre : 2023 URL Congrès : https://conferences.cirm-math.fr/2761.html
Données de citation
DOI : 10.24350/CIRM.V.20051803Citer cette vidéo: Hils, Martin (2023). Lang-Weil type bounds in finite difference fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20051803 URI : http://dx.doi.org/10.24350/CIRM.V.20051803 |
Voir aussi
- [Multi angle] Beautiful pairs revisited / Auteur de la Conférence Ye, Jinhe.
- [Multi angle] An Imaginary Ax-Kochen/Ershov principle: the equicharacteristic zero case / Auteur de la Conférence Vicaria, Mariana.
- [Multi angle] Existential closedness of $\mathbb{Q}^{alg}$ as a globally valued field / Auteur de la Conférence Szachniewicz, Michal.
- [Multi angle] Interpretable, definably semisimple groups in various valued fields / Auteur de la Conférence Peterzil, Ya'acov.
- [Multi angle] The existential closedness problem for analytic solutions of difference equations / Auteur de la Conférence Padgett, Adele.
- [Multi angle] Multi topological fields and NTP2 / Auteur de la Conférence Montenegro Guzman, Samaria.
- [Multi angle] A Hasse principle over Berkovich analytic curves / Auteur de la Conférence Mehmeti, Vlerë.
- [Multi angle] Definable convex and henselian valuations on ordered fields / Auteur de la Conférence Krapp, Lothar Sebastian.
- [Multi angle] Beyond the Fontaine-Wintenberger theorem / Auteur de la Conférence Kartas, Konstantinos.
- [Multi angle] Around NIP Noetherian domains / Auteur de la Conférence Johnson, Will.
- [Multi angle] An invitation to globally valued fields / Auteur de la Conférence Hrushovski, Ehud.
- [Multi angle] Residue field domination / Auteur de la Conférence Haskell, Deirdre.
- [Multi angle] Motivic integration with pseudo-finite residue field and fundamental lemma / Auteur de la Conférence Forey, Arthur.
- [Multi angle] Tropical functions on skeletons / Auteur de la Conférence Ducros, Antoine.
- [Multi angle] Contracting endomorphisms of valued fields / Auteur de la Conférence Dor, Yuval.
- [Multi angle] Ax-Kochen-Ershov principles for finitely ramified henselian fields / Auteur de la Conférence Dittmann, Philip.
- [Multi angle] Group construction in $C$-minimal structures / Auteur de la Conférence Delon, Françoise.
- [Multi angle] Homology groups in algebraically closed valued fields / Auteur de la Conférence Cubides Kovacsics, Pablo.
- [Multi angle] Existential uniform p-adic integration and descent for integrability and largest poles / Auteur de la Conférence Cluckers, Raf.
- [Multi angle] Polynomials that vanish on many sets of codimension 2 / Auteur de la Conférence Ben Yaacov, Itaï.
- [Multi angle] The model theory of Hardy fields and linear differential equations / Auteur de la Conférence Aschenbrenner, Matthias.
- [Multi angle] Transfer of decidability for existential theories of (valued) fields / Auteur de la Conférence Anscombe, Sylvy.
Bibliographie
- CAFURE, Antonio et MATERA, Guillermo. Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields and Their Applications, 2006, vol. 12, no 2, p. 155-185. - https://doi.org/10.1016/j.ffa.2005.03.003
- DOR, Yuval et HRUSHOVSKI, Ehud. Specialization of Difference Equations and High Frobenius Powers. arXiv preprint arXiv:2212.05366, 2022. - https://doi.org/10.48550/arXiv.2212.05366
- HRUSHOVSKI, Ehud. The elementary theory of the Frobenius automorphisms. arXiv preprint math/0406514, 2004. - https://doi.org/10.48550/arXiv.math/0406514
- ZOU, Tingxiang. Pseudofinite difference fields and counting dimensions. Journal of Mathematical Logic, 2021, vol. 21, no 01, p. 2050022. - https://doi.org/10.1142/S0219061320500221
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