Séminaires : Géométrie et Théorie des Modèles

Equipe(s) : lm,
Responsables :Zoé Chatzidakis, Raf Cluckers, Georges Comte
Email des responsables : zoe.chatzidakis@imj-prg.fr
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http://gtm.imj-prg.fr/

 

Pour recevoir le programme par e-mail, écrivez à : zoe.chatzidakis@imj-prg.fr
Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici : https://webusers.imj-prg.fr/~zoe.chatzidakis/papiers/MTluminy.dvi/MTluminy.dvi. Ces personnes peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.Retour ligne automatique
Les notes de quelques-uns des exposés sont disponibles.

Orateur(s) Will Johnson - Fudan University,
Titre Curve-excluding fields
Date21/05/2021
Horaire09:00 à 12:00
Diffusion
RésumeLet T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅.
We can use this theory to construct a field K with an interesting combination of properties. On the model-theoretic side, the theory of K is complete, decidable, model-complete, and algebraically bounded, and K is a “geometric structure” in the sense of Hrushovski and Pillay. Additionally, some classification-theoretic properties might hold in K. On the field-theoretic side, K is non-large---there is a smooth curve C such that C(K) is finite and non-empty. This is unusual; the vast majority of model-theoretically tractable fields are large or finite. On the other hand, K is “virtually large”---it has a finite extension which is large. In fact, every proper algebraic extension of K is pseudo algebraically closed (PAC). The absolute Galois group of K is an ω-free profinite group. This negatively answers a question of Junker and Koenigsmann (is every model-complete infinite field large?) and a question of Macintyre (does every model-complete field have a small Galois group?).
This is based on joint work with Erik Walsberg and Vincent Ye.
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