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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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) Tom Scanlon - UC Berkeley,
Titre The dynamical Mordell-Lang problem in positive characteristic
Date23/03/2018
Horaire14:15 à 15:45
Diffusion
RésumeThe dynamical Mordell-Lang conjecture in characteristic zero predicts that if f : X → X is a map of algebraic varieties over a field K of characteristic zero, Y ⊆ X is a closed subvariety and a in X(K) is a K-rational point on X, then the return set n in N : f^n(a) in Y(K) is a finite union of points and arithmetic progressions. For K a field of characteristic p > 0, it is necessary to allow for finite unions with sets of the form a + ∑_i=1^m p^n_i : (n_1, ... , n_m) in N^m and one might conjecture that all return sets are finite unions of points, arithmetic progressions and such p-sets. We studied the special case of the positive characteristic dynamical Mordell-Lang problem on semiabelian varieites and using our earlier results with Moosa on so-called F-sets reduced the problem to that of solving a class of exponential diophantine equations in characteristic zero. In so doing, under the hypothesis that X is a semiabelian variety and either Y has small dimension or f is sufficiently general, we prove the conjecture. However, we also show that our reduction to the exponential diiophantine problems may be reversed so that the positive characteristic dynamical Mordell-Lang conjecture in general is equivalent to a class of hard exponential diophantine problems which the experts consider to be out of reach given our present techniques.
(This is a report on joint work with Pietro Corvaja, Dragos Ghioca and Umberto Zannier available at arXiv:1802.05309.)
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