Algèbre et géométrie

Tilting perfectoid fields, developed by Scholze, allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for […]

In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, Casale, Freitag, Nagloo), the following picture has been obtained regarding the transcendence properties of their solutions: - (Strong minimality): outside of an exceptional set of parameters, the corresponding differential equations are strongly minimal, - (Geometric triviality): algebraic independence of several solutions is controlled by pairwise algebraic independence outside of this exceptional set of parameters, - (Multidimensionality): the differential equations defined by generic independent parameters are orthogonal. Are […]

For a ring R, a subset of a cartesian power of R is said to be Diophantine if it is positive existentially definable over R with parameters from R. In general, Diophantine sets over rings are not well-understood even in very natural situations; for instance, we do not know if the ring of integers Z is Diophantine in the field of rational numbers. To show that a set is Diophantine requires to produce a particular existential formula that defines it. However, to show that a set is not Diophantine is […]

Les squelettes sont des sous-ensembles linéaires par morceaux d'espaces analytiques non-archimédiens apparaissant naturellement dans nombre de situations. Nous présenterons un résultat général de finitude, obtenu en collaboration avec A. Ducros, E. Hrushovski et J. Ye, concernant le groupe abélien ordonné des fonctions tropicales sur les squelettes des analytifiés de Berkovich de variétés algébriques. Notre approche utilise la version modèle théorique de l'analytification (la complétion stable) développée dans un travail antérieur avec E. Hrushovski.

In 2004 I proved that that if C is a transcendental curve definable in the structure R_{an}, then the number of points on C with integer coordinates of modulus less than H, is bounded by k loglog H for some constsnt k depending only on C. (The situation is vastly different for rational points.) The proof used the fact that such sets C are, in fact, semi-analytic everywhere-including infinity-and so the crux of the matter was to bound the number of solutions to equations of the form (*)    F(1/n) = […]