Scolarité M2 (sauf MIC, MIDS)
- Mme Chatoux
- bureau 5055
- 01 57 27 93 06
- Mme Prudlo
- bureau 5055
- 01 57 27 93 06
The model theory of pseudo finite fields
8 ECTS, semester 2, 12 weeks
| Requirements | Galois theory. Rudiments of algebraic geometry and model theory would be welcome. Reminders will be done in class if necessary. |
| Program requirements | examen |
| Teacher | Silvain Rideau |
| Weekly hours | 4 h CM |
| Years | Master Logique et Fondements de l'Informatique |
Syllabus
The asymptotic properties of finite fields, i.e. the properties that are true in every sufficiently large finite field, can be understood by looking at pseudo finite fields: the infinite models of set of statements that hold in every finite field. This class was defined by Ax and he gave an algebraic characterisation of it: they are exactly the perfect, pseudo- algebraically closed fields with exactly one extension of any given degree.
Pseudo finite structures have lately played an important role in the model theoretic study of certain questions in combinatorics, among others in results of Hrushovski in additive combinatorics. These results find some of their roots in the work of Chatzidakis, van den Dries and Macintyre that gave a precise description of sets definable in a pseudo-finite field by, among other things, exhibiting a pseudo-finite equivalent of of the counting measure. The goal of
The goal of this class will be to introduce the results of Ax and Chatzidakis-van den Dries-Macintyre as well as introduce the algebraic notions necessary to understand them. We will then consider questions related to geometric model theory like the study of definable groups, the imaginaries or classification questions.
Contents
- A quick reminder of algebra and model theory
- Study of pseudo-algebraically closed (bounded) fields: model completeness, description of types.
- Ax’s theorem, a glimpse into Lang-Weil and Tchebotarev’s theorems.
- Chatzidakis-van den Dries-Macityre’s theorem: existence and definability of the pseudo-finite measure and dimension.
- Classification: simple theories, independence property.
- Imaginaries in bounded pseudo-algebraically closed fields.
- Definable groups: groups configuration, algebraic groups.
Bibliography
- J. Ax.The elementary theory of finite fields. Ann. of Math. (2) 88 (1968), pp.239-271.
- Z. Chatzidakis. Théorie des modèles des corps finis et pseudo-finis. Cours de M2. 1996.
- Z. Chatzidakis. L. van den Dries, and A. Macintyre. Definable sets over finite fields. J. Reine Angew. Math. 427 (1992), pp.107-135.
- M. D. Fried and M. Jarden. Field arithmetic. Third. Vol. 11. Ergeb. Math. Grenzgeb. (3). Revised by Jarden. Springer-Verlag, Berlin, 2008, pp. xxiv+792.
- E. Hrushovski. Pseudo-finite fields and related structures. In : Model theory and applications. Vol. 11. Quad. Mat. Aracne, Rome, 2002, pp. 151-212.