RequirementsGalois theory. Rudiments of algebraic geometry and model theory would be welcome. Reminders will be done in class if necessary.
Program requirementsexamen
TeacherSilvain Rideau
Weekly hours 4 h CM
Years Master Logique et Fondements de l'Informatique

Syllabus

From its early development, valued fields have always played an important role in model theory. One reason for this interest is that their strong connection to arithmetic and geometry has allowed the introduction of model theoretic techniques in other fields of mathematics, resulting most often in the resolution of open questions in that field.

One of the first example of such an interaction can be found in the work of Ax-Kochen and independently Ershov who gave a solution to Artin’s conjecture on the existence of solution to homogeneous equations over p-adic fields. One of the core concept of their proof is a reduction of a question on certain characteristic zero Henselian fields, in that specific case the description of its theory, to this very same question on their residue field and value group. The idea of this reduction can be found time and time again in later work on the model theory of valued fields.

The goal of this class will be, starting with questions of quantifier elimination and then moving on to Shelah’s classification theory and more « geometric » model theory, to show the ubiquity of this Ax-Kochen-Ershov principle in the model theory of valued fields. We will also try, as much as it is possible, to give an insight into some of the recent applications of the model theory of valued fields.

Contents

  • Quantifier elimination in algebraically closed valued fields and Henselian fields of characteristic zero
  • Ax-Kochen and Ershov’s theorem and the solution to Artin’s conjecture
  • Study of an important example: p-adic fields.
  • The independence property in valued fields.
  • Imaginaries in valued fields.
  • Geometric model theory of algebraically closed valued fields.

Bibliography