Workshop The geometry of the Frobenius automorphism

CIRM , 25 - 29 March 2013


List of participants
Slides and Notes

Aim of the workshop

To give a series of lectures explaining the proof by Ehud Hrushovski of the twisted Lang-Weil estimates for difference varieties. Link to the paper "The elementary theory of the Frobenius Automorphism" (preprint).
Here are some slides of some survey lectures given at the Orsay Workshop Model theory of difference fields and applications, in December 2011:
Non-standard Frobenius: Introduction, by M. Hils.
Plan détaillé de la preuve de Hrushovski, by G. Giabicani and Y. Laszlo.

Lectures will begin around 9:00 am on Monday 25 March and probably end around 4:00 pm on Friday 29 March.

Modus Operandi

Participants are supposed to have a working knowledge of the prerequisites (see below). We hope that most of them will participate actively in the programme.
A set of lecture notes, in a preliminary form, is expected to be available before the start of the workshop.


All participants are supposed to have a working knowledge of the basics both in model theory and in algebraic geometry.

For example, in model theory, be comfortable with at least the first six sections of Dave Marker's Orsay lecture notes:

  • Sections 1-3: Language, Structures and Theories, The Compactness Theorem, Ultraproducts and Compactness
  • Sections 4-6: Complete Theories, Quantifier Elimination, Algebraically Closed Fields
  • In Algebraic Geometry
    The basic definitions for example: Robin Hartshorne, Algebraic Geometry, Chap. I-II

    Some further suggestions

    Valued fields:
    - O. Zariski et P. Samuel: "Commutative algebra II" (Ch VI), GTM, Springer.
    - Notes of a course given by Z. Chatzidakis on valued fields (in French), chapter 1, and maybe parts of chapters 2 and 3.

    Difference algebra:
    - Richard Cohn, Difference algebra, (Ch 1-3).
    - Lecture Notes "Algebraic difference equations" by Michael Wibmer.

    Intersection theory: Basic definitions and results will be recalled but it could be useful to look at (or browse through) one of the following:
    - W. Fulton: "Introduction to intersection theory in algebraic geometry" (first 3 sections), Cbms Regional Conference Series in Mathematics, A.M.S. 1984.
    - I. R. Shafarevich: "Basic algebraic geometry 1" (Ch IV, sections 1 and 2)

    Advanced algebraic geometry (Around the Weil conjectures): Again, this is not required, but here is a possible reference:
    - B. Mazur: "Eigenvalues of Frobenius acting on algebraic varieties over finite fields", in: "Algebraic Geometry - Arcata 1974", Proc. of Symposia in Pure Mathematics, Vol XXIX, pp 231-262.


    If you wish to be put on the mailing list, please write an e-mail to

    Organisation Committee: Z. Chatzidakis, F. Loeser, T. Scanlon.
    Scientific Committee: J.-B. Bost, E. Bouscaren, Z. Chatzidakis, M. Hils, Y. Laszlo, F. Loeser, T. Scanlon.

    Last updated: 04/17/2013 10:05:49