Summerschool “Around the Zilber-Pink conjectures”

Paris, 25 June - 5 July 2012


Useful information
Satellite day


Introductory material

We will assume basic algebraic number theory, some exposure to algebraic geometry, and knowledge of sections 1-3 of Marker's notes on model theory.

Shimura varieties.
Pierre Deligne, Travaux de Griffiths, Séminaire BOURBAKI 22e année, 1969/70, no 376, mai-juin 1970.
Pierre Deligne, Travaux de Shimura, Séminaire BOURBAKI 23e année, 1970/71, no 389, février 1971.
J. S. Milne, Introduction to Shimura varieties, Clay Mathematical Proceedings Volume 4, 2005, 265-378.

Algebraic geometry
Marc Hindry, Introduction to Abelian Varieties and the Mordell-Lang Conjecture.
E. Bombieri, W. Gubler, Heights in diophantine Geometry, Cambridge University Press 2006.
M. Hindry, J. Silverman, Diophantine Geometry: an introduction, Springer 2000.

Model Theory.
In addition to sections 1-3 of Dave Marker's course notes, we suggest that people read sections 4-7 and the Appendix.
Course Model Theory for Algebra and Algebraic Geometry, by Dave Marker. Orsay, Spring 2010. Content:

  • Sections 1--3: Language, Structures and Theories, The Compactness Theorem, Ultraproducts and Compactness
  • Sections 4-6: Complete Theories, Quantifier Elimination, Algebraically Closed Fields
  • Section 7: Real Closed Fields and o-minimality
  • Section 8: The Pila-Zannier proof of the Manin-Mumford Conjecture
  • Appendix: Real Algebra
  • Exercises using the Compactness Theorem
  • More advanced material

    Thomas Scanlon, A proof of the André-Oort conjecture via mathematical logic [after Pila, Wilkie, Zannier]. April 2011, Bourbaki seminar.

    Antoine Chambert-Loir, Relations de dépendance et intersections exceptionnelles, Séminaire Bourbaki, 63e année, 2010-2011, exposé No 1032. arXiv:1011.4738

    L. van den Dries, Tame topology and o-minimal structures, Cambridge Univ. Press, New York, 1998

    U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, with Appendices by D. Masser (Annals of Mathematics Studies, Am-181), Princeton UP, 2012

    Handouts of the workshop The Zilber-Pink conjecture, Luminy, 16-20 May 2011. Content: