Cours de l'IHES 2023-2024

Point-Counting and the Zilber-Pink Conjecture (2/4)

by Jonathan Pila (University of Oxford & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres CS 40001 91893 Bures-sur-Yvette Cedex

The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.
Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.
The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.
We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

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Emmanuel Ullmo