Over the last 15 years a remarkable link between o-minimality and algebraic/arithmetic geometry has been unfolding following the discovery of Pila-Wilkie’s counting theorem and its applications around unlikely intersections, functional transcendence etc. While the counting theorem is nearly optimal in general, Wilkie has conjectured a much sharper form in the structure R_exp. There is a folklore expectation that such sharper bounds should hold in structures « coming from geometry », but for lack of a general formalism explicit conjectures have been made only for specific structures.
I will describe a refinement of the standard o-minimality theory aimed at capturing the finer « arithmetic tameness » that we expect to see in structures coming from geometry. After presenting the general framework I will discuss my result with Vorobjov showing that the restricted Pfaffian structure is sharply o-minimal, and how this was used in our recent work with Novikov and Zack to prove Wilkie’s conjecture for the restricted Pfaffian structure and for Wilkie’s original case of R_exp. I will also discuss some conjectures on the construction of larger sharply o-minimal structures, and some partial results in this direction. Finally I will explain the crucial role played by these results in my recent work with Schmidt and Yafaev on Galois orbit lower bounds for CM points in general Shimura varieties, and subsequently in the recent resolution of general André-Oort conjecture by Pila-Shankar-Tsimerman-(Esnault-Groechenig).
- Séminaire Géométrie et théorie des modèles