O-minimality and the Zilber-Pink conjecture for (pure) Shimura varieties

17 November 2016
Chris Daw

In this talk, we will explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture. In particular, we will explain how to generalise the work of Habegger and Pila on a product of modular curves. 
Habegger and Pila were able to prove that the Zilber-Pink conjecture holds in such a product if the so-called weak complex Ax and large Galois orbits conjectures are true. In fact, around the same time, Pila and Tsimerman proved a stronger statement than the weak complex Ax conjecture, namely, the Ax-Schanuel conjecture for the $j$-function. We will formulate Ax-Schanuel and large Galois orbits conjectures for general Shimura varieties and attempt to imitate the Habegger-Pila strategy. However, we will encounter an additional difficulty in bounding the height of a pre-special subvariety.

This is joint work with Jinbo Ren.

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