In these talks, we will give a brief introduction to the theory of Shimura varieties and explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture in this setting. 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.
Lecture 1: Wed 23/11 at 4pm in G.209.
Lecture 2: Thu 24/11 at 2pm in Frank Adams 1, when we usually have the Number Theory seminar.
Lecture 3: Fri 25/11 3pm in Frank Adams 2.