# Applications of the hyperbolic Ax-Schanuel conjecture

## TL;DR

This paper explores how the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a point counting problem, enabling progress via Pila-Wilkie theorem.

## Contribution

It demonstrates the application of the hyperbolic Ax-Schanuel conjecture to simplify the Zilber-Pink conjecture for Shimura varieties into a manageable point counting problem.

## Key findings

- Reduction of Zilber-Pink conjecture to point counting problem
- Use of Pila-Wilkie theorem in specific cases
- Connection between hyperbolic Ax-Schanuel and arithmetic conjectures

## Abstract

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila-Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila-Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber-Pink conjecture for curves in abelian varieties.