Hyperbolic Ax-Lindemann theorem in the cocompact case

TL;DR

This paper establishes a hyperbolic version of the Ax-Lindemann theorem for compact Shimura varieties, advancing the understanding of their algebraic and transcendental structures in number theory.

Contribution

It extends the classical Ax-Lindemann theorem to the setting of compact Shimura varieties, providing new tools for related conjectures.

Findings

Proves a hyperbolic Ax-Lindemann theorem for compact Shimura varieties

Supports strategies towards the unconditional proof of the André-Oort conjecture

Enhances understanding of the transcendental structure of Shimura varieties

Abstract

We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionally