Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is Pila’s modular Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the André-Oort conjecture.
In this talk we will discuss how an entirely new approach, using the model theory of differential fields, can be used to prove the ALW Theorem with derivatives for Fuchsian automorphic functions – a direct generalization of Pila’s ALW theorem. We will also explain how new cases of the André-Pink conjecture can be obtained using this new approach.
This is joint work with G. Casale and J. Freitag.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
