Ax-Schanuel for the \(j\)-function. (English) Zbl 1419.11093

Pila, Jonathan; Tsimerman, Jacob

Duke Math. J. 165, No. 13, 2587-2605 (2016).

Summary: In this paper we prove a functional transcendence statement for the \(j\)-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the \(j\)-function are governed by modular relations.

MSC:

11G18	Arithmetic aspects of modular and Shimura varieties
03C98	Applications of model theory

Keywords:

Shimura varieties; Ax-Schanuel theorem; transcendence; modular curve; number theory