Ax-Schanuel for the j-function

TL;DR

This paper proves a functional transcendence theorem for the j-function, showing that algebraic relations involving the j-function are governed by modular relations, analogous to Ax-Schanuel for the exponential.

Contribution

It establishes an Ax-Schanuel type theorem for the j-function, extending functional transcendence results to modular functions.

Findings

Proves an Ax-Schanuel type theorem for the j-function.

Shows algebraic relations are governed by modular relations.

Extends functional transcendence to modular functions.

Abstract

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.