Some functional transcendence results around the Schwarzian differential equation

TL;DR

This paper extends the Ax-Lindemann-Weierstrass theorem to functions satisfying Schwarzian differential equations, using model theory, Galois theory, and geometry to generalize previous results for genus zero Fuchsian groups.

Contribution

It introduces new variants of the ALW theorem for Schwarzian differential equations, broadening the scope of previous uniformizer results.

Findings

Generalized ALW theorem for Schwarzian equations

Applied model theory, Galois theory, and geometry techniques

Extended results beyond genus zero Fuchsian groups

Abstract

This paper centers around proving variants of the Ax-Lindemann-Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, galois theory and geometry.