Schwarzian equation, automorphic functions and functional transcendence

Abstract: By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation.

In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\’e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.

dynamical systems

Audience: researchers in the topic

BIRS workshop: Algebraic Dynamics and its Connections to Difference and Differential Equations

Series comments: The field of algebraic dynamics has emerged over the past two decades at the confluence of algebraic geometry, discrete dynamical systems, and diophantine geometry. In recent work, striking connections have been observed between algebraic dynamics and much older theories of difference and differential equations. This meeting brings together mathematicians with expertise in such diverse fields as ring theory, complex dynamics, differential and difference algebra, combinatorics and algebraic geometry. New work towards the dynamical Mordell-Lang and dense orbit conjectures as well as theorems on hypertranscendence and functional independence proven by connecting difference Galois theory, algebraic dynamics and other algebraic approaches to the study offunctional equations will be presented at this meeting.