Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

Some functional transcendence results around the Schwarzian differential equation.

“…By the genus of the group Γ we mean the genus of the unique non singular completion of Γ\H. In [5] we proved the ALW theorem for Γ\H genus zero. In Section 4.22 we drop the assumption that Γ\H is genus zero, but assume that Γ\H is given by an algebraic curve over Q alg .…”

“…For a survey of recent developments in this area mainly centering on approaches using o-minimality, see [11]. Our approach, started in [5], is much different.…”

“…The classification of such relations for a given system is one of the central preoccupations of differential Galois theory and the model theory of differential fields, two of the central tools we employ. In [5], we solved the bi-algebraicity problem with φ given by the map applying j Γ and its first two derivatives to any number of coordinates in H n , where j Γ is a uniformizing function associated with the quotient Γ\H where Γ is a Fuchsian group of the first kind and genus zero, and therefore Γ\H is a n-punctured sphere. In this case, denoting a coordinate in the domain by t, we have that j Γ (t) is a solution of the Schwarzian equation:…”

“…In the case of a Fuchsian group Γ, the rational function R Γ depends on the group Γ and the choice of a uniformizing function j Γ , and as there are only countably many Fuchsian groups of the first kind of genus zero, the results of [5] only apply to a very particular (and countable) class of rational functions R. The bi-algebraicity problem has an equivalent statement in terms of functional transcendence, see [5,26]. We will, following [26], refer to both forms as Ax-Lindemann-Weierstrass type theorems.…”

“…In this case, we give a complete solution to the problem of bi-algebraicity, even with different such analytic functions applied to each coordinate. The case of generalized triangle equations is generally interesting (it includes, for instance the ALW result for the j-function associated with elliptic curves), but it also allows for an important and interesting generalization of our ALW result from [5], which we describe next.…”