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.
Highlights
Let X and Y be algebraic varieties over C and let φ : Xan → Y an be a complex analytic map which is not algebraic
Bi-algebraic subvarieties should be rare and revealing of important geometric aspects of the analytic map φ. This manuscript centers around the problem of determining the bi-algebraic subvarieties of analytic maps and several related problems of functional transcendence
The case of generalized triangle equations is generally interesting, but it allows for an important and interesting generalization of our Ax–Lindemann– Weierstrass (ALW) result from [5], which we describe
Summary
Let X and Y be algebraic varieties over C and let φ : Xan → Y an be a complex analytic map which is not algebraic. In [5], we solved the bi-algebraicity problem with φ given by the map applying jΓ and its first two derivatives to any number of coordinates in Hn, where jΓ is a uniformizing function associated with the quotient Γ\H where Γ is a Fuchsian group of the first kind and genus zero, and Γ\H is a n-punctured sphere In this case, denoting a coordinate in the domain by t, we have that jΓ(t) is a solution of the Schwarzian equation: St(y). In this very general setting, we obtain slightly weaker results by characterizing bi-algebraic curves rather than all varieties.
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