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

2018/11/15 by Guy Casale, James Freitag, Casale, Guy +3 · 5 citations
Computer Science · Mathematics · #03C60 #11F03 #12H05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Meromorphic and Entire Functions #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1811.06583

openalex publication_date 2018/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the\nuniformizing functions of genus zero Fuchsian groups of the first kind. Our\nproof relies on differential Galois theory, monodromy of linear differential\nequations, the study of algebraic and Liouvillian solutions, differential\nalgebraic work of Nishioka towards the Painlev 'e irreducibility of certain\nSchwarzian equations, and considerable machinery from the model theory of\ndifferentially closed fields.\n Our techniques allow for certain generalizations of the\nAx-Lindemann-Weierstrass theorem which have interesting consequences. In\nparticular, we apply our results to answer a question of Painlev 'e (1895). We\nalso answer certain cases of the Andr 'e-Pink conjecture, namely in the case of\norbits of commensurators of Fuchsian groups.\n

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• Forking independence in differentially closed fields of positive characteristic
• Bialgebraic varieties of the Gamma function
• The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function
• Real geometric transcendence for uniformization maps of algebraic Riemann surfaces
• Algebraic independence of solutions to multiple Lotka-Volterra systems

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