A conjecture of Mahler on automorphic functions
type: Item
instance of: scholarly article
title: A conjecture of Mahler on automorphic functions
zbMATH Open document ID: 0684.10022
DOI: 10.1007/BF01194871
author: Keiji Nishioka
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: The author proves a conjecture of \textit{K. Mahler} [J. Aust. Math. Soc. 10, 445-450 (1969; Zbl 0207.083)] on algebraic differential equation satisfied by automorphic functions of Fuchsian type. The precise result is as follows: Let G be a discontinuous subgroup of SL(2;\({\mathbb{C}})\), which possesses at least three limit points. Let u be a nonzero complex number. Then any automorphic function of G satisfies no algebraic differential equation of second order over \({\mathbb{C}}(z,e^{uz}).\) As an application it is shown that Jacobi's theta functions \(\theta\) (q) satisfy no algebraic differential equation of second order over \({\mathbb{C}}(q)\).
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 4120252
zbMATH Keywords: discontinuous group
zbMATH Keywords: algebraic differential equation
zbMATH Keywords: automorphic functions of Fuchsian type
zbMATH Keywords: Jacobi's theta functions
Wikidata QID: Q123008404
MaRDI profile type: Publication
cites work: Q5680222
cites work: Q5568135
cites work: On algebraic differential equations satisfied by automorphic functions
cites work: Some analytic and geometric applications of the invariant theoretic method
cites work: Q3237360
cites work: A DIFFERENTIAL EQUATION FOR THE THETA FUNCTION
description: scientific article; zbMATH DE number 4120252