A conjecture of Mahler on automorphic functions (Q1825232)

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 \({\mathbb{C}}(q)\).