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A conjecture of Mahler on automorphic functions. (English) Zbl 0684.10022

The author proves a conjecture of K. Mahler [J. Aust. Math. Soc. 10, 445–450 (1969; Zbl 0207.08302)] on algebraic differential equation satisfied by automorphic functions of Fuchsian type. The precise result is as follows: Let \(G\) be a discontinuous subgroup of \(\mathrm{SL}(2; \mathbb{C})\), 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 \(\mathbb{C}(q)\).

MSC:

11F03 Modular and automorphic functions
11F27 Theta series; Weil representation; theta correspondences

Citations:

Zbl 0207.08302
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References:

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