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Automata-style proof of Voloch’s result on transcendence. (English) Zbl 0853.11060
The author gives a new proof of the transcendence of the period of the Tate elliptic curve, a result originally proved by J. P. Voloch [J. Number Theory 58, 55-59 (1996; Zbl 0853.11061)]. The author’s nice proof is based on a criterium of transcendence due to G. Christol [Theor. Comput. Sci. 9, 141-145 (1979; Zbl 0402.68044)], also known as the Christol, Kamae, Mendès-France and Rauzy criterium [G. Christol, T. Kamae, M. Mendès-France, and G. Rauzy, Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)], which states the equivalence between the algebraicity of a formal power series over a field of positive characteristic and the recognizability of the sequence of its coefficients by a finite automaton. Let us note that this method does not allow one to reach Voloch’s result on the transcendence of parameters of algebraic points.

11J89 Transcendence theory of elliptic and abelian functions
11B85 Automata sequences
14H52 Elliptic curves
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