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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.

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