Allouche, Jean-Paul; Thakur, Dinesh S. Automata and transcendence of the Tate period in finite characteristic. (English) Zbl 0926.11038 Proc. Am. Math. Soc. 127, No. 5, 1309-1312 (1999). This paper offers a new and short proof of the function field analogue of the Mahler-Manin conjecture, proved by J. F. Voloch [in J. Number Theory 58, 55-59 (1996; Zbl 0853.11061)]. This new proof is based on Christol’s criterion of algebraicity [G. Christol, Theor. Comput. Sci. 9, 141-145 (1979; Zbl 0402.68044)] which states that the sequence of coefficients of an algebraic series (with coefficients in a finite field) is automatic. More precisely, the authors study the transcendence of the power series associated to higher divisor functions \(\sigma_k(n)= \sum_{d\mid n}d^k\), by using sharp results by A. Cobham on frequencies of letters of automatic sequences [Math. Syst. Theory 6, 164-192 (1972; Zbl 0253.02029)]. Reviewer: V.BerthĂ© (Marseille) Cited in 5 Documents MSC: 11G07 Elliptic curves over local fields 11J89 Transcendence theory of elliptic and abelian functions 11B85 Automata sequences Keywords:Transcendence; periods; elliptic curves; automata; recognizability; function field analogue; Mahler-Manin conjecture; automatic sequences Citations:Zbl 0853.11061; Zbl 0402.68044; Zbl 0253.02029 PDF BibTeX XML Cite \textit{J.-P. Allouche} and \textit{D. S. Thakur}, Proc. Am. Math. Soc. 127, No. 5, 1309--1312 (1999; Zbl 0926.11038) Full Text: DOI OpenURL