Automata and transcendence of the Tate period in finite characteristic. (English) Zbl 0926.11038

Allouche, Jean-Paul; Thakur, Dinesh S.

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.Berthé (Marseille)

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