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Automata and transcendence of the Tate period in finite characteristic

Author(s): Jean-Paul Allouche; Dinesh S. Thakur
Journal: Proc. Amer. Math. Soc. 127 (1999), 1309-1312.
MSC (1991): Primary 11J89, 11G07, 68Q68, 11B85
Posted: January 27, 1999
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Abstract: Using the techniques of automata theory, we give another proof of the function field analogue of the Mahler-Manin conjecture and prove transcendence results for the power series associated to higher divisor functions $\sigma _k(n)=\sum _{d|n}d^k$.

References:

[BDGP96]
K. Barre-Sirieix, G. Diaz, F. Gramain, G. Philibert, Une preuve de la conjecture de Mahler-Manin, Invent. Math. 124 (1996), 1-9. MR 96j:11103

[C79]
G. Christol, Ensembles presque-périodiques $k$-reconnaissables, Theoret. Comput. Sci. 9 (1979), 141-145. MR 80e:68141

[CKMR80]
G. Christol, T. Kamae, M. Mendès France, G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), 401-419. MR 82e:10092

[Co72]
A. Cobham, Uniform tag sequences, Math. Systems Theory 6 (1972), 164-192. MR 56:15230

[K75]
N. Katz, Higher congruences between modular forms, Ann. Math. 101 (1975), 332-367. MR 54:5120

[R77]
C. Radoux, Divisibilité de $\sigma _k(n)$ par un nombre premier, Séminaire Delange-Pisot-Poitou, Théorie des Nombres, $19^{\text{e}}$ année, 1977/78, Exposé n$^{\circ}$ 3, (1978), 3-01-3-05. MR 80b:10064

[Ran61]
R. A. Rankin, The divisibility of divisor functions, Proc. Glasgow Math. Assoc. 5 (1961), 35-40. MR 26:2407

[S-D73]
H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences of modular forms, in: Modular functions of one variable III, Proc. Internat. Summer School, Univ. Antwerp 1972, Springer Lecture Notes in Math. 350 (1973), 1-55.

[T96]
D. Thakur, Automata-style proof of Voloch's result on transcendence, J. Number Theory 58 (1996), 60-63. MR 98a:11100

[V96]
J. F. Voloch, Transcendence of elliptic modular functions in characteristic $p$, J. Number Theory 58 (1996), 55-59. MR 98a:11099

[W96]
M. Waldschmidt, Sur la nature arithmétique des valeurs de fonctions modulaires, Séminaire Bourbaki, $49^{\text{e}}$ année, vol. 1996-1997, exposé 824, Nov. 96, Paris, pp. 824-1-824-36.

[Wat35]
G. N. Watson, Über Ramanujansche Kongruenzeigenschaften der Zerfällungsan-
zahlen (I), Math. Z. 39 (1935), 712-731.

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Additional Information:

Jean-Paul Allouche
Affiliation: CNRS, LRI, Bâtiment 490, Université d'Orsay F-91405 Orsay Cedex, France
Email: allouche@lri.fr

Dinesh S. Thakur
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: thakur@math.arizona.edu

DOI: 10.1090/S0002-9939-99-04650-X
PII: S 0002-9939(99)04650-X
Keywords: Transcendence, periods, elliptic curves, automata, recognizability
Received by editor(s): August 27, 1997
Posted: January 27, 1999
Additional Notes: The second author was supported in part by NSF grant DMS 9623187.
Communicated by: David E. Rohrlich
Copyright of article: Copyright 1999, American Mathematical Society

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