On a variant of Schanuel conjecture for the Carlitz exponential
Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 845-873.

Nous introduisons et décrivons une variante de la conjecture de Schanuel dans le cadre de l’exponentielle de Carlitz sur des algèbres de Tate et de fonctions similaires. Un autre objectif de ce travail est de stimuler des possibles investigations en transcendance et indépendance algébrique en caractéristique non nulle.

We introduce and discuss a variant of Schanuel conjecture in the framework of the Carlitz exponential function over Tate algebras and allied functions. Another purpose of the present paper is to widen the horizons of possible investigations in transcendence and algebraic independence in positive characteristic.

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Accepté le :
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DOI : 10.5802/jtnb.1004
Classification : 11M38
Mots clés : Multiple zeta values, Carlitz module, Schanuel’s conjecture.
Federico Pellarin 1

1 Institut Camille Jordan UMR 5208 Site de Saint-Etienne 23 rue du Dr. P. Michelon 42023 Saint-Etienne, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Federico Pellarin. On a variant of Schanuel conjecture for the Carlitz exponential. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 845-873. doi : 10.5802/jtnb.1004. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1004/

[1] Greg W. Anderson; W. Dale Brownawell; Matthew A. Papanikolas Determination of the algebraic relations among special Γ-values in positive characteristic, Ann. Math., Volume 160 (2004) no. 1, pp. 237-313 | DOI | Zbl

[2] Bruno Anglès; Federico Pellarin Functional identities for L-series values in positive characteristic, J. Number Theory, Volume 142 (2014), pp. 223-251 | DOI | Zbl

[3] Bruno Anglès; Federico Pellarin Universal Gauss-Thakur sums and L-series, Invent. Math., Volume 200 (2015) no. 2, pp. 653-669 | DOI | Zbl

[4] Bruno Anglès; Federico Pellarin; Floric Tavares Ribeiro Arithmetic of positive characteristic L-series values in Tate algebras, Compos. Math., Volume 152 (2016) no. 1, pp. 1-61 (with an appendix of Florent Demeslay) | DOI | Zbl

[5] Bruno Anglès; Federico Pellarin; Floric Tavares Ribeiro Anderson-Stark units for 𝔽 q [θ], Trans. Am. Math. Soc. (2017) https://doi.org/10.1090/tran/6994 (to appear in print) | DOI

[6] James Ax On Schanuel’s Conjectures, Ann. Math., Volume 93 (1971), pp. 252-268 | DOI | Zbl

[7] Frits Beukers A refined version of the Siegel-Shidlovskii theorem, Ann. Math., Volume 163 (2006) no. 1, pp. 369-379 | DOI | Zbl

[8] Siegfried Bosch; Ulrich Güntzer; Reinhold Remmert Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, 261, Springer, 1984, xii+436 pages | Zbl

[9] Leonard Carlitz On certain functions connected with polynomials in a Galois field, Duke Math. J., Volume 1 (1935), pp. 137-168 | DOI | Zbl

[10] Chieh-Yu Chang Linear independence of monomials of multizeta values in positive characteristic, Compos. Math., Volume 150 (2014) no. 11, pp. 1789-1808 | DOI | Zbl

[11] Chieh-Yu Chang; Jing Yu Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math., Volume 216 (2007) no. 1, pp. 321-345 | DOI | Zbl

[12] Laurent Denis Indépendance algébrique et exponentielle de Carlitz, Acta Arith., Volume 69 (1995) no. 1, pp. 75-89 | DOI | Zbl

[13] Laurent Denis Indépendance algébrique de logarithmes en caractéristique p, Bull. Aust. Math. Soc., Volume 74 (2006) no. 3, pp. 461-470 | DOI | Zbl

[14] David Goss Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 35, Springer, 1996, xiii+422 pages | Zbl

[15] Serge Lang Introduction to transcendental numbers, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Company, 1966, vi+105 pages | Zbl

[16] Alexander B. Levin Difference algebra, Algebra and Applications, 8, Springer, 2008, xi+519 pages | Zbl

[17] David Marker A remark on Zilber’s pseudoexponentiation, J. Symb. Log., Volume 71 (2006) no. 3, pp. 791-798 | DOI | Zbl

[18] Matthew A. Papanikolas Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Volume 171 (2008) no. 1, pp. 123-174 | DOI | Zbl

[19] Federico Pellarin Aspects de l’indépendance algébrique en caractéristique non nulle, Séminaire Bourbaki. Volume 2006/2007. Exposés 967–981 (Astérisque), Volume 317 (2008), p. 205-242, Exp no. 973 | Zbl

[20] Federico Pellarin On the generalized Carlitz module, J. Number Theory, Volume 133 (2013) no. 5, pp. 1663-1692 | DOI | Zbl

[21] Thomas Scanlon o-minimality as an approach to the André-Oort conjecture (To appear in Panor. Synth.)

[22] Alain Thiery Indépendance algébrique des périodes et quasi-périodes d’un module de Drinfeld, The arithmetic of function fields. Proceedings of the workshop at the Ohio State University (Ohio State University Mathematical Research Institute Publications), Volume 2, Walter de Gruyter, 1992, pp. 265-284 | Zbl

[23] Paul M. Voutier (Letter to the author, June, 20, 2016)

[24] L. I. Wade Certain quantities transcendental over GF(p n ,x), Duke Math. J., Volume 8 (1941), pp. 701-720 | DOI | Zbl

[25] Michel Waldschmidt Schanuel’s Conjecture: algebraic independence of transcendental numbers, Colloquium De Giorgi 2013 and 2014 (Colloquia), Volume 5 (xv+137), pp. 129-137 | Zbl

[26] Jing Yu Analytic homomorphisms into Drinfeld modules, Ann. Math., Volume 145 (1997) no. 2, pp. 215-233 | DOI | Zbl

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