On a variant of Schanuel conjecture for the Carlitz exponential

Pellarin, Federico

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.

Classification : 11M38
Mots clés : Multiple zeta values, Carlitz module, Schanuel’s conjecture.

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

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

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

[4] Anglès, Bruno; Pellarin, Federico; Tavares Ribeiro, Floric 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] Anglès, Bruno; Pellarin, Federico; Tavares Ribeiro, Floric Anderson-Stark units for ${𝔽}_q\left[\theta \right]$, Trans. Am. Math. Soc. (2017) https://doi.org/10.1090/tran/6994 (to appear in print) | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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