Search the site

Table of contents for this issue | Previous article | Next article
Federico Pellarin
On a variant of Schanuel conjecture for the Carlitz exponential
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 845-873
Article PDF
Class. Math.: 11M38
Keywords: Multiple zeta values, Carlitz module, Schanuel’s conjecture.

Résumé - Abstract

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.


[1] Greg W. Anderson, W. Dale Brownawell & Matthew A. Papanikolas, Determination of the algebraic relations among special $\Gamma $-values in positive characteristic, Ann. Math. 160 (2004), p. 237-313
[2] Bruno Anglès & Federico Pellarin, Functional identities for $L$-series values in positive characteristic, J. Number Theory 142 (2014), p. 223-251
[3] Bruno Anglès & Federico Pellarin, Universal Gauss-Thakur sums and $L$-series, Invent. Math. 200 (2015), p. 653-669
[4] Bruno Anglès, Federico Pellarin & Floric Tavares Ribeiro, Arithmetic of positive characteristic $L$-series values in Tate algebras, Compos. Math. 152 (2016), p. 1-61
[5] Bruno Anglès, Federico Pellarin & Floric Tavares Ribeiro, Anderson-Stark units for $\mathbb{F}_q[\theta ]$, Trans. Am. Math. Soc. (2017) Article
[6] James Ax, On Schanuel’s Conjectures, Ann. Math. 93 (1971), p. 252-268
[7] Frits Beukers, A refined version of the Siegel-Shidlovskii theorem, Ann. Math. 163 (2006), p. 369-379
[8] Siegfried Bosch, Ulrich Güntzer & Reinhold Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften 261, Springer, 1984
[9] Leonard Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), p. 137-168
[10] Chieh-Yu Chang, Linear independence of monomials of multizeta values in positive characteristic, Compos. Math. 150 (2014), p. 1789-1808
[11] Chieh-Yu Chang & Jing Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), p. 321-345
[12] Laurent Denis, Indépendance algébrique et exponentielle de Carlitz, Acta Arith. 69 (1995), p. 75-89
[13] Laurent Denis, Indépendance algébrique de logarithmes en caractéristique $p$, Bull. Aust. Math. Soc. 74 (2006), p. 461-470
[14] David Goss, Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 35, Springer, 1996
[15] Serge Lang, Introduction to transcendental numbers, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Company, 1966
[16] Alexander B. Levin, Difference algebra, Algebra and Applications 8, Springer, 2008
[17] David Marker, A remark on Zilber’s pseudoexponentiation, J. Symb. Log. 71 (2006), p. 791-798
[18] Matthew A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), p. 123-174
[19] Federico Pellarin, Aspects de l’indépendance algébrique en caractéristique non nulle, in Séminaire Bourbaki. Volume 2006/2007. Exposés 967–981, Astérisque, Société Mathématique de France, 2008, p. 205-242, Exp no. 973
[20] Federico Pellarin, On the generalized Carlitz module, J. Number Theory 133 (2013), p. 1663-1692
[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 2, Walter de Gruyter, 1992, p. 265–284
[23] Paul M. Voutier, Letter to the author, June, 20, 2016
[24] L. I. Wade, Certain quantities transcendental over $\operatorname{GF}(p^n,x)$, Duke Math. J. 8 (1941), p. 701-720
[25] Michel Waldschmidt, Schanuel’s Conjecture: algebraic independence of transcendental numbers, in Colloquium De Giorgi 2013 and 2014, Colloquia, Edizioni della Normale, xv+137, p. 129-137
[26] Jing Yu, Analytic homomorphisms into Drinfeld modules, Ann. Math. 145 (1997), p. 215-233