Aspects de l'indépendance algébrique en caractéristique non nulle [d'après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu,...]

Pellarin, Federico

Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 973, 38 p.

MR Zbl

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     author = {Pellarin, Federico},
     title = {Aspects de l'ind\'ependance alg\'ebrique en caract\'eristique non nulle [d'apr\`es {Anderson,} {Brownawell,} {Denis,} {Papanikolas,} {Thakur,} {Yu,...]}},
     booktitle = {S\'eminaire Bourbaki - Volume 2006/2007  - Expos\'es 967-981},
     series = {Ast\'erisque},
     note = {talk:973},
     pages = {205--242},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {317},
     year = {2008},
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     url = {http://www.numdam.org/item/AST_2008__317__205_0/}
}

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Pellarin, Federico. Aspects de l'indépendance algébrique en caractéristique non nulle [d'après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu,...], dans Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 973, 38 p. http://www.numdam.org/item/AST_2008__317__205_0/

Bibliographie
Cité par

[1] B. Adamczewski & J. Bell - Function fields in positive characteristic and Cobham's theorem, J. Algebra 319 (2008), p. 2337-2350. | DOI | MR | Zbl

[2] J.-P. Allouche - Sur la transcendance de la série formelle $Π$ , Sém. Théor. Nombres Bordeaux (2) 2 (1990), p. 103-117. | DOI | EuDML | Numdam | MR | Zbl

[3] J.-P. Allouche, Transcendence of the Carlitz-Goss gamma function at rational arguments, J. Number Theory 60 (1996), p. 318-328. | DOI | MR | Zbl

[4] G. W. Anderson - $t$ -motives, Duke Math. J. 53 (1986), p. 457-502. | DOI | MR | Zbl

[5] G. W. Anderson, W. D. Brownawell & M. Papanikolas - Determination of the algebraic relations among special $Γ$ -values in positive characteristic, Ann. of Math. (2) 160 (2004), p. 237-313. | DOI | MR | Zbl

[6] G. W. Anderson & D. S. Thakur - Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), p. 159-191. | DOI | MR | Zbl

[7] Y. André - Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Société Mathématique de France, 2004. | MR | Zbl

[8] R. M. Beals & D. S. Thakur - Computational classification of numbers and algebraic properties, Internat. Math. Res. Notices 15 (1998), p. 799-818. | DOI | MR | Zbl

[9] P.-G. Becker, W. D. Brownawell & R. Tubbs - Gel'fond's theorem for Drinfel'd modules, Michigan Math. J. 41 (1994), p. 219-233. | DOI | MR | Zbl

[10] V. Berthé - Combinaisons linéaires de $ζ (𝑠) / Π^{𝑠}$ sur $𝐅_{q} (x)$ , pour $1 \leq 𝑠 \leq q - 2$ , J. Number Theory 53 (1995), p. 272-299. | MR | Zbl

[11] D. Bertrand - Lemmes de zéros et nombres transcendants, Séminaire Bourbaki, vol. 1985/86, exp. n° 652, Astérisque 145-146 (1987), p. 21-44. | EuDML | Numdam | MR | Zbl

[12] D. Bertrand, Un analogue différentiel de la théorie de Kummer, in Approximations diophantiennes et nombres transcendants (Luminy, 1990), de Gruyter, 1992, p. 39-49. | MR | Zbl

[13] V. Bosser & F. Pellarin - Hyperdifferential properties of Drinfeld quasimodular forms, article ID rnn032, 56 pages, doi : 10.1093/imrn/rnn032, 2008. | MR | Zbl

[14] W. D. Brownawell - Transcendence in positive characteristic, in Number theory (Tiruchirapalli, 1996), Contemp. Math., vol. 210, Amer. Math. Soc., 1998, p. 317-332. | DOI | MR | Zbl

[15] W. D. Brownawell & M. Papanikolas - Linear independence of gamma values in positive characteristic, J. reine angew. Math. 549 (2002), p. 91-148. | MR | Zbl

[16] L. Carlitz - On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), p. 137-168. | DOI | JFM | MR

[17] L. Carlitz, An analogue of the von Staudt-Clausen theorem, Duke Math. J. 3 (1937), p. 503-517. | DOI | JFM | MR

[18] L. Carlitz, A class of polynomials, Trans. Amer. Math. Soc. 43 (1938), p. 167-182. | DOI | JFM | MR

[19] L. Carlitz, Some spécial functions over $G F (q, x)$ , Duke Math. J. 27 (1960), p. 139-158. | DOI | MR | Zbl

[20] C.-Y. Chang & J. Yu - Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), p. 321-345. | DOI | MR | Zbl

[21] G. Christol - Ensembles presque périodiques $k$ -reconnaissables, Theoret. Comput. Sci. 9 (1979), p. 141-145. | DOI | MR | Zbl

[22] G. Christol, T. Kamae, M. Mendès France & G. Rauzy - Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), p. 401-419. | DOI | EuDML | Numdam | MR | Zbl

[23] A. Cobham - Uniform tag sequences, Math. Systems Theory 6 (1972), p. 164-192. | DOI | MR | Zbl

[24] G. Damamme & Y. Hellegouarch - Transcendence of the values of the Carlitz zeta function by Wade's method, J. Number Theory 39 (1991), p. 257-278. | DOI | MR | Zbl

[25] S. David & L. Denis - Indépendance algébrique sur les $T$ -modules, Compositio Math. 122 (2000), p. 1-22. | DOI | MR | Zbl

[26] S. David & L. Denis, Périodes de modules de "l'indépendance quadratique en rang II", J. Ramanujan Math. Soc. 17 (2002), p. 65-83. | MR | Zbl

[27] P. Deligne - Valeurs de fonctions $L$ et périodes d'intégrales, in Automorphic forms, représentations and $L$ -functions (Oregon State Univ. 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., 1979, p. 313-346. | MR | Zbl

[28] L. Denis - Théorème de Baker et modules de Drinfel'd, J. Number Theory 43 (1993), p. 203-215. | DOI | MR | Zbl

[29] L. Denis, Dérivées d'un module de Drinfel'd et transcendance, Duke Math. J. 80 (1995), p. 1-13. | DOI | MR | Zbl

[30] L. Denis, Indépendance algébrique et exponentielle de Carlitz, Acta Arith. 69 (1995), p. 75-89. | DOI | EuDML | MR | Zbl

[31] L. Denis, Lemmes de multiplicités et $T$ -modules, Michigan Math. J. 43 (1996), p. 67-79. | DOI | MR | Zbl

[32] L. Denis, Un critère de transcendance en caractéristique finie, J. Algebra 182 (1996), p. 522-533. | DOI | MR | Zbl

[33] L. Denis, Indépendance algébrique de différents $π$ , C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), p. 711-714. | DOI | MR | Zbl

[34] L. Denis, Indépendance algébrique des dérivées d'une période du module de Carlitz, J. Austral Math. Soc. Ser. A 69 (2000), p. 8-18. | DOI | MR | Zbl

[35] L. Denis, Indépendance algébrique de logarithmes en caractéristique $p$ , Bull. Austral. Math. Soc. 74 (2006), p. 461-470. | DOI | MR | Zbl

[36] S. Dion - Théorème de Brownawell-Waldschmidt en caractéristique finie, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), p. 71-90. | DOI | EuDML | Numdam | MR | Zbl

[37] J. Fresnel & M. Van Der Put - Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser, 2004. | MR | Zbl

[38] E.-U. Gekeler - On the de Rham isomorphism for Drinfel'd modules, J. reine angew. Math. 401 (1989), p. 188-208. | EuDML | MR | Zbl

[39] E.-U. Gekeler, Quasi-periodic functions and Drinfel'd modular forms, Compositio Math. 69 (1989), p. 277-293. | EuDML | Numdam | MR | Zbl

[40] D. Goss - von Staudt for $𝐅_{q} [T]$ , Duke Math. J. 45 (1978), p. 885-910. | DOI | MR | Zbl

[41] D. Goss, $υ$ -adic zeta functions, $L$ -series and measures for function fields, Invent. Math. 55 (1979), p. 107-119. | DOI | EuDML | MR | Zbl

[42] D. Goss, The $Γ$ -function in the arithmetic of function fields, Duke Math. J. 56 (1988), p. 163-191. | DOI | MR | Zbl

[43] D. Goss, Drinfel'd modules : cohomology and spécial functions, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., 1994, p. 309-362. | MR | Zbl

[44] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 35, Springer, 1996. | MR | Zbl

[45] Y. Hamahata - Tensor products of Drinfel'd modules and $υ$ -adic représentations, Manuscripta Math. 79 (1993), p. 307-327. | DOI | EuDML | MR | Zbl

[46] C. Hardouin - Computation of the Galois groups attached to the Carlitz logarithms according to the method of M. Papanikolas, prépublication de l'I.W.R. (Heidelberg, avril 2006).

[47] C. Hardouin, Hypertranscendance des systèmes aux différences diagonaux, Compos. Math. 144 (2008), p. 565-581. | DOI | MR | Zbl

[48] D. R. Hayes - Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), p. 77-91. | DOI | MR | Zbl

[49] Y. Hellegouarch - Fonctions zêta en caractéristique positive et modules de Carlitz-Hayes, Séminaire Bourbaki, vol. 1997/98, exp. n° 837, Astérisque 252 (1998), p. 57-79. | EuDML | Numdam | MR | Zbl

[50] K. S. Kedlaya - Finite automata and algebraic extensions of function fields, J. Théor. Nombres Bordeaux 18 (2006), p. 379-420. | DOI | EuDML | Numdam | MR | Zbl

[51] B. De Mathan - Irrationality measures and transcendence in positive characteristic, J. Number Theory 54 (1995), p. 93-112. | DOI | MR | Zbl

[52] M. Mendès France & J.-Y. Yao - Transcendence and the Carlitz-Goss gamma function, J. Number Theory 63 (1997), p. 396-402. | DOI | MR | Zbl

[53] D. Mumford - An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, 1978, p. 115-153. | MR | Zbl

[54] M. Papanikolas - Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), p. 123-174. | DOI | MR | Zbl

[55] P. Philippon - Variétés abéliennes et indépendance algébrique. I, Invent. Math. 70 (1982/83), p. 289-318. | DOI | EuDML | MR | Zbl

[56] P. Philippon, Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), p. 355-383. | DOI | EuDML | Numdam | MR | Zbl

[57] P. Philippon, Critères pour l'indépendance algébrique dans les anneaux diophantiens, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), p. 511-515. | MR | Zbl

[58] P. Philippon, Une approche méthodique pour la transcendance et l'indépendance algébrique de valeurs de fonctions analytiques, J. Number Theory 64 (1997), p. 291-338. | DOI | MR | Zbl

[59] P. Philippon, Indépendance algébrique et $K$ -fonctions, J. reine angew. Math. 497 (1998), p. 1-15. | DOI | MR | Zbl

[60] M. Van Der Put & M. Reversat - Krichever modules for différence and differential equations, Astérisque 296 (2004), p. 207-225, Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I. | Numdam | MR | Zbl

[61] S. K. Sinha - Periods of $t$ -motives and transcendence, Duke Math. J. 88 (1997), p. 465-535. | DOI | MR | Zbl

[62] D. S. Thakur - Gamma functions for function fields and Drinfel'd modules, Ann. of Math. (2) 134 (1991), p. 25-64. | DOI | MR | Zbl

[63] D. S. Thakur, Drinfel'd modules and arithmetic in the function fields, Internat. Math. Res. Notices 9 (1992), p. 185-197. | DOI | MR | Zbl

[64] D. S. Thakur, Transcendence of gamma values for $𝐅_{q} [T]$ , Ann. of Math. (2) 144 (1996), p. 181-188. | DOI | MR | Zbl

[65] D. S. Thakur, Automata and transcendence, in Number theory (Tiruchirapalli, 1996), Contemp. Math., vol. 210, Amer. Math. Soc., 1998, p. 387-399. | DOI | Zbl

[66] D. S. Thakur, Function field arithmetic, World Scientific Publishing Co. Inc., 2004. | Zbl

[67] A. Thiéry - Indépendance algébrique des périodes et quasi-périodes d'un module de Drinfel'd, in The arithmetic of function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, 1992, p. 265-284. | Zbl

[68] A. Thiéry, Théorème de Lindemann-Weierstrass pour les modules de Drinfel'd, Compositio Math. 95 (1995), p. 1-42. | EuDML | Numdam | Zbl

[69] R. Tubbs - Analytic subgroups of $t$ -modules, Trans. Amer. Math. Soc. 349 (1997), p. 2605-2617. | DOI | Zbl

[70] L. I. Wade - Certain quantities transcendental over $G F (p^{n}, x)$ , Duke Math. J. 8 (1941), p. 701-720. | DOI | Zbl

[71] M. Waldschmidt - Nombres transcendants, Springer, 1974, Lecture Notes in Mathematics, Vol. 402. | Zbl

[72] M. Waldschmidt, Diophantine properties of the periods of the Fermat curve, in Number theory related to Fermat's last theorem, Proc. Conf., Prog. Math., vol. 26, 1982, p. 79-88. | DOI | Zbl

[73] M. Waldschmidt, Transcendence problems connected with Drinfel'd modules, İstanbul Üniv. Fen Fak. Mat. Derg. 49 (1990), p. 57-75 | Zbl

M. Waldschmidt, Transcendence problems connected with Drinfel'd modules, İstanbul Üniv. Fen Fak. Mat. Derg. 49 (1993), International Symposium on Algebra and Number Theory (Silivri, 1990). | Zbl

[74] M. Waldschmidt, Sur la nature arithmétique des valeurs de fonctions modulaires, Séminaire Bourbaki, vol. 1996/97, exp. n° 824, Astérisque 245 (1997), p. 105-140. | EuDML | Numdam | Zbl

[75] M. Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften, vol. 326, Springer, 2000, Transcendence properties of the exponential function in several variables. | Zbl

[76] M. Waldschmidt, Un demi-siècle de transcendance, in Development of mathematics 1950-2000, Birkhäuser, 2000, p. 1121-1186. | DOI | Zbl

[77] G. Wüstholz - Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Ann. of Math. (2) 129 (1989), p. 501-517. | DOI | Zbl

[78] J. Yu - Transcendental numbers arising from Drinfel'd modules, Mathematika 30 (1983), p. 61-66. | DOI | Zbl

[79] J. Yu, A six exponentials theorem in finite characteristic, Math. Ann. 272 (1985), p. 91-98. | DOI | EuDML | Zbl

[80] J. Yu, Transcendence and Drinfel'd modules, Invent. Math. 83 (1986), p. 507-517. | DOI | EuDML | Zbl

[81] J. Yu, Transcendence and Drinfeld modules : several variables, Duke Math. J. 58 (1989), p. 559-575. | DOI | Zbl

[82] J. Yu, On periods and quasi-periods of Drinfel'd modules, Compositio Math. 74 (1990), p. 235-245. | EuDML | Numdam | Zbl

[83] J. Yu, Transcendence and spécial zeta values in characteristic $p$ , Ann. of Math. (2) 134 (1991), p. 1-23. | DOI | Zbl

[84] J. Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), p. 215-233. | DOI | Zbl