Aspects de l'indépendance algébrique en caractéristique non nulle [d'après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu,...]
Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque no. 317  (2008), Talk no. 973, p. 205-242
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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},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {317},
     year = {2008},
     note = {talk:973},
     pages = {205-242},
     zbl = {1185.11048},
     mrnumber = {2487735},
     language = {fr},
     url = {http://www.numdam.org/item/AST_2008__317__205_0}
}
Pellarin, Federico. Aspects de l'indépendance algébrique en caractéristique non nulle [d'après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu,...], in Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Talk no. 973, pp. 205-242. http://www.numdam.org/item/AST_2008__317__205_0/

[1] B. Adamczewski & J. Bell - Function fields in positive characteristic and Cobham's theorem, J. Algebra 319 (2008), p. 2337-2350. | Article | MR 2388308 | Zbl 1151.11060

[2] J.-P. Allouche - Sur la transcendance de la série formelle Π , Sém. Théor. Nombres Bordeaux (2) 2 (1990), p. 103-117. | Article | Numdam | MR 1061761 | Zbl 0709.11067

[3] J.-P. Allouche, Transcendence of the Carlitz-Goss gamma function at rational arguments, J. Number Theory 60 (1996), p. 318-328. | Article | MR 1412966 | Zbl 0862.11039

[4] G. W. Anderson - t-motives, Duke Math. J. 53 (1986), p. 457-502. | Article | MR 850546 | Zbl 0679.14001

[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. | Article | MR 2119721 | Zbl 1064.11055

[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. | Article | MR 1059938 | Zbl 0713.11082

[7] Y. André - Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Société Mathématique de France, 2004. | MR 2115000 | Zbl 1060.14001

[8] R. M. Beals & D. S. Thakur - Computational classification of numbers and algebraic properties, Internat. Math. Res. Notices 15 (1998), p. 799-818. | Article | MR 1639559 | Zbl 0916.03029

[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. | Article | MR 1278429 | Zbl 0834.11027

[10] V. Berthé - Combinaisons linéaires de ζ(𝑠)/Π 𝑠 sur 𝐅 q (x), pour 1𝑠q-2, J. Number Theory 53 (1995), p. 272-299. | MR 1348764 | Zbl 0853.11062

[11] D. Bertrand - Lemmes de zéros et nombres transcendants, Séminaire Bourbaki, vol. 1985/86, exp. n° 652, Astérisque 145-146 (1987), p. 21-44. | Numdam | MR 880024 | Zbl 0613.14001

[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 1176522 | Zbl 0785.12002

[13] V. Bosser & F. Pellarin - Hyperdifferential properties of Drinfeld quasimodular forms, article ID rnn032, 56 pages, doi : 10.1093/imrn/rnn032, 2008. | MR 2428858 | Zbl 1151.11021

[14] W. D. Brownawell - Transcendence in positive characteristic, in Number theory (Tiruchirapalli, 1996), Contemp. Math., vol. 210, Amer. Math. Soc., 1998, p. 317-332. | Article | MR 1478501 | Zbl 0893.11025

[15] W. D. Brownawell & M. Papanikolas - Linear independence of gamma values in positive characteristic, J. reine angew. Math. 549 (2002), p. 91-148. | MR 1916653 | Zbl 1002.11051

[16] L. Carlitz - On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), p. 137-168. | Article | JFM 61.0127.01 | MR 1545872

[17] L. Carlitz, An analogue of the von Staudt-Clausen theorem, Duke Math. J. 3 (1937), p. 503-517. | Article | JFM 63.0879.03 | MR 1546006

[18] L. Carlitz, A class of polynomials, Trans. Amer. Math. Soc. 43 (1938), p. 167-182. | Article | JFM 64.0093.01 | MR 1501937

[19] L. Carlitz, Some spécial functions over G F ( q , x ) , Duke Math. J. 27 (1960), p. 139-158. | Article | MR 206348 | Zbl 0097.25904

[20] C.-Y. Chang & J. Yu - Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), p. 321-345. | Article | MR 2353259 | Zbl 1123.11025

[21] G. Christol - Ensembles presque périodiques k-reconnaissables, Theoret. Comput. Sci. 9 (1979), p. 141-145. | Article | MR 535129 | Zbl 0402.68044

[22] G. Christol, T. Kamae, M. Mendès France & G. Rauzy - Suites algébriques, automates et substitutions, Bull. Soc. Math. France 108 (1980), p. 401-419. | Article | Numdam | MR 614317 | Zbl 0472.10035

[23] A. Cobham - Uniform tag sequences, Math. Systems Theory 6 (1972), p. 164-192. | Article | MR 457011 | Zbl 0253.02029

[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. | Article | MR 1133556 | Zbl 0743.11070

[25] S. David & L. Denis - Indépendance algébrique sur les T-modules, Compositio Math. 122 (2000), p. 1-22. | Article | MR 1771450 | Zbl 0955.11024

[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 1906421 | Zbl 1008.11021

[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 546622 | Zbl 0449.10022

[28] L. Denis - Théorème de Baker et modules de Drinfel'd, J. Number Theory 43 (1993), p. 203-215. | Article | MR 1207500 | Zbl 0767.11029

[29] L. Denis, Dérivées d'un module de Drinfel'd et transcendance, Duke Math. J. 80 (1995), p. 1-13. | Article | MR 1360609 | Zbl 0848.11028

[30] L. Denis, Indépendance algébrique et exponentielle de Carlitz, Acta Arith. 69 (1995), p. 75-89. | Article | MR 1310843 | Zbl 0823.11032

[31] L. Denis, Lemmes de multiplicités et T-modules, Michigan Math. J. 43 (1996), p. 67-79. | Article | MR 1381600 | Zbl 0853.11050

[32] L. Denis, Un critère de transcendance en caractéristique finie, J. Algebra 182 (1996), p. 522-533. | Article | MR 1391598 | Zbl 0857.11029

[33] L. Denis, Indépendance algébrique de différents π, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), p. 711-714. | Article | MR 1660017 | Zbl 0922.11055

[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. | Article | MR 1767389 | Zbl 0961.11020

[35] L. Denis, Indépendance algébrique de logarithmes en caractéristique p , Bull. Austral. Math. Soc. 74 (2006), p. 461-470. | Article | MR 2273754 | Zbl 1116.11058

[36] S. Dion - Théorème de Brownawell-Waldschmidt en caractéristique finie, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), p. 71-90. | Article | Numdam | MR 1815941 | Zbl 1165.11327

[37] J. Fresnel & M. Van Der Put - Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser, 2004. | MR 2014891 | Zbl 1096.14014

[38] E.-U. Gekeler - On the de Rham isomorphism for Drinfel'd modules, J. reine angew. Math. 401 (1989), p. 188-208. | MR 1018059 | Zbl 0672.14011

[39] E.-U. Gekeler, Quasi-periodic functions and Drinfel'd modular forms, Compositio Math. 69 (1989), p. 277-293. | Numdam | MR 985916 | Zbl 0703.11066

[40] D. Goss - von Staudt for 𝐅 q [T], Duke Math. J. 45 (1978), p. 885-910. | Article | MR 518110 | Zbl 0404.12013

[41] D. Goss, υ-adic zeta functions, L-series and measures for function fields, Invent. Math. 55 (1979), p. 107-119. | Article | MR 553704 | Zbl 0402.12006

[42] D. Goss, The Γ-function in the arithmetic of function fields, Duke Math. J. 56 (1988), p. 163-191. | Article | MR 932861 | Zbl 0661.12006

[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 1265558 | Zbl 0827.11035

[44] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 35, Springer, 1996. | MR 1423131 | Zbl 0874.11004

[45] Y. Hamahata - Tensor products of Drinfel'd modules and υ-adic représentations, Manuscripta Math. 79 (1993), p. 307-327. | Article | MR 1223025 | Zbl 0797.11055

[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. | Article | MR 2422340 | Zbl 1183.39005

[48] D. R. Hayes - Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), p. 77-91. | Article | MR 330106 | Zbl 0292.12018

[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. | Numdam | MR 1685616 | Zbl 0934.11034

[50] K. S. Kedlaya - Finite automata and algebraic extensions of function fields, J. Théor. Nombres Bordeaux 18 (2006), p. 379-420. | Article | Numdam | MR 2289431 | Zbl 1161.11317

[51] B. De Mathan - Irrationality measures and transcendence in positive characteristic, J. Number Theory 54 (1995), p. 93-112. | Article | MR 1352639 | Zbl 0832.11022

[52] M. Mendès France & J.-Y. Yao - Transcendence and the Carlitz-Goss gamma function, J. Number Theory 63 (1997), p. 396-402. | Article | MR 1443770 | Zbl 0874.11042

[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 578857 | Zbl 0423.14007

[54] M. Papanikolas - Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), p. 123-174. | Article | MR 2358057 | Zbl 1235.11074

[55] P. Philippon - Variétés abéliennes et indépendance algébrique. I, Invent. Math. 70 (1982/83), p. 289-318. | Article | MR 683686 | Zbl 0503.10024

[56] P. Philippon, Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), p. 355-383. | Article | Numdam | MR 878242 | Zbl 0617.14001

[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 1181282 | Zbl 0781.13011

[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. | Article | MR 1453214 | Zbl 0901.11026

[59] P. Philippon, Indépendance algébrique et K-fonctions, J. reine angew. Math. 497 (1998), p. 1-15. | Article | MR 1617424 | Zbl 0887.11032

[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 2136011 | Zbl 1086.12001

[61] S. K. Sinha - Periods of t-motives and transcendence, Duke Math. J. 88 (1997), p. 465-535. | Article | MR 1455530 | Zbl 0887.11028

[62] D. S. Thakur - Gamma functions for function fields and Drinfel'd modules, Ann. of Math. (2) 134 (1991), p. 25-64. | Article | MR 1114607 | Zbl 0734.11036

[63] D. S. Thakur, Drinfel'd modules and arithmetic in the function fields, Internat. Math. Res. Notices 9 (1992), p. 185-197. | Article | MR 1185833 | Zbl 0756.11015

[64] D. S. Thakur, Transcendence of gamma values for 𝐅 q [T], Ann. of Math. (2) 144 (1996), p. 181-188. | Article | MR 1405948 | Zbl 0861.11043

[65] D. S. Thakur, Automata and transcendence, in Number theory (Tiruchirapalli, 1996), Contemp. Math., vol. 210, Amer. Math. Soc., 1998, p. 387-399. | Article | Zbl 0897.11026

[66] D. S. Thakur, Function field arithmetic, World Scientific Publishing Co. Inc., 2004. | Zbl 1061.11001

[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 0798.11021

[68] A. Thiéry, Théorème de Lindemann-Weierstrass pour les modules de Drinfel'd, Compositio Math. 95 (1995), p. 1-42. | Numdam | Zbl 0839.11025

[69] R. Tubbs - Analytic subgroups of t-modules, Trans. Amer. Math. Soc. 349 (1997), p. 2605-2617. | Article | Zbl 0912.11027

[70] L. I. Wade - Certain quantities transcendental over GF(p n ,x), Duke Math. J. 8 (1941), p. 701-720. | Article | Zbl 0063.08101

[71] M. Waldschmidt - Nombres transcendants, Springer, 1974, Lecture Notes in Mathematics, Vol. 402. | Zbl 0302.10030

[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. | Article | Zbl 0505.10017

[73] M. Waldschmidt, Transcendence problems connected with Drinfel'd modules, İstanbul Üniv. Fen Fak. Mat. Derg. 49 (1990), p. 57-75 | Zbl 0802.11023

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 0802.11023

[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. | Numdam | Zbl 0908.11029

[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 0944.11024

[76] M. Waldschmidt, Un demi-siècle de transcendance, in Development of mathematics 1950-2000, Birkhäuser, 2000, p. 1121-1186. | Article | Zbl 0977.11030

[77] G. Wüstholz - Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Ann. of Math. (2) 129 (1989), p. 501-517. | Article | Zbl 0675.10025

[78] J. Yu - Transcendental numbers arising from Drinfel'd modules, Mathematika 30 (1983), p. 61-66. | Article | Zbl 0523.12013

[79] J. Yu, A six exponentials theorem in finite characteristic, Math. Ann. 272 (1985), p. 91-98. | Article | Zbl 0574.12014

[80] J. Yu, Transcendence and Drinfel'd modules, Invent. Math. 83 (1986), p. 507-517. | Article | Zbl 0586.12010

[81] J. Yu, Transcendence and Drinfeld modules : several variables, Duke Math. J. 58 (1989), p. 559-575. | Article | Zbl 0687.12008

[82] J. Yu, On periods and quasi-periods of Drinfel'd modules, Compositio Math. 74 (1990), p. 235-245. | Numdam | Zbl 0703.11067

[83] J. Yu, Transcendence and spécial zeta values in characteristic p, Ann. of Math. (2) 134 (1991), p. 1-23. | Article | Zbl 0734.11040

[84] J. Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), p. 215-233. | Article | Zbl 0881.11055