Théories de Galois différentielles et transcendance
[Differential Galois theories and transcendence]
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2773-2803.

We survey recent work on the exponential and logarithmic cases of the functional Schanuel conjecture. Using various differential Galois theories, we present parallel (and sometimes new) proofs in the case of abelian varieties.

On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.

DOI: 10.5802/aif.2507
Classification: 12H05, 14K05, 03C60, 34M15, 11J95
Mot clés : théorie de Galois différentielle, indépendance algébrique, variétés abéliennes, cohomologie galoisienne, connexion de Gauss-Manin, dérivées logarithmiques
Keywords: Differential Galois theory, algebraic independence, abelian varieties, Galois cohomology, Gauss-Manin connections, logarithmic derivatives

Bertrand, Daniel 1

1 Université Paris VI Institut de Mathématiques Case 247 4, place Jussieu, Tour 45-46 75252 Paris Cedex 5 (France)
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Bertrand, Daniel. Théories de Galois différentielles et transcendance. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2773-2803. doi : 10.5802/aif.2507. https://aif.centre-mersenne.org/articles/10.5802/aif.2507/

[1] André, Y. Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compo Math., Volume 82 (1992), pp. 1-24 | Numdam | MR | Zbl

[2] André, Y. Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, Société Mathématique de France, 2004 no. 17 | MR | Zbl

[3] Ax, J. On Schanuel’s conjecture, Annals of Maths, Volume 93 (1971), pp. 252-268 (Voir aussi : Some topics in differential algebraic geometry I ; Amer. J. Maths, 94, 1972, 1195-1204) | DOI | Zbl

[4] Bays, M.; Kirby, J.; Wilkie, A. A Schanuel property for exponentially transcendental powers (submitted. Voir aussi : arXiv :0810.4457)

[5] Bertolin, C. Le groupe de Mumford-Tate des 1-motifs, Ann. Inst. Fourier, Volume 52 (2002), pp. 1041-1059 | DOI | Numdam | MR | Zbl

[6] Bertrand, D. Extensions de D-modules et groupes de Galois différentiels, p -adic analysis (Trento, 1989) (Lecture Notes in Math.), Volume 1454, Springer, Berlin, 1990, pp. 125-141 | MR | Zbl

[7] Bertrand, D. Manin’s theorem of the kernel : a remark on a paper of C-L. Chai, 2008 (accessible sur http://www.math.jussieu.fr/~bertrand/)

[8] Bertrand, D. Schanuel’s conjecture for non-isoconstant elliptic curves over function fields, Model theory with applications to algebra and analysis. Vol. 1 (London Math. Soc. Lecture Note Ser.), Volume 349, Cambridge Univ. Press, Cambridge, 2008, pp. 41-62 | MR

[9] Bertrand, D.; Pillay, A. A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields (à paraître au J. Amer. Math. Soc. Voir aussi arXiv : AG.0810.0383)

[10] Buium, A.; Cassidy, P. Differential algebraic geometry and differential algebraic groups, Selected works of E. Kolchin, AMS, 1999, pp. 567-636

[11] Buium, Alexandru Differential algebraic groups of finite dimension, Lecture Notes in Mathematics. 1506. Berlin etc. : Springer-Verlag. xv, 145 p., 1992 | MR | Zbl

[12] Cantat-F. Loray, S. Holomorphic dynamics, Painlevé VI equation and Character Varieties (Voir hal-00186558)

[13] Casale, Guy The Galois groupoid of Picard-Painlevé VI equation, Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies (RIMS Kôkyûroku Bessatsu, B2), Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, pp. 15-20 | MR

[14] Chai, C.-L. A note on Manin’s theorem of the kernel, Amer. J. Maths, Volume 113 (1991), pp. 387-389 | DOI | MR | Zbl

[15] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 (Théorie de Hodge III ; Publ. Math. IHES, 44, 1974, 5–77) | DOI | Numdam | MR | Zbl

[16] Deligne, Pierre Théorie de Hodge irrégulière ; I (1984)) ; II (2006), Correspondance Deligne-Malgrange-Ramis (Documents mathématiques), Volume 5, Société Mathématique de France, 2007

[17] Hardouin, C.; Singer, M. Differential Galois theory of linear difference equations, Math. Ann., Volume 342 (2008), pp. 333-377 | DOI | MR | Zbl

[18] Hien, Marco; Roucairol, Céline Integral representations for solutions of exponential Gauss-Manin systems, Bull. Soc. Math. France, Volume 136 (2008) no. 4, pp. 505-532 | Numdam | MR | Zbl

[19] Kolchin, E. R. Algebraic groups and algebraic dependence, Amer. J. Math., Volume 90 (1968), pp. 1151-1164 | DOI | MR | Zbl

[20] Kowalski, Piotr A note on a theorem of Ax, Ann. Pure Appl. Logic, Volume 156 (2008) no. 1, pp. 96-109 | DOI | MR | Zbl

[21] Malgrange, Bernard Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 (Monogr. Enseign. Math.), Volume 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR | Zbl

[22] Marker, David; Pillay, Anand Differential Galois theory. III. Some inverse problems, Illinois J. Math., Volume 41 (1997) no. 3, pp. 453-461 | MR | Zbl

[23] Pillay, Anand Differential Galois theory. I, Illinois J. Math., Volume 42 (1998) no. 4, pp. 678-699 | MR | Zbl

[24] Pillay, Anand Algebraic D-groups and differential Galois theory, Pacific J. Math., Volume 216 (2004) no. 2, pp. 343-360 | DOI | MR | Zbl

[25] Serre, Jean-Pierre Cohomologie galoisienne, Lecture Notes in Mathematics, fifth ed., 5, Springer-Verlag, Berlin, 1994 | MR | Zbl

[26] Umemura, Hiroshi Sur l’équivalence des théories de Galois différentielles générales, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 21-22, pp. 1155-1158 | MR

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