Differential Galois theories and transcendence

Daniel Bertrand[1]

  • [1] Université Paris VI Institut de Mathématiques Case 247 4, place Jussieu, Tour 45-46 75252 Paris Cedex 5 (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2773-2803
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Bertrand, Daniel. "Théories de Galois différentielles et transcendance." Annales de l’institut Fourier 59.7 (2009): 2773-2803. <http://eudml.org/doc/10471>.

@article{Bertrand2009,
abstract = {On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.},
affiliation = {Université Paris VI Institut de Mathématiques Case 247 4, place Jussieu, Tour 45-46 75252 Paris Cedex 5 (France)},
author = {Bertrand, Daniel},
journal = {Annales de l’institut Fourier},
keywords = {Differential Galois theory; algebraic independence; abelian varieties; Galois cohomology; Gauss-Manin connections; logarithmic derivatives},
language = {fre},
number = {7},
pages = {2773-2803},
publisher = {Association des Annales de l’institut Fourier},
title = {Théories de Galois différentielles et transcendance},
url = {http://eudml.org/doc/10471},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Bertrand, Daniel
TI - Théories de Galois différentielles et transcendance
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2773
EP - 2803
AB - On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.
LA - fre
KW - Differential Galois theory; algebraic independence; abelian varieties; Galois cohomology; Gauss-Manin connections; logarithmic derivatives
UR - http://eudml.org/doc/10471
ER -

References

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  1. Y. André, Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compo Math. 82 (1992), 1-24 Zbl0770.14003MR1154159
  2. Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), (2004), Société Mathématique de France Zbl1060.14001MR2115000
  3. J. Ax, On Schanuel’s conjecture, Annals of Maths 93 (1971), 252-268 Zbl0232.10026
  4. M. Bays, J. Kirby, A. Wilkie, A Schanuel property for exponentially transcendental powers Zbl1223.11093
  5. C. Bertolin, Le groupe de Mumford-Tate des 1-motifs, Ann. Inst. Fourier 52 (2002), 1041-1059 Zbl1001.14017MR1926672
  6. D. Bertrand, Extensions de D -modules et groupes de Galois différentiels, -adic analysis (Trento, 1989) 1454 (1990), 125-141, Springer, Berlin Zbl0732.13008MR1094849
  7. D. Bertrand, Manin’s theorem of the kernel : a remark on a paper of C-L. Chai, (2008) 
  8. D. Bertrand, Schanuel’s conjecture for non-isoconstant elliptic curves over function fields, Model theory with applications to algebra and analysis. Vol. 1 349 (2008), 41-62, Cambridge Univ. Press, Cambridge Zbl1215.14050MR2441374
  9. D. Bertrand, A. Pillay, A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields Zbl1276.12003
  10. A. Buium, P. Cassidy, Differential algebraic geometry and differential algebraic groups, Selected works of E. Kolchin (1999), 567-636, AMS 
  11. Alexandru Buium, Differential algebraic groups of finite dimension, (1992), Lecture Notes in Mathematics. 1506. Berlin etc. : Springer-Verlag. xv, 145 p. Zbl0756.14028MR1176753
  12. S. Cantat-F. Loray, Holomorphic dynamics, Painlevé VI equation and Character Varieties Zbl1204.34123
  13. Guy Casale, The Galois groupoid of Picard-Painlevé VI equation, Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies (2007), 15-20, Res. Inst. Math. Sci. (RIMS), Kyoto Zbl1219.34113MR2310018
  14. C.-L. Chai, A note on Manin’s theorem of the kernel, Amer. J. Maths 113 (1991), 387-389 Zbl0759.14017MR1109343
  15. Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), 5-57 Zbl0219.14007MR498551
  16. Pierre Deligne, Théorie de Hodge irrégulière ; I (1984)) ; II (2006), Correspondance Deligne-Malgrange-Ramis 5 (2007), Société Mathématique de France 
  17. C. Hardouin, M. Singer, Differential Galois theory of linear difference equations, Math. Ann. 342 (2008), 333-377 Zbl1163.12002MR2425146
  18. Marco Hien, Céline Roucairol, Integral representations for solutions of exponential Gauss-Manin systems, Bull. Soc. Math. France 136 (2008), 505-532 Zbl1171.32019MR2443034
  19. E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151-1164 Zbl0169.36701MR240106
  20. Piotr Kowalski, A note on a theorem of Ax, Ann. Pure Appl. Logic 156 (2008), 96-109 Zbl1155.03019MR2474444
  21. Bernard Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 38 (2001), 465-501, Enseignement Math., Geneva Zbl1033.32020MR1929336
  22. David Marker, Anand Pillay, Differential Galois theory. III. Some inverse problems, Illinois J. Math. 41 (1997), 453-461 Zbl0927.03065MR1458184
  23. Anand Pillay, Differential Galois theory. I, Illinois J. Math. 42 (1998), 678-699 Zbl0916.03028MR1649893
  24. Anand Pillay, Algebraic D -groups and differential Galois theory, Pacific J. Math. 216 (2004), 343-360 Zbl1093.12004MR2094550
  25. Jean-Pierre Serre, Cohomologie galoisienne, 5 (1994), Springer-Verlag, Berlin Zbl0812.12002MR1324577
  26. Hiroshi Umemura, Sur l’équivalence des théories de Galois différentielles générales, C. R. Math. Acad. Sci. Paris 346 (2008), 1155-1158 Zbl1204.12009MR2464256

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