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A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields


Authors: Daniel Bertrand and Anand Pillay
Journal: J. Amer. Math. Soc. 23 (2010), 491-533
MSC (2000): Primary 12H05, 14K05, 03C60, 34M15, 11J95
DOI: https://doi.org/10.1090/S0894-0347-09-00653-5
Published electronically: December 2, 2009
MathSciNet review: 2601041
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a $ \mathbb{Q}$-linearly independent set of algebraic numbers are algebraically independent), replacing $ \mathbb{Q}^{alg}$ by $ \mathbb{C}(t)^{alg}$ and $ \mathbb{G}_{m}^{n}$ by a semi-abelian variety over $ \mathbb{C}(t)^{alg}$. Both the formulations of our results and the methods are differential algebraic in nature.


References [Enhancements On Off] (What's this?)

  • 1. Y. André: Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part; Compositio Math. 82, 1992, 1-24. MR 1154159 (93b:14026)
  • 2. J. Ax: Some topics in differential algebraic geometry I: Analytic subgroups of algebraic groups; Amer. J. Math. 94, 1972, 1195-1204. See also: On Schanuel's conjecture; Annals of Math. 93, 1971, 252-268. MR 0435088 (55:8050); MR 0277482 (43:3215)
  • 3. A. Bertapelle: Deligne's duality on the de Rham realizations of 1-motives; http://arxiv. org/abs/math.AG/0506344
  • 4. D. Bertrand: Schanuel's conjecture for non-isoconstant elliptic curves over function fields; Model Theory with Applications to Algebra and Analysis, vol. 1, eds. Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie, LMS Lecture Note Series 349, Cambridge University Press, 2008. MR 2441374
  • 5. D. Bertrand: Manin's theorem of the kernel: A remark on a paper of C-L. Chai; June 2008.
  • 6. M. Brion: Anti-affine algebraic groups; J. Algebra 321, 2009, 934-952. MR 2488561 (2009j:14059)
  • 7. D. Brownawell, K. Kubota: Algebraic independence of Weierstrass functions; Acta Arithm. 33, 1977, 111-149. MR 0444582 (56:2932)
  • 8. J-L. Brylinski: ``1-motifs'' et formes automorphes; Publ. Math. Univ. Paris VII, 15, 1983, 43-106. MR 723182 (85g:11047)
  • 9. A. Buium: Differential algebraic groups of finite dimension; Springer LN 1506, 1992. MR 1176753 (93i:12010)
  • 10. A. Buium: Differential Algebra and Diophantine Geometry; Herman, Paris, 1994. MR 1487891 (99k:12010)
  • 11. E. Cattani, P. Deligne, A. Kaplan: On the locus of Hodge classes, JAMS 8, 1995, 483-506. MR 1273413 (95e:14003)
  • 12. C-L. Chai: A note on Manin's theorem of the kernel; Amer. J. Math. 113, 1991, 387-389. MR 1109343 (93b:14036)
  • 13. R. Coleman: Manin's proof of the Mordell conjecture over functions fields; L'Ens. Math. 36, 1990, 393-427. MR 1096426 (92e:11069)
  • 14. R. Coleman: The universal vectorial biextension and $ p$-adic heights; Invent. Math. 103, 1991, 631-650. See also: Duality for the de Rham cohomology of an abelian scheme; Ann. Fourier 48, 1998, 1379-1393. MR 1091621 (92k:14021); MR 1662247 (2000j:14032)
  • 15. P. Deligne: Théorie de Hodge II & III; Publ. math. IHES 40, 1971, 5-57 & 44, 1974, 5-77. MR 0498551 (58:16653a); MR 0498552 (58:16653b)
  • 16. N. Feldman, Y. Nesterenko: Number Theory IV, eds. A. Parshin, I. Shafarevich, Encyclopedia of Mathematical Sciences, vol. 44, Springer Verlag, 1998. MR 1603604 (99a:11088a)
  • 17. E. Hrushovski: The Mordell-Lang conjecture for function fields, JAMS 9, 1996, 667-690. MR 1333294 (97h:11154)
  • 18. N. Katz, T. Oda: On the differentiation of De Rham cohomology classes with respect to parameters; J. Math. Kyoto Univ. 8, 1968, 199-213. MR 0237510 (38:5792)
  • 19. J. Kirby: The theory of exponential differential equations; Ph. D. thesis, Oxford, 2006.
  • 20. E. Kolchin: Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. MR 0568864 (58:27929)
  • 21. E. Kolchin: Differential Algebraic Groups, Academic Press, New York, 1985. MR 776230 (87i:12016)
  • 22. E. Kolchin: Algebraic groups and algebraic dependence, Amer. J. Math. 90, 1151-1164. MR 0240106 (39:1460)
  • 23. P. Kowalski, A. Pillay: Quantifier elimination for algebraic $ D$-groups, TAMS 358, 2006, 167-181. MR 2171228 (2006i:03051)
  • 24. B. Malgrange: Differential algebraic groups; Notes of an ICTP course, Alexandria, 2007. See also: Systèmes différentiels involutifs; Panoramas et Synthèses, vol. 19, SMF, 2005. MR 2187078 (2006m:58004)
  • 25. Y. Manin: Rational points of algebraic curves over function fields; Izv. AN SSSR Mat. 27, 1963, 1395-1440, or AMS Transl. 37, 1966, 189-234. MR 0157971 (28:1199)
  • 26. D. Marker: Manin kernels, Quaderni Math. vol 6, Napoli, 2000, 1-21. MR 1930680 (2003g:12009)
  • 27. D. Marker and A. Pillay: Differential galois theory III: Some inverse problems; Illinois Journal of Math. 41, 1997, 453-461. MR 1458184 (99m:12011)
  • 28. B. Mazur and W. Messing: Universal extensions and one dimensional crystalline cohomology; Springer LN 370, 1974. MR 0374150 (51:10350)
  • 29. A. Pillay: Some foundational questions concerning differential algebraic groups; Pacific J. of Math. 179, 1997, 179-200. MR 1452531 (98g:12008)
  • 30. A. Pillay: Differential algebraic groups and the number of countable differentially closed fields, in Model Theory of Fields, eds. D. Marker, M. Messmer, A. Pillay, Lecture Notes in Logic 5, 2nd edition, ASL, A.K. Peters, 2006. MR 2215060 (2006k:03063)
  • 31. A. Pillay: Algebraic $ D$-groups and differential Galois theory; Pacific J. Math. 216, 2004, 343-360. MR 2094550 (2005k:12007)
  • 32. A. Pillay, M. Ziegler: Jet spaces of varieties over differential and difference fields, Selecta Math. 9, 2003, 579-599. MR 2031753 (2004m:12011)
  • 33. J. Steenbrink, S. Zucker: Variations of mixed Hodge structures I; Invent. Math. 80, 1985, 489-542. MR 791673 (87h:32050a)

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Additional Information

Daniel Bertrand
Affiliation: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Case 247, 4, Place Jussieu, F-75252 Paris Cedex 05, France

Anand Pillay
Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

DOI: https://doi.org/10.1090/S0894-0347-09-00653-5
Keywords: Algebraic independence, differential algebraic groups, logarithmic derivatives, Gauss-Manin connections, differential Galois theory.
Received by editor(s): October 24, 2008
Published electronically: December 2, 2009
Additional Notes: The second author was supported by a Marie Curie Chair and an EPSRC grant
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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