Dans la première partie de ce texte, on établit au moyen de l’application de Manin un énoncé de finitude reliant les sections d’un schéma elliptique et les solutions des équations de Painlevé VI. Le reste de l’article concerne le théorème du noyau de Manin dans le cadre d’un schéma abélien sur une courbe, et passe en revue les divers énoncés connus sous cette appellation.
In the first part of the paper, we use Manin’s map to establish a finiteness result linking rational sections of an elliptic scheme and solutions of Painlevé VI equations. The rest of the paper concerns abelian schemes over curves, and presents a survey of the various statements encompassed by Manin’s theorem of the kernel.
Mots clés : abelian varieties, Manin maps, Gauss–Manin connections, Mumford–Tate groups, Painlevé VI equations
@article{AFST_2020_6_29_5_1301_0,
author = {Bertrand, Daniel},
title = {Revisiting {Manin{\textquoteright}s} theorem of the kernel},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1301--1318},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 29},
number = {5},
year = {2020},
doi = {10.5802/afst.1662},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1662/}
}
TY - JOUR AU - Bertrand, Daniel TI - Revisiting Manin’s theorem of the kernel JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2020 SP - 1301 EP - 1318 VL - 29 IS - 5 PB - Université Paul Sabatier, Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1662/ DO - 10.5802/afst.1662 LA - en ID - AFST_2020_6_29_5_1301_0 ER -
%0 Journal Article %A Bertrand, Daniel %T Revisiting Manin’s theorem of the kernel %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2020 %P 1301-1318 %V 29 %N 5 %I Université Paul Sabatier, Toulouse %U http://www.numdam.org/articles/10.5802/afst.1662/ %R 10.5802/afst.1662 %G en %F AFST_2020_6_29_5_1301_0
Bertrand, Daniel. Revisiting Manin’s theorem of the kernel. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 5, pp. 1301-1318. doi : 10.5802/afst.1662. http://www.numdam.org/articles/10.5802/afst.1662/
[1] Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math., Volume 82 (1992) no. 1, pp. 1-24 | Numdam | MR | Zbl
[2] Groupes de Galois motiviques et périodes, Séminaire Bourbaki. Volume 2015/2016 (Astérisque), Volume 390, Société Mathématique de France, 2015, pp. 1-26 (exposé n° 1104) | Zbl
[3] The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao) (2018) (https://arxiv.org/abs/1802.03204)
[4] Extensions de -modules et groupes de Galois différentiels, p-adic analysis (Lecture Notes in Mathematics), Volume 1454, Springer, 1990, pp. 125-141 | DOI | Zbl
[5] Manin’s theorem of the kernel : a remark on a paper of C-L. Chai (2008) (unpublished, webusers.imj-prg.fr/~daniel.bertrand/)
[6] Galois descent in Galois theories, Arithmetic and Galois theories of differential equations (Séminaires et Congrès), Volume 23, Société Mathématique de France, 2011, pp. 1-24 | MR | Zbl
[7] A Lindemann–Weierstrass Theorem for semiabelian varieties over function fields, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 491-533 | DOI | Zbl
[8] Galois theory, functional Lindemann-Weierstrass and Manin maps, Pac. J. Math., Volume 281 (2016) no. 1, pp. 51-82 | DOI | MR | Zbl
[9] Differential algebra and diophantine geometry, Actualités Mathématiques, Hermann, 1994 | Zbl
[10] The Galois groupoid of Picard–Painlevé VI equation, RIMS Kôkyûroku Bessatsu, Volume B2 (2007), pp. 15-20 | MR | Zbl
[11] Correction to [12] (available on www.math.upenn.edu/~chai/papers.html)
[12] A note on Manin’s theorem of the kernel, Am. J. Math., Volume 113 (1991) no. 3, pp. 387-389 | DOI | MR | Zbl
[13] Manin’s proof of the Mordell conjecture over function fields, Enseign. Math., Volume 36 (1990) no. 3-4, pp. 393-427 | MR | Zbl
[14] Théorie de Hodge II, Publ. Math., Inst. Hautes Étud. Sci., Volume 40 (1971), pp. 5-58 | DOI | Numdam | Zbl
[15] Théorie de Hodge III, Publ. Math., Inst. Hautes Étud. Sci., Volume 44 (1974), pp. 5-77 | DOI | Numdam | Zbl
[16] Arakelov theorem for abelian varieties, Invent. Math., Volume 73 (1983), pp. 337-347 | DOI | MR
[17] Unipotent radicals of Tannakian Galois groups in positive characteristic, Arithmetic and Galois theories of differential equations (Séminaires et Congrès), Volume 23, Société Mathématique de France, 2011, pp. 283-299 | MR | Zbl
[18] Rational points of algebraic curves over functional fields, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 27 (1963) no. 6, pp. 1395-1440 translation in Am. Math. Soc., Transl. 50 (1966), p. 189–234 | Zbl
[19] Letter to the editor, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 53 (1989) no. 2, pp. 447-448 translation in Math. USSR, Izv. 34 (1990), n° 2, p. 465-466 | Zbl
[20] Sixth Painlevé equation, universal elliptic curve, and mirror of , Am. Math. Soc., Transl., Volume 186 (1998) no. 39, pp. 131-151 | Zbl
[21] Galois theory and Painlevé equations, Théories asymptotiques et équations de Painlevé (Séminaires et Congrès), Volume 14, Société Mathématique de France, 2006, pp. 299-339 | MR | Zbl
Cité par Sources :
