Revisiting Manin’s theorem of the kernel
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 5, pp. 1301-1318.

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.

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.

Published online:
DOI: 10.5802/afst.1662
Classification: 14K05, 32G20, 11G10, 12H05, 34M55
Keywords: abelian varieties, Manin maps, Gauss–Manin connections, Mumford–Tate groups, Painlevé VI equations
Bertrand, Daniel 1

1 Sorbonne Université & UMR 7586 du CNRS, Institut de Mathématiques de Jussieu-PRG, Case 247, 75 252 Paris Cédex 05, France
@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, Serie 6, Volume 29 (2020) no. 5, pp. 1301-1318. doi : 10.5802/afst.1662. http://www.numdam.org/articles/10.5802/afst.1662/

[1] André, Yves 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] André, Yves 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] André, Yves; Corvaja, Pietro; Zannier, Umberto 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] Bertrand, Daniel Extensions de D-modules et groupes de Galois différentiels, p-adic analysis (Lecture Notes in Mathematics), Volume 1454, Springer, 1990, pp. 125-141 | DOI | Zbl

[5] Bertrand, Daniel Manin’s theorem of the kernel : a remark on a paper of C-L. Chai (2008) (unpublished, webusers.imj-prg.fr/~daniel.bertrand/)

[6] Bertrand, Daniel 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] Bertrand, Daniel; Pillay, Anand 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] Bertrand, Daniel; Pillay, Anand Galois theory, functional Lindemann-Weierstrass and Manin maps, Pac. J. Math., Volume 281 (2016) no. 1, pp. 51-82 | DOI | MR | Zbl

[9] Buium, Alexandru Differential algebra and diophantine geometry, Actualités Mathématiques, Hermann, 1994 | Zbl

[10] Casale, Guy The Galois groupoid of Picard–Painlevé VI equation, RIMS Kôkyûroku Bessatsu, Volume B2 (2007), pp. 15-20 | MR | Zbl

[11] Chai, Ching-Li Correction to [12] (available on www.math.upenn.edu/~chai/papers.html)

[12] Chai, Ching-Li A note on Manin’s theorem of the kernel, Am. J. Math., Volume 113 (1991) no. 3, pp. 387-389 | DOI | MR | Zbl

[13] Coleman, Robert F. Manin’s proof of the Mordell conjecture over function fields, Enseign. Math., Volume 36 (1990) no. 3-4, pp. 393-427 | MR | Zbl

[14] Deligne, Pierre Théorie de Hodge II, Publ. Math., Inst. Hautes Étud. Sci., Volume 40 (1971), pp. 5-58 | DOI | Numdam | Zbl

[15] Deligne, Pierre Théorie de Hodge III, Publ. Math., Inst. Hautes Étud. Sci., Volume 44 (1974), pp. 5-77 | DOI | Numdam | Zbl

[16] Faltings, Gerd Arakelov theorem for abelian varieties, Invent. Math., Volume 73 (1983), pp. 337-347 | DOI | MR

[17] Hardouin, Charlotte 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] Manin, Yu. I. 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] Manin, Yu. I. 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] Manin, Yu. I. Sixth Painlevé equation, universal elliptic curve, and mirror of 2 , Am. Math. Soc., Transl., Volume 186 (1998) no. 39, pp. 131-151 | Zbl

[21] Umemura, Hiroshi 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

Cited by Sources: