[Sur la conjecture de Pink sur les intersections exceptionnelles et les familles de variétés semi-abéliennes]
Le torseur de Poincaré d’une famille de Shimura de variétés abéliennes s’interprète à la fois comme une famille de variétés semi-abéliennes et comme une variété de Shimura mixte. Nous montrons que ses sous-variétés spéciales en ce deuxième sens ne peuvent pas toutes se décrire en termes de sous-schémas en groupes. Cela donne un contre-exemple à la conjecture de Manin-Mumford relative, mais témoigne aussi de la pertinence de la conjecture de Pink sur les intersections exceptionnelles dans les variétés de Shimura mixtes. L’essentiel de l’article porte sur les structures de Hodge mixtes, mais d’autres approches, de nature plus géométrique, sont aussi abordées.
The Poincaré torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the subgroup schemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformisation of the Poincaré torsor, but other, more geometric, approaches are also discussed.
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Keywords: Semi-abelian varieties, Poincaré biextensions, mixed Shimura varieties, Manin-Mumford conjecture, André-Oort conjecture, Zilber-Pink conjecture
Mot clés : Variétés semi-abéliennes, bi-extensions de Poincaré, variétés de Shimura mixtes, conjecture de Manin-Mumford, conjecture d’André-Oort, conjecture de Zilber-Pink
Daniel Bertrand 1 ; Bas Edixhoven 2
CC-BY 4.0
@article{JEP_2020__7__711_0,
author = {Daniel Bertrand and Bas Edixhoven},
title = {Pink{\textquoteright}s conjecture on unlikely intersections and families of semi-abelian varieties},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {711--742},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
doi = {10.5802/jep.126},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.126/}
}
TY - JOUR AU - Daniel Bertrand AU - Bas Edixhoven TI - Pink’s conjecture on unlikely intersections and families of semi-abelian varieties JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 711 EP - 742 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.126/ DO - 10.5802/jep.126 LA - en ID - JEP_2020__7__711_0 ER -
%0 Journal Article %A Daniel Bertrand %A Bas Edixhoven %T Pink’s conjecture on unlikely intersections and families of semi-abelian varieties %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 711-742 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.126/ %R 10.5802/jep.126 %G en %F JEP_2020__7__711_0
Daniel Bertrand; Bas Edixhoven. Pink’s conjecture on unlikely intersections and families of semi-abelian varieties. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 711-742. doi : 10.5802/jep.126. https://jep.centre-mersenne.org/articles/10.5802/jep.126/
[1] - “Special points and Poincaré bi-extensions”, 2011, with an appendix by Bas Edixhoven | arXiv
[2] - “Extensions panachées autoduales”, J. K-Theory 11 (2013) no. 2, p. 393-411 | DOI | Zbl
[3] - “Relative Manin-Mumford for semi-Abelian surfaces”, Proc. Edinburgh Math. Soc. (2) 59 (2016) no. 4, p. 837-875 | DOI | MR | Zbl
[4] - “Unlikely intersections in semiabelian surfaces”, Algebra Number Theory 13 (2019) no. 6, p. 1455-1473 | DOI | MR | Zbl
[5] - Néron models, Ergeb. Math. Grenzgeb. (3), vol. 21, Springer-Verlag, Berlin, 1990 | DOI | Zbl
[6] - “Biextensions alternées”, Compositio Math. 63 (1987) no. 1, p. 99-122 | Numdam | MR | Zbl
[7] - “Géométrie d’Arakelov et hauteurs canoniques sur des variétés semi-abéliennes”, Math. Ann. 314 (1999) no. 2, p. 381-401 | DOI | Zbl
[8] - “Théorie de Hodge. II”, Publ. Math. Inst. Hautes Études Sci. 40 (1971), p. 5-57 | DOI | Numdam | Zbl
[9] - “Théorie de Hodge. III”, Publ. Math. Inst. Hautes Études Sci. 44 (1974), p. 5-77 | DOI | Numdam | Zbl
[10] - “Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques”, in Automorphic forms, representations and -functions (Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, American Mathematical Society, Providence, RI, 1979, p. 247-289 | Zbl
[11] - Degeneration of abelian varieties, Ergeb. Math. Grenzgeb. (3), vol. 22, Springer-Verlag, Berlin, 1990 | DOI | MR | Zbl
[12] - “Conducteur, descente et pincement”, Bull. Soc. math. France 131 (2003) no. 4, p. 553-585 | DOI | Numdam | MR | Zbl
[13] - “A special point problem of André-Pink-Zannier in the universal family of Abelian varieties”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 17 (2017) no. 1, p. 231-266 | MR | Zbl
[14] - “Higher genus counterexamples to relative Manin-Mumford”, ALGANT Master Thesis, Univ. Leiden, July 2012, available at https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/howemaster.pdf
[15] - Elliptic curves, Graduate Texts in Math., vol. 111, Springer-Verlag, New York, 2004 | MR | Zbl
[16] - “Deficient points on extensions of abelian varieties by ”, J. Number Theory 25 (1987) no. 2, p. 133-151 | DOI
[17] - “Hodge loci and atypical intersections: conjectures”, 2017 | arXiv
[18] - “Bi-algebraic geometry and the André-Oort conjecture”, in Algebraic geometry (Salt Lake City, 2015), Proc. Sympos. Pure Math., vol. 97, American Mathematical Society, Providence, RI, 2018, p. 319-359
[19] - “Pink’s conjecture on semi-abelian varieties”, MSc. thesis, Univ. Leiden, September 2014, available at https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/lopuhaamaster.pdf
[20] - “Linearity properties of Shimura varieties. I”, J. Algebraic Geom. 7 (1998) no. 3, p. 539-567 | MR | Zbl
[21] - Pinceaux de variétés abéliennes, Astérisque, vol. 129, Société Mathématique de France, Paris, 1985 | Numdam | Zbl
[22] - Abelian varieties, Tata Institute of Fundamental Research Studies in Math., vol. 5, Hindustan Book Agency, New Delhi, 2008 | MR | Zbl
[23] - Arithmetical compactification of mixed Shimura varieties, Bonner Math. Schriften, vol. 209, Universität Bonn, Mathematisches Institut, Bonn, 1990, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn,1989, https://people.math.ethz.ch/~pink/dissertation.html | MR | Zbl
[24] - “A combination of the conjectures of Mordell-Lang and André-Oort”, in Geometric methods in algebra and number theory, Progress in Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, p. 251-282 | DOI | MR | Zbl
[25] - “A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang”, 2005, preprint, https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf | Zbl
[26] - “Cohomological realization of a family of -motives”, J. Number Theory 25 (1987) no. 2, p. 152-161 | DOI
[27] - Groupes algébriques et corps de classes, Publications de l’Institut Mathématique de l’Université de Nancago, vol. 7, Hermann, Paris, 1984 | Zbl
[28] - Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux (SGA 3.II), Lect. Notes in Math., vol. 152, Springer-Verlag, Berlin-New York, 1970, Séminaire de Géométrie Algébrique du Bois Marie 1962/64. Dirigé par M. Demazure et A. Grothendieck | Zbl
[29] - “The André-Oort conjecture for ”, Ann. of Math. (2) 187 (2018) no. 2, p. 379-390 | DOI | MR
[30] - Some problems of unlikely intersections in arithmetic and geometry, Annals of Math. Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012 | MR | Zbl
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