Lattices with skew-Hermitian forms over division algebras and unlikely intersections

Christopher Daw; Martin Orr

doi:10.5802/jep.240

Lattices with skew-Hermitian forms over division algebras and unlikely intersections
[Réseaux munis de formes anti-hermitiennes sur des algèbres à division et intersections atypiques]

Christopher Daw ¹ ; Martin Orr ²

¹ Department of Mathematics and Statistics, University of Reading Whiteknights, PO Box 217, Reading, Berkshire RG6 6AH, United Kingdom
² Department of Mathematics, The University of Manchester Alan Turing Building, Oxford Road, Manchester M13 9PL, United Kingdom

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1097-1156.

Résumé
Abstract

Cet article a deux objectifs. Nous étudions d’abord les réseaux munis de formes anti-hermitiennes sur des algèbres à division avec involutions positives. Pour les algèbres à division de type I et II dans la classification d’Albert, nous montrons qu’un tel réseau contient une base « orthogonale » pour un sous-réseau d’indice borné de manière effective. Ensuite, nous appliquons ce résultat pour obtenir de nouveaux résultats dans la théorie des intersections atypiques. En particulier, nous prouvons la conjecture de Zilber–Pink pour l’intersection de courbes avec les sous-variétés spéciales de type PEL simple I et II en supposant la conjecture des grandes orbites galoisiennes vraie. De plus, nous prouvons cette conjecture sur les orbites galoisiennes dans certains cas de type II.

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an “orthogonal” basis for a sublattice of effectively bounded index. Second, we apply this result to obtain new results in the field of unlikely intersections. More specifically, we prove the Zilber–Pink conjecture for the intersection of curves with special subvarieties of simple PEL type I and II under a large Galois orbits conjecture. We also prove this Galois orbits conjecture for certain cases of type II.

Reçu le : 2021-11-25
Accepté le : 2023-05-10
Publié le : 2023-05-25

DOI : 10.5802/jep.240

Classification : 11E39, 11G18
Keywords: Division algebras, Hermitian forms, abelian varieties, Zilber–Pink conjecture, unlikely intersections
Mot clés : Algèbres à division, formes hermitiennes, variétés abéliennes, conjecture de Zilber–Pink, intersections atypiques

Affiliations des auteurs :

Christopher Daw ¹ ; Martin Orr ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Lattices with {skew-Hermitian} forms over division algebras and unlikely intersections},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Christopher Daw; Martin Orr. Lattices with skew-Hermitian forms over division algebras and unlikely intersections. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1097-1156. doi : 10.5802/jep.240. https://jep.centre-mersenne.org/articles/10.5802/jep.240/

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