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.
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.
Accepted:
Published online:
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
Christopher Daw 1; Martin Orr 2
@article{JEP_2023__10__1097_0, author = {Christopher Daw and Martin Orr}, title = {Lattices with {skew-Hermitian} forms over division algebras and unlikely intersections}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1097--1156}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.240}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.240/} }
TY - JOUR AU - Christopher Daw AU - Martin Orr TI - Lattices with skew-Hermitian forms over division algebras and unlikely intersections JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 1097 EP - 1156 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.240/ DO - 10.5802/jep.240 LA - en ID - JEP_2023__10__1097_0 ER -
%0 Journal Article %A Christopher Daw %A Martin Orr %T Lattices with skew-Hermitian forms over division algebras and unlikely intersections %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 1097-1156 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.240/ %R 10.5802/jep.240 %G en %F JEP_2023__10__1097_0
Christopher Daw; Martin Orr. Lattices with skew-Hermitian forms over division algebras and unlikely intersections. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1097-1156. doi : 10.5802/jep.240. https://jep.centre-mersenne.org/articles/10.5802/jep.240/
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