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

Daw, Christopher; Orr, Martin

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

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.

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

[And89] André, Y. $G$-functions and geometry, Aspects of Math., E13, Friedr. Vieweg & Sohn, Braunschweig, 1989 | DOI

[AQ97] Anderson, G. D.; Qiu, S.-L. A monotoneity property of the gamma function, Proc. Amer. Math. Soc., Volume 125 (1997) no. 11, pp. 3355-3362 | DOI | MR | Zbl

[BG06] Bombieri, E.; Gubler, W. Heights in Diophantine geometry, New Math. Monographs, 4, Cambridge University Press, Cambridge, 2006 | DOI

[BHC62] Borel, A.; Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. of Math. (2), Volume 75 (1962), pp. 485-535 | DOI | MR | Zbl

[BL04] Birkenhake, C.; Lange, H. Complex abelian varieties, Grundlehren Math. Wissen., 302, Springer-Verlag, Berlin, 2004 | DOI

[BM21] Binyamini, G.; Masser, D. Effective André–Oort for non-compact curves in Hilbert modular varieties, Comptes Rendus Mathématique, Volume 359 (2021) no. 3, pp. 313-321 | DOI | Numdam | Zbl

[Cas97] Cassels, J. W. S. An introduction to the geometry of numbers, Classics in Math., Springer-Verlag, Berlin, 1997

[CR62] Curtis, C. W.; Reiner, I. Representation theory of finite groups and associative algebras, Pure and Applied Math., XI, Interscience Publishers, New York-London, 1962

[DCD00] Del Corso, I.; Dvornicich, R. Relations among discriminant, different, and conductor of an order, J. Algebra, Volume 224 (2000) no. 1, pp. 77-90 | DOI | MR | Zbl

[DO21] Daw, C.; Orr, M. Unlikely intersections with $E\times \mathrm{CM}$ curves in ${𝒜}_2$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 22 (2021) no. 4, pp. 1705-1745 | MR

[DO22] Daw, C.; Orr, M. Quantitative reduction theory and unlikely intersections, Internat. Math. Res. Notices (2022) no. 20, pp. 16138-16195 | DOI | Zbl | MR

[DR18] Daw, C.; Ren, J. Applications of the hyperbolic Ax–Schanuel conjecture, Compositio Math., Volume 154 (2018), pp. 1843-1888 | DOI | Zbl | MR

[Hal35] Hall, P. On representatives of subsets, J. London Math. Soc., Volume 10 (1935) no. 1, pp. 26-30 | DOI | MR | Zbl

[HJ85] Horn, R. A.; Johnson, C. R. Matrix analysis, Cambridge University Press, Cambridge, 1985 | DOI

[HP12] Habegger, P.; Pila, J. Some unlikely intersections beyond André–Oort, Compositio Math., Volume 148 (2012) no. 1, pp. 1-27 | DOI | Zbl

[HP16] Habegger, P.; Pila, J. O-minimality and certain atypical intersections, Ann. Sci. École Norm. Sup. (4), Volume 49 (2016) no. 4, pp. 813-858 | DOI | MR | Zbl

[KUY16] Klingler, B.; Ullmo, E.; Yafaev, A. The hyperbolic Ax-Lindemann-Weierstrass conjecture, Publ. Math. Inst. Hautes Études Sci., Volume 123 (2016) no. 1, pp. 333-360 | DOI | MR | Zbl

[Mil99] Milne, J. S. Lefschetz classes on abelian varieties, Duke Math. J., Volume 96 (1999) no. 3, pp. 639-675 | DOI | MR | Zbl

[Mil05] Milne, J. S. Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties (Clay Math. Proc.), Volume 4, American Mathematical Society, Providence, RI, 2005, pp. 265-378 | MR | Zbl

[Mum74] Mumford, D. Abelian varieties, T.I.F.R. Studies in Math., 5, Oxford University Press, 1974

[MW94] Masser, D. W.; Wüstholz, G. Endomorphism estimates for abelian varieties, Math. Z., Volume 215 (1994) no. 4, pp. 641-653 | DOI | Zbl | MR

[MW95] Masser, D.W.; Wüstholz, G. Factorization estimates for abelian varieties, Publ. Math. Inst. Hautes Études Sci., Volume 81 (1995), pp. 5-24 | Numdam | Zbl | DOI

[Orr15] Orr, M. Families of abelian varieties with many isogenous fibres, J. reine angew. Math., Volume 705 (2015), pp. 211-231 | DOI | MR | Zbl

[Orr18] Orr, M. Height bounds and the Siegel property, Algebra Number Theory, Volume 12 (2018) no. 2, pp. 455-478 | DOI | Zbl | MR

[Orr21] Orr, M. Unlikely intersections with Hecke translates of a special subvariety, J. Eur. Math. Soc. (JEMS), Volume 23 (2021) no. 1, pp. 1-28 | DOI | MR | Zbl

[Pin05] Pink, R. A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang (2005) http://www.math.ethz.ch/~pink/ftp/AOMMML.pdf (preprint, available at http://www.math.ethz.ch/~pink/ftp/AOMMML.pdf)

[PS10] Peterzil, Y.; Starchenko, S. Tame complex analysis and o-minimality, Proceedings of the ICM. Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 58-81 | Zbl

[Rei75] Reiner, I. Maximal orders, London Math. Soc. Monographs, 5, Academic Press, London-New York, 1975, xii+395 pages

[Shi63] Shimura, G. On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2), Volume 78 (1963), pp. 149-192 | DOI | Zbl | MR

[Ull14] Ullmo, E. Applications du théorème d’Ax-Lindemann hyperbolique, Compositio Math., Volume 150 (2014) no. 2, pp. 175-190 | DOI | Zbl

[Voi21] Voight, J. Quaternion algebras, Graduate Texts in Math., 288, Springer, Cham, 2021 | DOI

[Wey40] Weyl, H. Theory of reduction for arithmetical equivalence, Trans. Amer. Math. Soc., Volume 48 (1940), pp. 126-164 | DOI | MR | Zbl