Generalised André–Pink–Zannier conjecture for Shimura varieties of abelian type

Rodolphe Richard; Andrei Yafaev

doi:10.5802/crmath.751

Research article - Algebraic geometry, Number theory

Generalised André–Pink–Zannier conjecture for Shimura varieties of abelian type

Rodolphe Richard ¹; Andrei Yafaev ¹

¹UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 873-878

Abstract (Original language)
Résumé

This note describes the results of [6]. The main result is the proof of the Generalised André–Pink–Zannier conjecture in Shimura varieties of abelian type. The core result is a lower bound, in terms of height functions defined in [7], for the sizes of Galois orbits of points in generalised Hecke orbits, which is unconditional for Shimura varieties of abelian type.

Dans cette note nous décrivons les résultats de [6]. Le résultat principal est la preuve de la Conjecture d’André–Pink–Zannier genéralisée pour les variétés de Shimura de type abélien. Le résultat central énonce des bornes inférieures, en termes des fontions hauteurs de [7], pour la taille des orbites galoisiennes dans une orbite de Hecke généralisée, qui sont inconditionnelles pour les variétés de Shimura de type abelien.

Received: 2022-12-21
Revised: 2023-11-17
Accepted: 2025-04-16
Published online: 2025-07-21

DOI: 10.5802/crmath.751

Classification: 03C64, 11G18, 11G50, 11F80, 14L30, 20G35, 15A16, 14G35
Keywords: Shimura varieties, Hecke orbits, Zilber–Pink, heights, Siegel sets, Mumford–Tate conjecture, adelic linear groups
Mots-clés : Variétés de Shimura, orbites de Hecke, Zilber–Pink, hauteurs, ensembles de Siegel, conjecture de Mumford–Tate, groupes linéaires adéliques

Author's affiliations:

Rodolphe Richard ¹ ; Andrei Yafaev ¹

¹ UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Generalised {Andr\'e{\textendash}Pink{\textendash}Zannier} conjecture for {Shimura} varieties of abelian type},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {873--878},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.751},
     language = {en},
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Rodolphe Richard; Andrei Yafaev. Generalised André–Pink–Zannier conjecture for Shimura varieties of abelian type. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 873-878. doi: 10.5802/crmath.751

References
Cited by

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[8] Rodolphe Richard; Andrei Yafaev On the Generalised André–Pink–Zannier conjecture (To appear in C. R. Math.)

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