On the generalised André–Pink–Zannier conjecture.

Richard, Rodolphe; Yafaev, Andrei

Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1717-1722

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the “weakly adelic Mumford–Tate hypothesis” and prove the generalised André–Pink–Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila–Zannier strategy.

On introduit et étudie la notion d’orbite de Hecke généralisée dans une variété de Shimura. On définit une notion de hauteur sur une telle orbite et étudie ses properiétés. On obtient une borne inférieure pour la taille des orbites Galoisiennes de points dans une orbite de Hecke généralisee en termes de cette fonction hauteur en admettant « la conjecture de Mumford–Tate faiblement adélique » et on démontre la conjecture de André–Pink–Zannier généralisée (un cas particulier de la conjecture de Zilber-Pink) en utilisant la stratégie de Pila–Zannier.

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

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