The André-Pink conjecture : Hecke orbits and weakly special subvarieties
type: Dataset
description: The André-Pink conjecture predicts that a subvariety of a Shimura variety which has dense intersection with a Hecke orbit is weakly special. We prove this conjecture for curves in a Shimura variety of abelian type, as well as for certain cases for subvarieties of higher dimension. This is a special case of the Zilber-Pink conjecture. It generalises theorems of Edixhoven and Yafaev when the Hecke orbit consists of special points, of Pink when the Hecke orbit consists of Galois generic points, and of Habegger and Pila when the Shimura variety is a product of modular curves. Our proof of the André-Pink conjecture for curves in the moduli space of principally polarised abelian varieties is based on the Pila-Zannier method, using a strong form of the Pila-Wilkie counting theorem. The necessary Galois bounds are obtained from the Masser-Wüstholz isogeny theorem. In order to relate isogeny bounds to heights, we also prove various bounds concerning the arithmetic of Hermitian forms over the endomorphism ring of an abelian variety. In order to extend the result on the André-Pink conjecture to curves in Shimura varieties of abelian type and to some cases of higher-dimensional subvarieties, we study the functorial properties of Hecke orbits and variations thereof. One chapter concerns the ranks of Mumford-Tate groups of complex abelian varieties. We prove a lower bound for these ranks in terms of the dimension of the abelian variety, subject to the condition that the simple abelian subvarieties are pairwise non-isogenous.
landing page: https://theses.hal.science/tel-00879010
identifier: NNT: 2013PA112189
keyword: abelian-varieties
keyword: conjecture-de-zilber-pink
keyword: groupes-de-mumford-tate
keyword: infoeu-reposemanticsdoctoralthesis
keyword: isogenies
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: mumford-tate-groups
keyword: shimura-varieties
keyword: theses
keyword: varietes-abeliennes
keyword: varietes-de-shimura
keyword: zilber-pink-conjecture
issued: 2026-05-09T04:29:42.986335
modified: 2026-05-09T04:29:42.986340
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

The André-Pink conjecture : Hecke orbits and weakly special subvarieties
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00879010
issued: 2026-05-09T04:29:42.988685
modified: 2026-05-09T04:29:42.961181