Bi-algebraic geometry and the André-Oort
                    conjecture

Klingler, Bruno; Ullmo, Emmanuel; Yafaev, Andrei

doi:10.1090/pspum/097.2/01709

Cited by 30 publications

(33 citation statements)

References 48 publications

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“…The André-Oort conjecture has been proven when Sh 0 K (G, X ) is of abelian type, for instance for Sh 0 K (G, X ) = A g . We refer to [17] for a survey on the André-Oort conjecture. While the André-Oort conjecture is an "atypical intersection" statement in the sense of [29], Corollary 1.7 may be thought of as a "typical intersection" statement.…”

Section: Example 1: Subvarieties Of Shimura Varietiesmentioning

confidence: 99%

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On the closure of the Hodge locus of positive period dimension

Klingler

Otwinowska

2021

Invent. math.

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Given $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

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Section: Example 1: Subvarieties Of Shimura Varietiesmentioning

confidence: 99%

“…Let us start by recalling the general functional transcendence context of "bialgebraic geometry" (see [17], [15,Section 7]): Definition 4.16 A bi-algebraic structure on a connected complex algebraic variety S is a pair (D :S → X, ρ : π 1 (S) → Aut(X ))…”

Section: Bi-algebraic Geometry For (S V)mentioning

confidence: 99%

On the closure of the Hodge locus of positive period dimension

Klingler

Otwinowska

2021

Invent. math.

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“…Recall that we have notions of being special and weakly special for both subvarieties of Shimura and abelian varieties, in particular we denote by CM ⊂ S(Q) the subset of special points of S. In this section we combine the two as follows. For an overview about special subvarieties and the André-Oort conjecture, we refer the reader to [13].…”

Section: Mixed Shimura Varieties and The Zilber-pink Conjecturementioning

confidence: 99%

On a conjecture of Buium and Poonen

Baldi¹

2020

Annales De l'Institut Fourier

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Given a correspondence between a modular curve S and an elliptic curve A, we prove that the intersection of any finite-rank subgroup of A with the set of points on A corresponding to an isogeny class on S is finite. The question was proposed by A. Buium and B. Poonen in 2009. We follow the strategy proposed by the authors, using a result about the equidistribution of Hecke points on Shimura varieties and Serre's open image theorem. The result is an instance of the Zilber-Pink conjecture.Résumé. -Étant donnée une correspondance entre une courbe modulaire S et une courbe elliptique A, nous prouvons que l'intersection de tout sous-groupe de rang fini de A avec l'ensemble de points de A d'une classe isogénie sur S est fini. Le question a été posée par A. Buium et B. Poonen en 2009. Nous suivons la stratégie proposée par les auteurs, utilisant un résultat sur l'équidistribution des points de Hecke sur les variétés de Shimura et le théorème de l'image ouverte de Serre. Le résultat est un cas particulier de la conjecture de Zilber-Pink.

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“…We will be especially interested in the case that X is the universal cover of Y , where open domains such as H, the complex upper half-plane arise naturally. For a survey of recent developments in this area mainly centering on approaches using o-minimality, see [11]. Our approach, started in [5], is much different.…”

Section: Introductionmentioning

confidence: 99%