A special point problem of André-Pink-Zannier in the universal family of abelian varieties

Gao, Ziyang

doi:10.2422/2036-2145.201510_013

Cited by 16 publications

(28 citation statements)

References 17 publications

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“…The proof of the next proposition is similar to the arguments appearing in Theorem 3.3, 5.3 and 5.7 of the preprint [23]. See also [11,Section 8] and [22], where it is explained that André-Pink(-Zannier) for mixed Shimura varieties implies Mordell-Lang ([22, Theorem 5.4]). We first fix some notations.…”

Section: The Conjecturementioning

confidence: 68%

“…For recent developments, using O-minimality, towards the André-Pink-Mordell-Lang we point out to the reader the main theorems of G. Dill, see [8]. See also the main theorems of [11]. Indeed, as mentioned in the introduction, Conjecture 5.4 formally follows from Gao's André-Pink-Zannier (3) .…”

Section: Mixed Shimura Varieties and The Zilber-pink Conjecturementioning

confidence: 94%

“…This approach relies on the Pila-Wilkie counting theorem, and it was used to prove both the Manin-Mumford and the André-Oort conjecture. For example Z. Gao, in [11], obtained important results towards what he calls the André-Pink-Zannier conjecture. After the paper was written, it was pointed out to the author that Gabriel Dill employed such strategy to obtain results about…”

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: The Conjecturementioning

confidence: 68%

Section: Mixed Shimura Varieties and The Zilber-pink Conjecturementioning

confidence: 94%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

On a conjecture of Buium and Poonen

Baldi¹

2020

Annales De l'Institut Fourier

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“…Compared to similar earlier results, a new aspect is that at once V is allowed to be of arbitrary dimension and Γ of arbitrary rank. So far, results have been obtained only in the cases when V is a curve (Dill [8], Gao [14], Lin-Wang [32]) or Γ contains only torsion points (Gao [14], Habegger [21], Pila [45]). See also [2] and [47] for related results.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 is an instance of the following conjecture, which was formulated in [8] as Conjecture 1.1. It is a slightly modified version of Gao's Conjecture 1.2 in [14], which he calls the André-Pink-Zannier conjecture, in the case of a base curve. The field K is now again arbitrary of characteristic 0 and we place no restrictions on A → S or A 0 .…”

Section: Introductionmentioning

confidence: 99%

Unlikely intersections with isogeny orbits in a product of elliptic schemes

Dill

2019

Preprint

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Fix an elliptic curve E0 without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of E g 0 , also defined over the algebraic numbers, under all isogenies between E g 0 and some fiber of the g-th fibered power A of the elliptic scheme, where g is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside A that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-Rémond inequality with the Pila-Zannier strategy. Contents 1. Introduction 1 2. Preliminaries and notation 5 3. Reduction to the non-degenerate case 6 4. Height bounds 9 5. Application of the Pila-Zannier strategy 26 6. Proof of Theorem 1.3 30 Acknowledgements 31 Appendix A. Generalized Vojta-Rémond inequality 32 References 33

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Unlikely intersections with isogeny orbits in a product of elliptic schemes

Dill

2020

Math. Ann.

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A special point problem of André-Pink-Zannier in the universal family of abelian varieties

Cited by 16 publications

References 17 publications

On a conjecture of Buium and Poonen

On a conjecture of Buium and Poonen

Unlikely intersections with isogeny orbits in a product of elliptic schemes

Unlikely intersections with isogeny orbits in a product of elliptic schemes

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