Unlikely intersections with isogeny orbits in a product of elliptic schemes

Dill, Gabriel Andreas

doi:10.1007/s00208-020-02024-2

Cited by 6 publications

(7 citation statements)

References 41 publications

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“…Among the problems of unlikely intersections for families of abelian varieties, we shall also mention the recent result of Dill . Namely, Dill proved that, in the same setting of Theorem , given a fixed fibre

A_{0}

scriptA

and a finite rank subgroup

normalΓ

A_{0}

, there are at most finitely many points

c \in C (C)

that lie in the image of

normalΓ

under an isogeny, unless

scriptC

is contained in a translate of a torsion curve by a constant section of the constant part of

scriptA

, generalizing an earlier result of Gao .…”

Section: Introductionmentioning

confidence: 77%

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Barroero

Capuano

2019

Proc. London Math. Soc.

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Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme scriptA over S and a curve scriptC inside scriptA, both defined over k. In previous works, we proved that, when scriptA is a fibred product of elliptic schemes, if scriptC is not contained in a proper subgroup scheme of scriptA, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g⩾2. This, combined with the above‐mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes which has applications in the study of solvability of almost‐Pell equations in polynomials.

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A_{0}

scriptA

and a finite rank subgroup

normalΓ

A_{0}

, there are at most finitely many points

c \in C (C)

that lie in the image of

normalΓ

under an isogeny, unless

scriptC

is contained in a translate of a torsion curve by a constant section of the constant part of

scriptA

, generalizing an earlier result of Gao .…”

Section: Introductionmentioning

confidence: 77%

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Barroero

Capuano

2019

Proc. London Math. Soc.

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“…If everything is defined over , then the purely qualitative statement of ‘Manin–Mumford with isogenies’ is known in this case by [Hab13]. Furthermore, if the subvariety is a curve or the fixed abelian variety is a power of an elliptic curve without complex multiplication (CM), then the qualitative statement of ‘Mordell–Lang with isogenies’ is also known by [Dil20, Dil21]. The new features of the results we present here are their full or partial explicitness and sometimes their effectivity.…”

Section: Introductionmentioning

confidence: 99%

“…It is a consequence of Pink's more general Conjecture 1.6 in [Pin05a] on intersections of subvarieties of mixed Shimura varieties with generalized Hecke orbits. Special cases of (variants of) the André–Pink–Zannier conjecture and Conjecture 1.1 have been proven by Habegger in [Hab13], by Pila in [Pil14], by Lin and Wang in [LW15], by Gao in [Gao17a], and by the author in [Dil20, Dil21].…”

Section: Introductionmentioning

confidence: 99%

Torsion points on isogenous abelian varieties

Dill¹

2022

Compositio Math.

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Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

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“…In [4], we apply our generalized Vojta inequality to a relative version of the Mordell-Lang problem in an abelian scheme A π → S, where S is an irreducible variety and everything is defined over Q. In the problem, one fixes an abelian variety A 0 , defined over Q, a finite rank subgroup Γ ⊂ A 0 ( Q) and an irreducible closed subvariety V ⊂ A and studies the points p ∈ V of the form φ(γ) for an isogeny φ : A 0 → A π(p) , A π(p) denoting the fiber of the abelian scheme over π(p), and γ ∈ Γ.…”

Section: Introductionmentioning

confidence: 99%

“…1 really is a generalization of Rémond's work (up to the factor 2 and the slightly different definitions of Λ and c 1 ). In the application in [4], the fact that c (i) 3 depends only on h(X i ) and δ i and not on h(X j ) or δ j (j = i) is crucial. Ange's version of the inequality is therefore not sufficient for the application.…”

Section: Introductionmentioning

confidence: 99%