Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

Gao, Ziyang; Habegger, Philipp

doi:10.4007/annals.2019.189.2.3

Cited by 18 publications

(24 citation statements)

References 28 publications

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“…Betti map was first studied and used in [Zan12]. Then it was used to study the relative Manin–Mumford conjecture by Bertrand, Corvaja, Masser, Pillay, and Zannier in a series of works [MZ12, MZ14, MZ15, BMPZ16, CMZ18, MZ18], to prove the geometric Bogomolov conjecture over char by Gao and Habegger [GH19] and Cantat, Gao, Habegger and Xie [CGHX20] with (1.1) for , and to prove the denseness of torsion points on sections of Lagrangian fibrations by Voisin [Voi18] using André, Corvaja and Zannier's result [ACZ20] on (1.1) for .…”

Section: Introductionmentioning

confidence: 99%

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Generic rank of Betti map and unlikely intersections

Gao¹

2020

Compositio Math.

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Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$ . For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$ , the real torus of dimension $2g$ , by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

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Section: Introductionmentioning

confidence: 99%

“…Definition 1.5 is closely related to the generically special subvarieties defined in [GH19, Definition 1.2]. See Appendix A for some discussion.…”

Section: Introductionmentioning

confidence: 99%

Generic rank of Betti map and unlikely intersections

Gao¹

2020

Compositio Math.

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“…(vii) Betti coordinates also occur in a forthcoming work by Z. Gao and Ph. Habegger on the geometric Bogomolov conjecture [12].…”

Section: 3mentioning

confidence: 99%

The Betti map associated to a section of an abelian scheme

2020

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Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section).We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3, and study in detail the case of jacobians of families of hyperelliptic curves.Our main application, obtained in collaboration with Z. Gao, states that if A → S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ −1 Ators is dense in S(C).

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“…The second step already appeared in [14] and [8], but the final argument was based on Pila-Zannier's counting strategy. Here, we import ideas from dynamical systems, and in particular a result of Muchnik [20].…”

Section: 23mentioning

confidence: 99%

“…In characteristic 0, Cinkir had proved the geometric Bogomolov conjecture when X is a curve of arbitrary genus (see [3], and [7] when the genus is small). Recently, the second and the third-named authors [8] proved the conjecture in the case char k = 0 and dim B = 1. This last reference, as well as the present article, make use of the Betti map and its monodromy: the idea comes from [14], in which the third-named author gave a new proof of the conjecture in characteristic 0 when A is the power of an elliptic curve and dim B = 1.…”

Section: Introductionmentioning

confidence: 96%