Generic rank of Betti map and unlikely intersections

Abstract

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.

Betti maps, Pell equation in polynomials and almost Belyi maps

“…In [CMZ18] the authors studied the rank of Betti maps associated to abelian surface schemes in the context of a relative Manin-Mumford problem. This work initiated more general investigations that led to [ACZ20], in which the authors conjectured a sufficient condition for the maximality of rk β, proving it under some quite general and natural hypotheses (see also [Gao20] for further results on this topic).…”

Uniformity in Mordell–Lang for curves

“…André, Corvaja, and Zannier [ACZ20] recently began the study of the maximal rank of the Betti map, especially the submersivity, using a slightly different definition. A full study of this maximal rank was realized in [Gao20a]. Closely related to the Betti map is the Betti form, a semi-positive (1, 1)-form on A an , which was first introduced in Mok [Mok91].…”

“…The generalization has two parts: generalizing the inequality itself under the non-degeneracy condition and generalizing the criterion of non-degenerate subvarieties. We execute the first part in the current paper while the second part was done by the second-named author in [Gao20a]. Let us explain the setup.…”

“…Note that [GH19, Theorem 1.4] is, up to some minor reduction, precisely Theorem 1.6 for dim S = 1 together with the criterion for X to be non-degenerate when dim S = 1. In general, the degeneracy behavior of X is fully studied in [Gao20a]. See [Gao20a, Theorem 1.1] for the criterion.…”

Corrigendum: Generic rank of Betti map and unlikely intersections

“…1. Theorem 1.3(ii) of [Gao20] should read Indeed, Theorem 1.3(ii) is proved by applying Theorem 10.1(ii) to , which says If is quasi-finite, so is , and hence .…”