Generic rank of Betti map and unlikely intersections

Let A → S be an abelian scheme over an irreducible variety over C of relative dimension g. For any simply-connected subset Δ of S

^an one can define the Betti map from A

_Δ to T

^2g, the real torus of dimension 2g, by identifying each closed fiber of A

_Δ → Δ with T

^2g via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety X of 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.