Generic rank of Betti map and unlikely intersections

2020/12/01 by Ziyang Gao · 2 citations
Mathematics · Computer Science · #Tensor decomposition and applications #Polynomial and algebraic computation #Algebraic Geometry and Number Theory

paper · doi:10.1112/s0010437x20007435

Abstract

Let \mathcal A → S be an abelian scheme over an irreducible variety over \mathbb C of relative dimension g . For any simply-connected subset Δ of S^\mathrm an one can define the Betti map from \mathcal A_Δ to \mathbb T^2g , the real torus of dimension 2g , by identifying each closed fiber of \mathcal A_Δ → Δ 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.

Citations

• Courbes sur une vari�t� ab�lienne et points de torsion
• Torsion points on families of products of elliptic curves

Cited by

• Likely intersections in powers of the multiplicative group
• Singular intersections in families of abelian varieties

Related

• The Betti map associated to a section of an abelian scheme (with an\n appendix by Z. Gao)
• Intersection homology Betti numbers
• The Relative Manin-Mumford Conjecture
• Estimates for Betti numbers and relative Hermite-Minkowski theorem for perverse sheaves
• Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms