Generic rank of Betti map and unlikely intersections

authored by

Ziyang Gao

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 characteristic \(0\) 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 enough fibered power (if \(X\) satisfies some conditions): it is an important step to prove the bound for the number of rational points on curves [DGH20]. Another application is to answer a question of Andr\'e-Corvaja-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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Compositio Math.

Volume

156

Pages

2469-2509

No. of pages

41

Publication date

12.2020

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.48550/arXiv.1810.12929 (Access: Open)
https://doi.org/10.1112/S0010437X20007435 (Access: Closed)
https://doi.org/10.1112/S0010437X21007673 (Access: Open)