Generic rank of Betti map and unlikely intersections

verfasst von

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.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

Compositio Math.

Band

156

Seiten

2469-2509

Anzahl der Seiten

41

Publikationsdatum

12.2020

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.48550/arXiv.1810.12929 (Zugang: Offen)
https://doi.org/10.1112/S0010437X20007435 (Zugang: Geschlossen)
https://doi.org/10.1112/S0010437X21007673 (Zugang: Offen)