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Generic rank of Betti map and unlikely intersections

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  12 January 2021

Ziyang Gao
Ziyang Gao*
Affiliation:
CNRS, IMJ-PRG, 4 place Jussieu, 75005 Paris, France ziyang.gao@imj-prg.fr
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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.

Keywords

Abelian schemeBetti mapBetti rankunlikely intersections

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 11G50: Heights 14G25: Global ground fields 14K15: Arithmetic ground fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 12 , December 2020 , pp. 2469 - 2509
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Copyright
© The Author(s) 2021

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