TY - JOUR

T1 - Generic rank of Betti map and unlikely intersections

AU - Gao, Ziyang

PY - 2020/12

Y1 - 2020/12

N2 - 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.

AB - 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.

KW - math.NT

KW - math.AG

KW - 11G10, 11G50, 14G25, 14K15

KW - Abelian scheme

KW - Betti rank

KW - Betti map

KW - unlikely intersections

UR - http://www.scopus.com/inward/record.url?scp=85099353378&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1810.12929

DO - 10.48550/arXiv.1810.12929

M3 - Article

VL - 156

SP - 2469

EP - 2509

JO - Compositio Math.

JF - Compositio Math.

IS - 12

ER -