Details
| Original language | English |
|---|---|
| Pages (from-to) | 2469-2509 |
| Number of pages | 41 |
| Journal | Compositio Math. |
| Volume | 156 |
| Issue number | 12 |
| Publication status | Published - Dec 2020 |
Abstract
Keywords
- math.NT, math.AG, 11G10, 11G50, 14G25, 14K15, Abelian scheme, Betti rank, Betti map, unlikely intersections
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Compositio Math., Vol. 156, No. 12, 12.2020, p. 2469-2509.
Research output: Contribution to journal › Article › Research › peer review
}
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 -