Details
Original language | English |
---|---|
Pages (from-to) | 2469-2509 |
Number of pages | 41 |
Journal | Compositio Mathematica |
Volume | 156 |
Issue number | 12 |
Publication status | Published - Dec 2020 |
Abstract
Let A → S be an abelian scheme over an irreducible variety over C of relative dimension g. For any simply-connected subset Δ of S an one can define the Betti map from A Δ to T 2g, the real torus of dimension 2g, by identifying each closed fiber of A Δ → Δ with T 2g via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety X of 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
- 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 Mathematica, 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
N1 - Publisher Copyright: © Foundation Compositio Mathematica 2021.
PY - 2020/12
Y1 - 2020/12
N2 - Let A → S be an abelian scheme over an irreducible variety over C of relative dimension g. For any simply-connected subset Δ of S an one can define the Betti map from A Δ to T 2g, the real torus of dimension 2g, by identifying each closed fiber of A Δ → Δ with T 2g via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety X of 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.
AB - Let A → S be an abelian scheme over an irreducible variety over C of relative dimension g. For any simply-connected subset Δ of S an one can define the Betti map from A Δ to T 2g, the real torus of dimension 2g, by identifying each closed fiber of A Δ → Δ with T 2g via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety X of 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.
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 Mathematica
JF - Compositio Mathematica
SN - 1570-5846
IS - 12
ER -