Details
| Original language | English |
|---|---|
| Pages (from-to) | 1405-1456 |
| Number of pages | 52 |
| Journal | Compositio mathematica |
| Early online date | 30 Jun 2020 |
| Publication status | Published - 2020 |
Abstract
The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian -variety there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in. This conjecture has been shown by Habegger in the case where is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on. This allows us to demonstrate the conjecture for general semiabelian varieties.
Keywords
- 11G50, 14G40, 14K15, 2010 Mathematics Subject Classification
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Compositio mathematica, 2020, p. 1405-1456.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The bounded height conjecture for semiabelian varieties
AU - Kühne, Lars
PY - 2020
Y1 - 2020
N2 - The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian -variety there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in. This conjecture has been shown by Habegger in the case where is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on. This allows us to demonstrate the conjecture for general semiabelian varieties.
AB - The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian -variety there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in. This conjecture has been shown by Habegger in the case where is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on. This allows us to demonstrate the conjecture for general semiabelian varieties.
KW - 11G50
KW - 14G40
KW - 14K15
KW - 2010 Mathematics Subject Classification
UR - http://www.scopus.com/inward/record.url?scp=85088321512&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1703.03891
DO - 10.48550/arXiv.1703.03891
M3 - Article
AN - SCOPUS:85088321512
SP - 1405
EP - 1456
JO - Compositio mathematica
JF - Compositio mathematica
SN - 0010-437X
ER -