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The bounded height conjecture for semiabelian varieties

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

Published online by Cambridge University Press:  30 June 2020

Lars Kühne
Lars Kühne*
Affiliation:
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland email lars.kuehne@unibas.ch
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Abstract

The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ 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 $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ 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 $G$. This allows us to demonstrate the conjecture for general semiabelian varieties.

MSC classification

Primary: 11G50: Heights
Secondary: 14K15: Arithmetic ground fields 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1405 - 1456
Copyright
© The Author 2020

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Footnotes

This work was supported by an Ambizione Grant of the Swiss National Science Foundation.

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