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Authors: Javanpeykar, Ariyan
Levin, Aaron
Title: Urata's theorem in the logarithmic case and applications to integral points
Online publication date: 2-Feb-2023
Year of first publication: 2022
Language: english
Abstract: Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata’s theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field 𝐾 with only finitely many𝐿,𝑇-points for any number field 𝐿∕𝐾and any finite set of finite places 𝑇 of 𝐿 has, in fact, only finitely many points in any given ℤ-finitely generated integral domain of characteristic zero. We use this latter result in combination with Green’s criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green’s criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over ℂ which is not hyperbolically embeddable.
DDC: 510 Mathematik
510 Mathematics
Institution: Johannes Gutenberg-Universität Mainz
Department: FB 08 Physik, Mathematik u. Informatik
Place: Mainz
Version: Published version
Publication type: Zeitschriftenaufsatz
Document type specification: Scientific article
License: CC BY-NC-ND
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Journal: Bulletin of the London Mathematical Society
Pages or article number: 1772
Publisher: John Wiley & Sons, Ltd
Publisher place: Oxford
Issue date: 2022
ISSN: 1469-2120
Publisher DOI: 10.1112/blms.12655
Appears in collections:DFG-491381577-H

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