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POINTS OF SMALL HEIGHT ON AFFINE VARIETIES DEFINED OVER FUNCTION FIELDS OF FINITE TRANSCENDENCE DEGREE

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  05 October 2020

DRAGOS GHIOCA Open the ORCID record for DRAGOS GHIOCA [Opens in a new window]  and
DAC-NHAN-TAM NGUYEN Open the ORCID record for DAC-NHAN-TAM NGUYEN [Opens in a new window]
DRAGOS GHIOCA*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
DAC-NHAN-TAM NGUYEN
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

Keywords

function fieldsWeil heightBogomolov conjecture

MSC classification

Primary: 11G50: Heights
Secondary: 11G10: Abelian varieties of dimension $> 1$ 14G17: Positive characteristic ground fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 3 , June 2021 , pp. 418 - 427
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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