Relative Bogomolov extensions

Robert Grizzard

Relative Bogomolov extensions

Robert Grizzard

Acta Arithmetica (2015)

Volume: 170, Issue: 1, page 1-13
ISSN: 0065-1036

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Abstract

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A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of

K^{\times}

has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of

L^{\times} ∖ K^{\times}

. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.

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Robert Grizzard. "Relative Bogomolov extensions." Acta Arithmetica 170.1 (2015): 1-13. <http://eudml.org/doc/286067>.

@article{RobertGrizzard2015,
abstract = {A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^\{×\}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^\{×\}∖K^\{×\}$. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.},
author = {Robert Grizzard},
journal = {Acta Arithmetica},
keywords = {Bogomolov property; heights; infinite algebraic extensions},
language = {eng},
number = {1},
pages = {1-13},
title = {Relative Bogomolov extensions},
url = {http://eudml.org/doc/286067},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Robert Grizzard
TI - Relative Bogomolov extensions
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 1
SP - 1
EP - 13
AB - A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{×}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{×}∖K^{×}$. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.
LA - eng
KW - Bogomolov property; heights; infinite algebraic extensions
UR - http://eudml.org/doc/286067
ER -

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