scholarly journals A Bogomolov property for curves modulo algebraic subgroups

2009 ◽  
Vol 137 (1) ◽  
pp. 93-125 ◽  
Author(s):  
Philipp Habegger
Keyword(s):  
Bogomolov Property

scholarly journals On the Northcott property and other properties related to polynomial mappings

2013 ◽  
Vol 155 (1) ◽  
pp. 1-12
Author(s):  
SARA CHECCOLI ◽  
MARTIN WIDMER
Keyword(s):  
Galois Extension ◽  
Solvable Groups ◽  
Galois Extensions ◽  
Open Problems ◽  
Polynomial Mappings ◽  
Bogomolov Property ◽  
Finite Exponent ◽  
Uniformly Bounded

AbstractWe prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

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