Relative Bogomolov extensions

Robert Grizzard¹

¹ Department of Mathematics University of Wisconsin-Madison 480 Lincoln Dr. Madison, WI 53706-1325, U.S.A.

Acta Arithmetica, Tome 170 (2015) no. 1, pp. 1-13.

Résumé

A subfield $K \subseteq \overline {\mathbb {Q}}$ has the Bogomolov property if there exists a positive $\varepsilon $ such that no non-torsion point of $K^\times $ has absolute logarithmic height below $\varepsilon $. We define a relative extension $L/K$ to be Bogomolov if this holds for points of $L^\times \setminus 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 $K$.

Détail

DOI : 10.4064/aa170-1-1

Mots-clés : subfield subseteq overline mathbb has bogomolov property there exists positive varepsilon non torsion point times has absolute logarithmic height below nbsp varepsilon define relative extension bogomolov holds points times setminus times construct various examples extensions which are bogomolov prove ramification criterion property extensions always constructed rational prime has bounded ramification index