Fields generated by finite rank subgroups of ℚ¯∗

Abstract

Let [Formula: see text] be a finite rank subgroup of [Formula: see text]. We prove that the multiplicative group of the field generated by all elements in the divisible hull of [Formula: see text] is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond’s generalized Lehmer conjecture is satisfied.

Points de petite hauteur sur une variété semi‐abélienne de la forme Gmn×A$\mathbb {G}_m^n \times A$

“…Clairement, $$$$\begin{equation*} 0 \leqslant \inf \lbrace h(\gamma \alpha _{\phi (1)}), \gamma \in \langle b \rangle _\mathrm{sat}\rbrace = \inf \lbrace h(\gamma \alpha _{\phi (n)}), \gamma \in \langle b \rangle _\mathrm{sat}\rbrace \leqslant h(\alpha _{\phi (n)}). \end{equation*}$$$$Par le théorème des gendarmes, on en conclut que $$$$\begin{equation*} \inf \lbrace h(\gamma \alpha _{\phi (1)}), \gamma \in \langle b \rangle _\mathrm{sat}\rbrace =0, \end{equation*}$$$$ce qui ne peut se produire que si $$\alpha _{\phi (1)}\in \langle b \rangle _\mathrm{sat}$$ d'après [21, Lemma 2.4]. On a donc $$\beta _i= \alpha _{\phi (1)} \;\zeta (\phi (1)) \; b^{y_{\phi (1)}} \in \langle b\rangle _\mathrm{sat}$$.…”