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On fields with Property (B)

Authors: Francesco Amoroso, Sinnou David and Umberto Zannier
Journal: Proc. Amer. Math. Soc. 142 (2014), 1893-1910
MSC (2010): Primary 11G50; Secondary 12E30
Published electronically: March 3, 2014
MathSciNet review: 3182009
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Abstract: Let $ K$ be a number field and let $ L/K$ be an infinite Galois extension with Galois group $ G$. Let us assume that $ G/Z(G)$ has finite exponent. We show that $ L$ has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on $ L^*$ is bounded from below outside the set of roots of unity by an absolute constant. We also discuss some features of Property (B): stability by algebraic extensions and relations with field arithmetic. As a side result, we prove that the Galois group over $ \mathbb{Q}$ of the compositum of all totally real fields is torsion free.

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Francesco Amoroso
Affiliation: Laboratoire de mathématiques Nicolas Oresme, CNRS UMR 6139, Université de Caen, Campus II, BP 5186, 14032 Caen Cedex, France

Sinnou David
Affiliation: Institut de Mathématiques, CNRS UMR 7586, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05, France

Umberto Zannier
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri, 56126 Pisa, Italy

Received by editor(s): January 18, 2012
Received by editor(s) in revised form: July 4, 2012
Published electronically: March 3, 2014
Additional Notes: The first and second authors were partially supported by ANR “HaMoT”
The third author was partially supported by ERC “Diophantine Problems”
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society

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