Small Height and Infinite Non-Abelian Extensions

Details Small Height and Infinite Non-Abelian Extensions Small Height and Infinite Non-Abelian Extensions

2011-09-27

arxiv.org/abs/1109.5859v1

Mathematics

Number Theory

Scientific paper

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ and let $F$ denote the field generated by all torsion points of $E$. If $E$ does not have complex multiplication then $F$ is an infinite non-abelian Galois extension of the rationals. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the N\'eron-Tate height has a similar gap on $E(F)$.

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