Room 326

Points of small height in certain nonabelian extensions

S. Sahu

CMI

Let $E$ be an elliptic curve without complex
multiplication defined over a number field $K$ which has at least one real
embedding. The field $F$ generated by all torsion points of $E$ over $K$ is an
infinite, non-abelian Galois extension of the ground field which has unbounded, wild
ramification above all primes. Following the treatment in Habegger's paper titled
" Small Height and Infinite Nonabelian Extensions",
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 $E$ and $K$. We also show that
the N\'eron-Tate height has a similar gap on $E(F)$ .