#### 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 $E(F)$ .

Done