Points of Small Heights in Certain Nonabelian Extensions

Abstract

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 'Small Height And Infinite Nonabelian Extensions' by P. Habegger 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 Néron-Tate height has a similar gap on $E(F)$. In appendix-A we have included some new results about Galois properties of division points of formal groups which are generalizations of results proved in chapter 2. This work is my M.Sc. thesis submitted to Chennai Mathematical Institute.

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