A lower bound for the canonical height on elliptic curves over abelian extensions

Let E/K be an elliptic curve defined over a number field, let be the canonical height on E, and let Kab/K be the maximal abelian extension of K. Extending work of M. Baker (IMRN 29 (2003) 1571–1582), we prove that there is a constant C(E/K)>0 so that every nontorsion point P E(Kab) satisfies .