A lower bound for the canonical height on abelian varieties over abelian extensions

Let A be an abelian variety defined over a number field K and let be the canonical height function on A( K) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A, K, L) > 0 such that (P ) C for all nontorsion points P A(K ab ), where K ab is the maximal abelian extension of K.