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Let A be an abelian variety defined over a number field K and let ĥ 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 ĥ(P ) ≥ C for all nontorsion points P ∈ A(K), where K is the maximal abelian extension of K.

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