A Bogomolov property for curves modulo subgroups of codimension 2.

Let X be a curve embedded in the algebraic torus G_m^n and assume X is not contained in the translate of a proper algebraic subgroup. Bombieri, Masser and Zannier showed that X contains only finitely many points which are in an algebraic subgroup of codimension 2. On the other hand Zhang proved that there are only
finitely many points on X of small height. In this talk we combine these two results and show that there exists an epsilon > 0 such that there are only finitely many points on X which are a product a.b where a is in an algebraic subgroup of
codimension 2 and the height of b is at most epsilon. The proof applies a recent lower bound for heights on varieties by Amoroso and David. We also discuss some potential applications to the abelian case.