Faltings’s theorem on rational points on subvarieties of abelian varieties can be used to show that al butfinitely many algebraic points on a curve arise in families parametrized byP1or positive rank abelian va-rieties; we call these finitely many exceptions isolated points. We study how isolated points behave undermorphisms and then specialize to the case of modular curves. We show that isolated points onX1(n) pushdown to isolated points on a modular curve whose level is bounded by a constant that depends only on thej-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.