Isolated points on modular curves

Let C be an algebraic curve over a number field. Faltings’s theorem on rational points on subvarieties of abelian varieties implies that all algebraic points on C arise in algebraic families, with finitely many exceptions.  These exceptions are known as isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves.  We show that isolated points on X_1(n) push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the j-invariant of the isolated point.  This is joint work with A.
Bourdon, O. Ejder, Y. Liu, and F. Odumodu.