Faltings's theorem on rational points on subvarieties of abelian varieties can be used to show that al but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^{1}$ or positive rank abelian varieties, we call these finitely many exceptions 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 aj only on the $j$-invariant of the isolated point.
This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.[-]