Points of bounded height in images of morphisms of weighted projective stacks: With applications to counting elliptic curves

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results for counting elliptic curves over number fields with prescribed level structures, including the cases of for , for , and for . In all cases we give an asymptotic with an expression for the leading coefficient, and in many cases we also give a power‐saving error term.