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Asymptotics are given for the number of rational points in the domain of amorphism of weighted projective stacks whose images have bounded height andsatisfy a (possibly infinite) set of local conditions. As a consequence weobtain results for counting elliptic curves over number fields with prescribedlevel structures, including the cases of $\Gamma(N)$ for $N\in\{1,2,3,4,5\}$,$\Gamma_1(N)$ for $N\in\{1,2,\dots,10,12\}$, and $\Gamma_0(N)$ for$N\in\{1,2,4,6,8,9,12,16,18\}$. In all cases we give an asymptotic with anexpression for the leading coefficient, and in many cases we also give apower-saving error term.

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