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

Abstract

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 $\Gamma(N)$ 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 an expression for the leading coefficient, and in many cases we also give a power-saving error term.