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

TL;DR

This paper develops asymptotic formulas for counting rational points of bounded height in the images of morphisms between weighted projective stacks, with applications to enumerating elliptic curves over number fields with specific level structures.

Contribution

It introduces a new framework for counting rational points in weighted projective stacks and applies it to obtain asymptotics for counting elliptic curves with prescribed level structures.

Findings

Asymptotic formulas with leading coefficients for rational points of bounded height.

Results include counts for elliptic curves with various level structures over number fields.

Many cases feature power-saving error terms.

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.