Counting elliptic curves with prescribed level structures over number fields

Abstract

Harron and Snowden (J. reine angew. Math. 729 (2017), 151-170) counted the number of elliptic curves over Q$\mathbb {Q}$ up to height X$X$ with torsion group G$G$ for each possible torsion group G$G$ over Q$\mathbb {Q}$. In this paper, we generalize their result to all number fields and all level structures G$G$ such that the corresponding modular curve XG$X_G$ is a weighted projective line P(w0,w1)$\mathbb {P}(w_0,w_1)$ and the morphism XG -> X(1)$X_G\rightarrow X(1)$ satisfies a certain condition. In particular, this includes all modular curves X1(m,n)$X_1(m,n)$ with coarse moduli space of genus 0. We prove our results by defining a size function on P(w0,w1)$\mathbb {P}(w_0,w_1)$ following unpublished work of Deng (Preprint, ), and working out how to count the number of points on P(w0,w1)$\mathbb {P}(w_0,w_1)$ up to size X$X$.