Counting elliptic curves with prescribed level structures over number fields

TL;DR

This paper extends the counting of elliptic curves with specific level structures from rational numbers to general number fields, focusing on cases where the associated modular curve is a weighted projective line.

Contribution

It generalizes Harron and Snowden's results to all number fields and certain level structures, including all genus 0 modular curves X_1(m,n).

Findings

Established a size function on weighted projective lines for counting points.

Counted elliptic curves with prescribed level structures over number fields.

Included all modular curves X_1(m,n) with genus 0 in the analysis.

Abstract

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 We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.