Counting elliptic curves with prescribed level structures over number fields

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 We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.

20 pages, final version, to appear Journal of the LMS

References in corpus (1)

• Counting elliptic curves with a rational $N$-isogeny for small $N$

Cited by in corpus (3)

• Counting elliptic curves with a rational $N$-isogeny for small $N$
• Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves
• Points of bounded height on weighted projective spaces over global function fields