Counting elliptic curves with prescribed level structures over number fields

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 . In this paper, we generalize their result all fields level structures such that corresponding modular curve $X_G$ is a weighted projective line P ( w 0 , 1 ) {P}(w_0,w_1)$ morphism ? $X_G\rightarrow X(1)$ satisfies certain condition. particular, includes m n $X_1(m,n)$ coarse moduli space genus 0. We prove our results by defining size function on following unpublished work Deng (Preprint, https://arxiv.org/abs/math/9812082), working out how count points