Counting elliptic curves with prescribed level structures over number fields

Journal of the London Mathematical Society, vol. 105, pp. 2415–2435

Abstract

Harron and Snowden (J.reine angew.Math.729 (2017), 151-170) counted the number of elliptic curves over ℚ up to height 𝑋 with torsion group 𝐺 for each possible torsion group 𝐺 over 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 ℙ(𝑤 0 , 𝑤 1 ) and the morphism 𝑋 𝐺 → 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 We prove our results by defining a size function on ℙ(𝑤 0 , 𝑤 1 ) following unpublished work of Deng (Preprint, https://arxiv.org/abs/ math/9812082), and working out how to count the number of points on ℙ(𝑤 0 , 𝑤 1 ) up to size 𝑋.

Authors 2

1. Peter Bruin corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut Universiteit Leiden The Netherlands

   Mathematisch Instituut, Universiteit Leiden, The Netherlands

2. Filip Najman

   University of Zagreb

   Affiliation as printed

   Faculty of Science Department of Mathematics University of Zagreb Croatia

   Faculty of Science, Department of Mathematics, University of Zagreb, Croatia