Counting elliptic curves over $\mathbb{Q}$ with bounded naive height

Abstract

In this paper, we give exact and asymptotic formulas for counting elliptic curves $E_{A,B}:y^2=x^3+Ax+B$ with $A,B\in \mathbb{Z}$, ordered by naive height. We study the family of all such curves and also several natural subfamilies, including those with fixed $j$-invariant and those with complex multiplication (CM). In particular, we provide formulas for two commonly used normalizations of the naive height appearing in the literature: the calibrated naive height, defined by \[ H^{\mathrm{cal}}(E_{A,B}) := \max\{ 4|A|^3, 27B^2 \}, \] and the uncalibrated naive height, defined by \[ H^{\mathrm{ncal}}(E_{A,B}) := \max\{ |A|^3, B^2 \}. \] In fact, we prove our theorems with respect to the more general naive height $H_{\hat{I}\pm ,{\hat{I}}^2}(E_{A,B}):=\max \{\hat{I}\pm |A{|}^3,{\hat{I}}^2{B}^2\}$, defined for arbitrary positive real numbers $\hat{I}\pm ,{\hat{I}}^2\in {\mathbb{R}}_{>0}$. As part of our approach, we give a completely explicit parametrization of the set of curves $E_{A,B}$ with fixed $j$-invariant and bounded naive height, describing them as twists of the curve $E_{A_j,B_j}$ of minimal naive height for the given $j$-invariant. We also include tables comparing and verifying our theoretical predictions with exact counts obtained via exhaustive computer searches, and we compute data for CM elliptic curves of naive height up to $10^{30}$. Code in SageMath is provided to compute all exact and asymptotic formulas appearing in the paper.

27 pages, 7 tables, 1 figure; code available on GitHub to reproduce all computations