Counting elliptic curves over $\mathbb{Q}$ with bounded naive height
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909657586466816 |
|---|---|
| author | Barquero-Sanchez, Adrian Mora-Mora, Daniel |
| author_facet | Barquero-Sanchez, Adrian Mora-Mora, Daniel |
| contents | In this paper, we give exact and asymptotic formulas for counting elliptic curves $ E_{A,B} \colon 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 $H_{α, β}(E_{A,B}) := \max\{ α|A|^3, βB^2 \}$, defined for arbitrary positive real numbers $α, β\in \mathbb{R}_{> 0}$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18874 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting elliptic curves over $\mathbb{Q}$ with bounded naive height Barquero-Sanchez, Adrian Mora-Mora, Daniel Number Theory Algebraic Geometry 11G05 (Primary) 11G15, 11N45 (Secondary) In this paper, we give exact and asymptotic formulas for counting elliptic curves $ E_{A,B} \colon 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 $H_{α, β}(E_{A,B}) := \max\{ α|A|^3, βB^2 \}$, defined for arbitrary positive real numbers $α, β\in \mathbb{R}_{> 0}$. 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. |
| title | Counting elliptic curves over $\mathbb{Q}$ with bounded naive height |
| topic | Number Theory Algebraic Geometry 11G05 (Primary) 11G15, 11N45 (Secondary) |
| url | https://arxiv.org/abs/2506.18874 |