Boundedness of average rank of elliptic curves ordered by the coefficients
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910995166789632 |
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| author | Janbazi, Fatemehzahra |
| author_facet | Janbazi, Fatemehzahra |
| contents | We study the average rank of elliptic curves $E_{A,B} : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$, ordered by the height function $h(E_{A,B}) := \text{max}(|A|, |B|)$. Understanding this average rank requires estimating the number of irreducible integral binary quartic forms under the action of $\mathrm{GL}_2(\mathbb{Z})$, where the invariants $I$ and $J$ are bounded by $X$. A key challenge in this estimation arises from working within regions of the quartic form space that expand non-uniformly, with volume and projection of the same order. To address this, we develop a new technique for counting integral points in these regions, refining existing methods and overcoming the limitations of Davenport's lemma. This leads to a bound on the average size of the 2-Selmer group, yielding an upper bound of 1.5 for the average rank of elliptic curves ordered by $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness of average rank of elliptic curves ordered by the coefficients Janbazi, Fatemehzahra Number Theory We study the average rank of elliptic curves $E_{A,B} : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$, ordered by the height function $h(E_{A,B}) := \text{max}(|A|, |B|)$. Understanding this average rank requires estimating the number of irreducible integral binary quartic forms under the action of $\mathrm{GL}_2(\mathbb{Z})$, where the invariants $I$ and $J$ are bounded by $X$. A key challenge in this estimation arises from working within regions of the quartic form space that expand non-uniformly, with volume and projection of the same order. To address this, we develop a new technique for counting integral points in these regions, refining existing methods and overcoming the limitations of Davenport's lemma. This leads to a bound on the average size of the 2-Selmer group, yielding an upper bound of 1.5 for the average rank of elliptic curves ordered by $h$. |
| title | Boundedness of average rank of elliptic curves ordered by the coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.07089 |