The average number of integral points on the congruent number curves
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866913499450441728 |
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| author | Chan, Stephanie |
| author_facet | Chan, Stephanie |
| contents | We show that the total number of non-torsion integral points on the elliptic curves $\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\log N)^{-1/4+ε})$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the $2$-Selmer group of the curves in this family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_01615 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The average number of integral points on the congruent number curves Chan, Stephanie Number Theory 11G05, 11N45, 11R45 We show that the total number of non-torsion integral points on the elliptic curves $\mathcal{E}_D:y^2=x^3-D^2x$, where $D$ ranges over positive squarefree integers less than $N$, is $O( N(\log N)^{-1/4+ε})$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the $2$-Selmer group of the curves in this family. |
| title | The average number of integral points on the congruent number curves |
| topic | Number Theory 11G05, 11N45, 11R45 |
| url | https://arxiv.org/abs/2112.01615 |