Murmurations of Mestre-Nagao sums
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911814192726016 |
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| author | Bujanović, Zvonimir Kazalicki, Matija Novak, Lukas |
| author_facet | Bujanović, Zvonimir Kazalicki, Matija Novak, Lukas |
| contents | This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum
\[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}} \frac{a_p(E)\log{p}}{p}, \]
where $a_p(E)$ is defined as $p + 1 - \#E(\mathbb{F}_p)$ for an elliptic curve $E/\mathbb{Q}$ with good reduction at prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer conjecture.
Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves. Surprisingly, this suggests that in some cases, opting for a smaller value of $B$ yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of $[40\,000,45\,000]$, the rank classification based on $a_p$'s with $p < B = 3\,200$ produces better results compared to using $B = 50\,000$. This phenomenon finds partial explanation in the recent work of Zubrilina. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17626 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Murmurations of Mestre-Nagao sums Bujanović, Zvonimir Kazalicki, Matija Novak, Lukas Number Theory This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum \[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}} \frac{a_p(E)\log{p}}{p}, \] where $a_p(E)$ is defined as $p + 1 - \#E(\mathbb{F}_p)$ for an elliptic curve $E/\mathbb{Q}$ with good reduction at prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer conjecture. Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves. Surprisingly, this suggests that in some cases, opting for a smaller value of $B$ yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of $[40\,000,45\,000]$, the rank classification based on $a_p$'s with $p < B = 3\,200$ produces better results compared to using $B = 50\,000$. This phenomenon finds partial explanation in the recent work of Zubrilina. |
| title | Murmurations of Mestre-Nagao sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.17626 |