Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913885937729536 |
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| author | Bujanović, Zvonimir Kazalicki, Matija Vlah, Domagoj |
| author_facet | Bujanović, Zvonimir Kazalicki, Matija Vlah, Domagoj |
| contents | Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums Bujanović, Zvonimir Kazalicki, Matija Vlah, Domagoj Number Theory Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods. |
| title | Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.07967 |