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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.20175 |
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| _version_ | 1866913911925637120 |
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| author | Lakshmanan, K. |
| author_facet | Lakshmanan, K. |
| contents | We show that no arithmetic invariant strictly smaller than the conductor of an elliptic curve over \( \mathbb{Q} \) can appear in a functional equation governing the analytic continuation of an associated \( L \)-function of degree two. In particular, any attempt to define a modified \( L \)-function for an elliptic curve with a smaller invariant in place of the conductor leads to a contradiction with the Modularity Theorem. As a consequence, the classical upper bound \( \operatorname{rank}(E) \ll \log N_E \) is analytically optimal: no refinement replacing the conductor \( N_E \) with a smaller arithmetic quantity is possible. We further derive a conditional corollary: if a sub-conductor invariant were to govern the rank in an unbounded family of elliptic curves, the ranks must be unbounded - placing our results in connection with deep open questions concerning the distribution of ranks over \( \mathbb{Q} \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Minimality of the Conductor in Rank Bounds for Elliptic Curves Lakshmanan, K. Number Theory 11G05, 11G40 We show that no arithmetic invariant strictly smaller than the conductor of an elliptic curve over \( \mathbb{Q} \) can appear in a functional equation governing the analytic continuation of an associated \( L \)-function of degree two. In particular, any attempt to define a modified \( L \)-function for an elliptic curve with a smaller invariant in place of the conductor leads to a contradiction with the Modularity Theorem. As a consequence, the classical upper bound \( \operatorname{rank}(E) \ll \log N_E \) is analytically optimal: no refinement replacing the conductor \( N_E \) with a smaller arithmetic quantity is possible. We further derive a conditional corollary: if a sub-conductor invariant were to govern the rank in an unbounded family of elliptic curves, the ranks must be unbounded - placing our results in connection with deep open questions concerning the distribution of ranks over \( \mathbb{Q} \). |
| title | On the Minimality of the Conductor in Rank Bounds for Elliptic Curves |
| topic | Number Theory 11G05, 11G40 |
| url | https://arxiv.org/abs/2506.20175 |