Watkins's conjecture for elliptic curves with a rational torsion
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909267298091008 |
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| author | Bhakta, Subham Krishnamoorthy, Srilakshmi |
| author_facet | Bhakta, Subham Krishnamoorthy, Srilakshmi |
| contents | Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $ν_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime $\ell$, we quantify the elliptic curves featuring a rational $\ell$-torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq ν_2(m_E)$ for any positive integer $M$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Watkins's conjecture for elliptic curves with a rational torsion Bhakta, Subham Krishnamoorthy, Srilakshmi Number Theory Primary 11F30, 11L07, Secondary 11F52, 11F80 Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $ν_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime $\ell$, we quantify the elliptic curves featuring a rational $\ell$-torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq ν_2(m_E)$ for any positive integer $M$. |
| title | Watkins's conjecture for elliptic curves with a rational torsion |
| topic | Number Theory Primary 11F30, 11L07, Secondary 11F52, 11F80 |
| url | https://arxiv.org/abs/2407.17680 |