Modular degree and a conjecture of Watkins
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911266865414144 |
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| author | Bhakta, Subham Krishnamoorthy, Srilakshmi Pasupulati, Sunil Kumar |
| author_facet | Bhakta, Subham Krishnamoorthy, Srilakshmi Pasupulati, Sunil Kumar |
| contents | Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a surjective morphism $ϕ_E: X_0(N) \to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{deg}(ϕ_E)$ and shed some light on Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))} \mid \mathrm{deg}(ϕ_E)$. Moreover, for any elliptic curve over $\mathbb{F}_q(T)$, we have an analogous modular parametrization relating to the Drinfeld modular curves. In this case, we also discuss growth and the divisibility properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_13708 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular degree and a conjecture of Watkins Bhakta, Subham Krishnamoorthy, Srilakshmi Pasupulati, Sunil Kumar Number Theory Primary 11F30, 11L07, Secondary 11F52, 11F80 Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a surjective morphism $ϕ_E: X_0(N) \to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{deg}(ϕ_E)$ and shed some light on Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))} \mid \mathrm{deg}(ϕ_E)$. Moreover, for any elliptic curve over $\mathbb{F}_q(T)$, we have an analogous modular parametrization relating to the Drinfeld modular curves. In this case, we also discuss growth and the divisibility properties. |
| title | Modular degree and a conjecture of Watkins |
| topic | Number Theory Primary 11F30, 11L07, Secondary 11F52, 11F80 |
| url | https://arxiv.org/abs/2308.13708 |