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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.08321 |
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| _version_ | 1866918096665575424 |
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| author | Hatley, Jeffrey Kundu, Debanjana |
| author_facet | Hatley, Jeffrey Kundu, Debanjana |
| contents | Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell--Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq \mathbb{Z}^r \oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins' Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins' Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins' conjecture Hatley, Jeffrey Kundu, Debanjana Number Theory Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell--Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq \mathbb{Z}^r \oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins' Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins' Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two. |
| title | Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins' conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.08321 |