On the Elliptic Curve $X_0(49)$ over Quadratic Extensions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908618932092928 |
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| author | Dombrowsky, Charlotte |
| author_facet | Dombrowsky, Charlotte |
| contents | We study the rank of the modular curve $X_0(49)$ over quadratic extensions. Assuming the Birch and Swinnerton-Dyer Conjecture, we show that the rank over $\mathbb{Q}(\sqrt{d})$ is positive if and only if the number of solutions of two explicit ternary quadratic forms is the same. Following the approach of Tunnell, we apply a theorem due to Waldpurger which relates twisted $L$-functions of integer weight modular forms to coefficients of half-integral weight modular forms. To find suitable functions of half-integral weight, we use a decomposition described by Ueda. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_25251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Elliptic Curve $X_0(49)$ over Quadratic Extensions Dombrowsky, Charlotte Number Theory We study the rank of the modular curve $X_0(49)$ over quadratic extensions. Assuming the Birch and Swinnerton-Dyer Conjecture, we show that the rank over $\mathbb{Q}(\sqrt{d})$ is positive if and only if the number of solutions of two explicit ternary quadratic forms is the same. Following the approach of Tunnell, we apply a theorem due to Waldpurger which relates twisted $L$-functions of integer weight modular forms to coefficients of half-integral weight modular forms. To find suitable functions of half-integral weight, we use a decomposition described by Ueda. |
| title | On the Elliptic Curve $X_0(49)$ over Quadratic Extensions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.25251 |