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arXiv · 2510.25251

On the Elliptic Curve $X_0(49)$ over Quadratic Extensions

Charlotte Dombrowsky
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Abstract

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.

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Charlotte Dombrowsky. 2025-10-29. On the Elliptic Curve $X_0(49)$ over Quadratic Extensions. https://arxiv.org/abs/2510.25251

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