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

TL;DR

This paper investigates the rank of the modular curve X_0(49) over quadratic fields, linking it to solutions of quadratic forms and modular form coefficients, under the Birch and Swinnerton-Dyer conjecture.

Contribution

It establishes a criterion for positive rank over quadratic fields based on quadratic form solutions and modular form analysis, extending previous approaches.

Findings

Rank over quadratic fields relates to solutions of specific quadratic forms.

Application of Waldpurger's theorem connects L-functions to modular form coefficients.

Uses Ueda's decomposition to identify relevant half-integral weight modular forms.

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.