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

TL;DR

The paper analyzes the rank of the modular curve $X_0(49)$ over quadratic fields by leveraging the BSD conjecture and Waldspurger's theorem, which links central values of twists of a weight-2 modular form to coefficients of half-integral weight forms. Through Ueda's decomposition and Shimura lifts, the authors construct half-integral weight forms with the same Hecke eigenstructure as the weight-2 form associated to $E_1$, and express their coefficients via theta series of explicit ternary quadratic forms. They derive a concrete criterion: for positive, squarefree $d$ coprime to $7$, $E(\,\mathbb{Q}(\sqrt{d})\,)$ has non-torsion points iff the number of solutions to two specific ternary quadratic forms equal each other, under BSD. The work also notes symmetry with $E(\mathbb{Q}(\sqrt{-7d}))$ for all $d$, and provides a framework to obtain computable tests for ranks of quadratic twists using central $L$-values and representation numbers of ternary 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.

Theorems & Definitions (33)

• Theorem 1.1
• Conjecture 2.1: Birch and Swinnerton-Dyer
• Theorem 2.2: Gross-Zagier, Kolyvagin
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Remark 2.5
• Theorem 3.1: Waldspurger
• Proposition 3.2