A construction of everywhere good \(\mathbb{Q}\)-curves with \(p\)-isogeny
Let \(E\) be an elliptic curve defined over the algebraic closure \(\overline{\mathbb Q}\) of the field of rational numbers \(\mathbb Q\). We call \(E\) a \(\mathbb Q\)-curve if \(E\) and its Galois conjugate \(E^{\sigma}\) are isogenous over \(\overline{\mathbb Q}\) for any \(\sigma\in\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). It is conjectured by Ribet that \(\mathbb Q\)-curves are modular in the sense that each should be a factor over \(\overline{\mathbb Q}\) of the jacobian of the modular curve \(X_1(N)\) for some \(N\). In this paper, a \(\mathbb Q\)-curve \(E\) with the following properties is considered: For a real quadratic field \(K\) and a rational prime \(p\), (1) \(E\) is defined over \(K\), (2) \(E\) has everywhere good reduction over \(K\), and (3) \(E\) has an isogeny to its conjugate \(E^{\sigma}\) of degree \(p\). A method of constructing \(\mathbb Q\)-curves satisfying the above three properties is described. Then several examples of such \(\mathbb Q\)-curves are presented, and their modularity is established.
scientific article; zbMATH DE number 1207093