\(\mathbb{Q}\)-curves over quadratic fields (Q1382658)

According to Gross and Ribet, an elliptic curve is called a \(\mathbb{Q}\)-curve if it is isogenous to all its Galois conjugates. First using a result of Elkies that \(\mathbb{Q}\)-curves of degree \(D\) are parametrized by the rational modular curve \(X_0^*(D)\), the author constructs families of \(\mathbb{Q}\)-curves of degree \(p\) over \(\mathbb{Q}(\sqrt d)\) when \(X_0(p)\) has genus zero (i.e., \(p= 2,3,5,7,13\)). He further proves that every \(\mathbb{Q}\)-curve over \(\mathbb{Q}(\sqrt d)\) of degree \(p\) is the twist of a curve he constructs. Some condition has to be put on \(d\) when \(p=5\), and 13 (really necessary). Then he studies the isogenies between the curves and their Galois conjugates, in particular the definition field of those isogenies, which are certainly arithmetically interesting. Finally, he verifies that some of the \(\mathbb{Q}\)-curves he constructs are indeed modular, giving more evidence to a conjecture of Ribet, which says that every \(\mathbb{Q}\)-curve is modular. Relatedly, \textit{B. Roberts} and \textit{L. Washington} have given an algorithm to check modularity of \(\mathbb{Q}\)-curves over a quadratic field [Compos. Math. 111, 35-49 (1998)].