Families of elliptic \(\mathbb{Q}\)-curves defined over number fields with large degrees (Q1130269)

The aim of this article is to give some explicit examples of families of elliptic \({\mathbb Q}\)-curves in the sense of K. Ribet, defined over \((2,\ldots,2)\)-extensions of \({\mathbb Q}\) (for instance of degree \(4\) and \(8\)). Explicit examples of \({\mathbb Q}\)-curves were constructed before by \textit{Y. Hasegawa} [Manuscr. Math. 94, 347-364 (1997; Zbl 0909.11017)], but they were defined only over quadratic extensions of \({\mathbb Q}\). The authors essentially apply a result which is at this time still unpublished, and which is due to \textit{N. Elkies} [Remark on elliptic \(K\)-curves (1993)], making a precise one-to-one correspondence between rational points of \(X_0^*(N)\) (the quotient of the modular curve \(X_0(N)\) by the Atkin-Lehner involution) and elliptic \({\mathbb Q}\)-curves of degree \(N\). They specialise, more precisely, to squarefree rational integers \(N\) such that \(X_0(N)\) is hyperelliptic in order to find explicit equations for \(X_0^*(N)\).