Rational and elliptic parametrizations of \(\mathbb Q\)-curves
An elliptic curve \(E\) defined over \(\overline{\mathbb Q}\) is called \(\mathbb Q\)-curve if \(E\) and its Galois conjugates \(E^\sigma\) are isogenous over \(\mathbb Q\) for any \(\sigma\in \text{Gal} (\overline{\mathbb Q}/\mathbb Q)\). Let \(N\) be a square free positive integer and \(B(N)\) be the group generated by the Atkin-Lehner involutions of the modular curve \(X_0(N)\). Every non-cusp rational point of the curve \(X^*(N)= X_0(N)/B(N)\) lifts to \(X_0(N)\) giving \(\mathbb Q\)-curves defined over abelian extensions of type \((2,\dots, 2)\). In this paper the authors first determine all square-free values of \(N\) such that \(X_0(N)\) has genus \(g=0\) or 1. In case where \(g=0\), they describe the construction of a Hauptmodul on \(X^*(N)\) and obtain families of \(\mathbb Q\)-curves over quadratic, biquadratic and triquadratic extensions. As application of these results some parametric families of \(\mathbb Q\)-curves are given, obtained from the rational curves \(X^*(n)\), where \(n= 6,11,23,30\). In case where \(g=1\), they obtain explicit modular parametrizations of a reduced Neron model of \(X^*(N)\) and obtain families of \(\mathbb Q\)-curves parametrized by its Mordell-Weil group defined over quadratic, biquadratic, triquadratic and tetraquadratic extensions. Furthermore, they illustrate this method giving some examples from the curves \(X^*(n)\), where \(n= 37, 74, 82\).
scientific article; zbMATH DE number 1229760