A moduli approach to quadratic \(\mathbb{Q}\)-curves realizing projective mod \(p\) Galois representations. (English) Zbl 1206.11077

An elliptic curve \(E\) is called \(\mathbb Q\)-curve if it is defined over \(\overline{\mathbb Q}\), it does not have complex multiplication and is isogenous to each of its conjugates \(^\sigma E\) for every \(\sigma\in G_{\mathbb Q}=\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\). A quadratic \(\mathbb Q\)-curve is defined over a quadratic number field. For a fixed odd prime \(p\) and a representation \(\rho\) of \(G_{\mathbb Q}\) into the projective group \(\text{PGL}_2({\mathbb F}_p)\) the author provides the twisted modular curves whose rational points supply the quadratic \(\mathbb Q\)-curves of degree \(N\) prime to \(p\) that realize \(\rho\) through the Galois action on their \(p\)-torsion modules. The modular curve to twist is either the fiber product of \(X_0(N)\) and \(X(p)\) or a certain quotient of Atkin-Lehner type, depending on the value of \(N\) modulo \(p\).


11G15 Complex multiplication and moduli of abelian varieties
14G35 Modular and Shimura varieties
11G05 Elliptic curves over global fields
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