A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations

Abstract

For a fixed odd prime p and a representation \varrho of the absolute Galois group of \mathbb{Q} into the projective group {\rm PGL}_2(\mathbb{F}_p) , we provide the twisted modular curves whose rational points supply the quadratic \mathbb{Q} -curves of degree N prime to p that realize \varrho 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 mod p . For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphisms. By performing some genus computations, we obtain as a by-product some finiteness results on the number of quadratic \mathbb{Q} -curves of a given degree N realizing \varrho .