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 $\varrho$.