# A moduli approach to quadratic $Q$-curves realizing projective mod $p$ Galois representations

### Julio Fernández

Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain

## Abstract

For a fixed odd prime $p$ and a representation $ϱ$ of the absolute Galois group of $Q$ into the projective group $PGL_{2}(F_{p})$, we provide the twisted modular curves whose rational points supply the quadratic $Q$-curves of degree $N$ prime to $p$ that realize $ϱ$ 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 $Q$-curves of a given degree $N$ realizing $ϱ$.

## Cite this article

Julio Fernández, A moduli approach to quadratic $Q$-curves realizing projective mod $p$ Galois representations. Rev. Mat. Iberoam. 24 (2008), no. 1, pp. 1–30

DOI 10.4171/RMI/527