# Revista Matemática Iberoamericana

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**Volume 24, Issue 1, 2008, pp. 1–30**

**DOI: 10.4171/RMI/527**

Published online: 2008-04-30

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

Julio Fernández^{[1]}(1) Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain

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 $\varrho$.

*Keywords: *mod $p$ Galois representations, elliptic curves, $p$-torsion points, quadratic $\mathbb{Q}$-curves, twisted modular curves, moduli problem

Fernández Julio: A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations. *Rev. Mat. Iberoamericana* 24 (2008), 1-30. doi: 10.4171/RMI/527