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
.