For a fixed odd prime and a representation of the absolute Galois group of into the projective group , we provide the twisted modular curves whose rational points supply the quadratic -curves of degree prime to that realize through the Galois action on their -torsion modules. The modular curve to twist is either the fiber product of and or a certain quotient of Atkin-Lehner type, depending on the value of mod . 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 -curves of a given degree realizing .
Cite this article
Julio Fernández, A moduli approach to quadratic -curves realizing projective mod Galois representations. Rev. Mat. Iberoam. 24 (2008), no. 1, pp. 1–30DOI 10.4171/RMI/527