JournalsrmiVol. 24, No. 1pp. 1–30

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

  • Julio Fernández

    Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain
A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations cover
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Abstract

For a fixed odd prime pp and a representation ϱ\varrho of the absolute Galois group of Q\mathbb{Q} into the projective group PGL2(Fp){\rm PGL}_2(\mathbb{F}_p), we provide the twisted modular curves whose rational points supply the quadratic Q\mathbb{Q}-curves of degree NN prime to pp that realize ϱ\varrho through the Galois action on their pp-torsion modules. The modular curve to twist is either the fiber product of X0(N)X_0(N) and X(p)X(p) or a certain quotient of Atkin-Lehner type, depending on the value of NN mod pp. 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\mathbb{Q}-curves of a given degree NN realizing ϱ\varrho.

Cite this article

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

DOI 10.4171/RMI/527