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

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

2005-06-03

arxiv.org/abs/math/0506070v1

Mathematics

Number Theory

Scientific paper

For a fixed odd prime p and a representation \rho of the absolute Galois group of Q into the projective group PGL(2,p), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize \rho through the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves 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 from Faltings' theorem some finiteness results on the number of quadratic Q-curves of a given degree N realizing \rho.

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