Surface abéliennes à multiplication quaternionique munie d’une structure de niveau Γ0(N)

Resumen

A theorem of Mazur gives the set of possible prime degrees for rational isogenies between elliptic curves. In this paper, we are working on a similar problem in the case of abelian surfaces of $GL_2-$ type over $\mathbb{Q}$ (with $\mathbb{Q}$ $=Q$ ) with quaternionic multiplication (over $\bar{\mathds{Q}}$ ) endowed with a $\Gamma_0(N)$ level structure. We prove the following result: for a fixed indefinite quaternion algebra $B_D$ of discriminant $D$ and a fixed quadratic imaginary field $K=\mathds{Q}(\sqrt{-d})$ , there exists an effective bound $M=O(lcm(d,D)^{22})$ such that for a prime number $N\geqslantM$ , not dividing the conductor of the order $End_{\mathds{Q}}(A)$ , there do not exist abelian surfaces $A/\mathds{Q}$ such that $End_{\mathds{K}}(A)=End_{\mathds{\bar{Q}}}(A)$ is a maximal order of $B_D$ and $A$ is endowed with a $\Gamma_0(N)$ level structure.