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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  (with $=Q$) 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.

Autores: Gillibert, F.

Journal: International Journal of Number Theory

Journal Volume:

Journal Issue:

Journal Page: 1-17

Tipo de publicación: ISI

Fecha de publicación: 2016

Topics: Abelian surfaces, Galois representations, quaternionic multiplications, Shimura curves

URL de la publicación: http://www.worldscientific.com/doi/pdf/10.1142/S1793042117500725?src=recsys

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