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

Abstract

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 [Formula: see text]-type over [Formula: see text] with quaternionic multiplication (over [Formula: see text]) endowed with a [Formula: see text] level structure. We prove the following result: for a fixed indefinite quaternion algebra [Formula: see text] of discriminant [Formula: see text] and a fixed quadratic imaginary field [Formula: see text], there exists an effective bound [Formula: see text] such that for a prime number [Formula: see text], not dividing the conductor of the order [Formula: see text], there do not exist abelian surfaces [Formula: see text] such that [Formula: see text] is a maximal order of [Formula: see text] and [Formula: see text] is endowed with a [Formula: see text] level structure.