Surjectivity of Galois representations associated with quadratic {\mathbb {Q}}-curves

We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM \({\mathbb {Q}}\)-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur’s method, isogeny theorems, Runge’s method and analytic estimates of sums of L-functions.