Effective open image theorem and a Linnik type problem for elliptic curves

Abstract

We study an effective open image theorem for families of elliptic curves and products of elliptic curves ordered by conductor. Unconditionally, we prove that for $100\%$ of pairs of elliptic curves, the largest prime $\ell$, for which the associated mod $\ell$ Galois representation fails to be surjective, is small. Additionally, for semistable families, our bound on $\ell$ is comparable to the result obtained under the Generalized Riemann Hypothesis. We reduce the problem to a Linnik type problem for modular forms and apply the zero density estimates. This method, together with an analysis of the local Galois representations of elliptic curves, allows us to show similar results for single elliptic curves.