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

TL;DR

This paper establishes an effective open image theorem for elliptic curves, showing that for most pairs, the largest prime where the Galois representation isn't surjective is small, using zero density estimates and Linnik type problems.

Contribution

It introduces a new approach linking open image theorems for elliptic curves to Linnik type problems for modular forms, providing unconditional bounds comparable to GRH assumptions.

Findings

For 100% of pairs, the largest prime with non-surjective Galois representation is small.

Unconditional bounds are comparable to GRH-based results for semistable families.

Results extend to single elliptic curves through analysis of local Galois representations.

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.