Galois representations and composite moduli

Abstract

It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$ co-prime to $30,$ the induced Galois representation ${\rho }_{E,m}:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})⟶{\text{GL}}_2(\mathbb{Z}/m\mathbb{Z})$ is surjective if and only if ${\rho }_{E,\ell }$ is surjective for any prime $\ell |m.$ In this article, we shall discuss some generalizations, applications, and variants of this phenomenon.

arXiv commentThe title is modified, and the abstract is updated. A few arguments are rephrased