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 $\rho_{E,m}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \text{GL}{2}(\mathbb{Z}/m\mathbb{Z})$ 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.