On the possible images of the mod ell representations associated to elliptic curves over Q

Consider a non-CM elliptic curve $E$ defined over $\mathbb{Q}$. For each prime $\mathrm{\ell }$, there is a representation $\rho_{E,\ell}: G \to GL_2($\mathbb{F}$_\ell)$thatdescribestheGaloisactiononthe$\ell$-torsion points of $E$, where $G$ is the absolute Galois group of $\mathbb{Q}$. A famous theorem of Serre says that ${\rho }_{E,\mathrm{\ell }}$ is surjective for all large enough $\mathrm{\ell }$. We will describe all known, and conjecturally all, pairs $(E,\mathrm{\ell })$ such that ${\rho }_{E,\mathrm{\ell }}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E\mathrm{/}\mathbb{Q}$, outputs the set of such exceptional primes $\mathrm{\ell }$ and describes all the groups ${\rho }_{E,\mathrm{\ell }}(G)$ up to conjugacy. Much of the paper is dedicated to computing various modular curves of genus $0$ with their morphisms to the $j$-line.