A classification of curious Galois groups as direct products

Abstract

Let $N$ be a positive integer. Let $H$ be a group of level $N$ and let $E$ be an elliptic curve defined over the rationals with ${\text{j}}_E\ne 0,1728$. Then the image ${\overline{\rho }}_{E,N}\left(\mathrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$, of the mod-$N$ Galois representation attached to $E$, is conjugate to a subgroup of $H$ if and only if $E$ corresponds to a non-cuspidal rational point on the modular curve $X_H$ generated by $H$. In this article, we are interested when ${\overline{\rho }}_{E,N}(G_{\mathbb{Q}})$ is precisely $H$. More precisely, we classify all groups $H$ that are direct products of subgroups $H$ for which $X_H$ contains infinitely many non-cuspidal rational points but there is no elliptic curve $E/\mathbb{Q}$ such that ${\overline{\rho }}_{E,N}\left(\mathrm{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$ is conjugate to $H$.

30 pages, comments very welcome