Galois Images Parametrization Conjecture

Let $G_1,\ldots,G_n$ be open subgroups of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$, and for each $i$ let $T_{G_i}$ be its group of toric operators, $\overline{T}_{G_i}$ the induced group on coarse spaces, and $G_i^{\mathrm{ag}}$ its agreeable closure. For a point $x\in X_{G_i^{\mathrm{ag}}}(\mathbb{Q})$, let $\chi_x\colon \operatorname{Gal}_\mathbb{Q}\to\overline{T}_{G_i}$ describe the action on the fiber of $\pi_{G_i}\colon X_{G_i}\to X_{G_i^{\mathrm{ag}}}$.

Galois Images Parametrization Conjecture. There exist finitely many explicit open subgroups $G_1,\ldots,G_n$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ such that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$, there are an index $i$ and a point $x\in X_{G_i^{\mathrm{ag}}}(\mathbb{Q})$ with $j(x)=j(E)$ for which $G_E$ is conjugate to $G_i^\chi$, where $\chi\colon \operatorname{Gal}_\mathbb{Q}\to T_{G_i}$ lifts $\chi_x$.

Thus the groups $G_E$, as $E$ varies, are parameterized by rational points on finitely many modular curves. The conjecture gives a finite geometric framework for the possible adelic Galois images of non-CM elliptic curves over $\mathbb{Q}$; its resolution is not supplied here.