On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves

We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely, ${\mathrm{GL}}_2({\mathbb{F}}_3),S{D}_{16},D_6,D_4$ and $C_2^2$. In the cyclotomic case, we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.