Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields

Abstract: We prove that many representations $\overline{\rho} : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_2(\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p = 3$ is replaced by $p = 2$ or $p = 5$. As a consequence, we prove that a positive proportion of elliptic curves over any CM field not containing a 5th root of unity are modular.