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

Abstract

We prove that many representations $\overline{\rho }:\mathrm{Gal}(\overline{K}/K)\to {\mathrm{GL}}_2({\mathbb{F}}_3)$, 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.

Minor changes and corrections in response to comments of the referee. This is the accepted manuscript

References in corpus (2)

• Modularity of nearly ordinary 2-adic residually dihedral Galois representations
• Local-global compatibility for regular algebraic cuspidal automorphic representation when $\ell \ne p$

Cited by in corpus (1)

• Modularity of ${\mathrm{PGL}}_2({\mathbb{F}}_p)$-representations over totally real fields