## On Artin representations and nearly ordinary Hecke algebras over totally real fields

### Summary

Summary: We prove many new cases of the strong Artin conjecture for two-dimensional, totally odd, insoluble (icosahedral) representations $\mathrm{Gal}(\overline{F}/F)\rightarrow GL_2(\mathbf{C})$ of the absolute Galois group of a totally real field $F$.

### Mathematics Subject Classification

11G80, 11F33, 11F41, 14G22, 14G35

### Keywords/Phrases

Galois representations, p-adic modular forms, Hilbert modular varieties