In this paper, we study the level lowering problem for mod 2 representations of the absolute Galois group of a totally real field ${F}$. In the case ${F}=\Bbb Q$, this was done by Buzzard; here, we generalise some of Buzzard's results to higher weight and arbitrary totally real fields, using Rajaei's generalisation of Ribet's theorem and previous work of Fujiwara and the author.

11F33, 11F80, 11F41, 11G18, 14G35

Hilbert modular forms, totally real fields, Galois representations