Mod $p$ Hilbert modular forms of parallel weight one: the ramified case

We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at all places above $p$, then it arises from a mod $p$ Hilbert modular form of parallel weight one. This (mostly) resolves the weight one part of Serre’s conjecture for totally real fields.