The Breuil--Mezard conjecture for quaternion algebras

We formulate a version of the Breuil–Mezard conjecture for quaternion algebras, and show that it follows from the Breuil–Mezard conjecture for GL_2. In the course of the proof we establish a mod p analogue of the Jacquet–Langlands correspondence for representations of GL_2(k), k a finite field of characteristic p.