I am looking for the reference for the following fact (used, for example, in the proof of theorem 4.4. in Breuil's expose about local-global compatibility at Bourbaki):
For $f$ a modular cuspidal form of weight $k \geq 2$, let $\rho _f$ be the associated Galos representation and let $$\pi \pi _p (\rho f _{f, |{Gal(\overline{\mathbb{Q}} _p/\mathbb{Q} _p)})$ p/\mathbb{Q}_p)})$ be the smooth representation associated to $\rho _f$ by the Local Langlands correspondance. Then $\rho _f {f, |_{Gal(\overline{\mathbb{Q}} Gal(\overline{\mathbb{Q}} _p/\mathbb{Q} _p)}$ is crystalline iff $ \pi _p (\rho f _{f, | {Gal(\overline{\mathbb{Q}} Gal(\overline{\mathbb{Q}} _p/\mathbb{Q} p / \mathbb{Q} _p)}) ^{GL_2(\mathbb{Z} _p)}$ is non-zero.
Additional question: does there exist a generalisation of this fact? For instance, to the totally real setting?
