# Reference on a result on local Galois representation associated to classic modular form in p-adic Hodge theory

At the end of Fontaine’s rings and p-adic L-functions, P. Colmez states a Theorem 8.4.8 (click here) of Faltings-Tusji-Saito without references.

So I am wondering is there any references for this theorem?

The three articles referenced presented in logical order of exposition are respectively

Faltings, Gerd Hodge-Tate structures and modular forms Math. Ann. 278 (1987)

Tsuji, Takeshi $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137 (1999)

and

Saito, Takeshi Modular forms and $p$-adic Hodge theory. Invent. Math. 129 (1997)

• Thanks for your help! It seems that Faltings just proved that the represntation is Hodge-Tate (instead of de Rham, so as he did in his paper "p-adic Hodge Theory"), so did I missed anything or there is some further references? – Bonbon May 25 '17 at 18:31
• Can you see me?@Olivier – Bonbon May 26 '17 at 20:17
• @Bonbon It depends what you want to do. If you want to retrace the history of the subject, many other references can (and should) be mentioned. But if your aim is to understand a proof of the theorem, then you can read Falting's article (proving $\rho_f|G_{\mathbb Q_p}$ is Hodge-Tate), then Tsuji's (proving it is potentially semi-stable hence de Rham) and finally Saito's (proving the Weil-Deligne representation is the one attached to $\pi(f)_p$ by the Local Langlands Correspondence. – Olivier May 27 '17 at 7:45
• @Oliver Oh I got it ,grateful for your help!!! – Bonbon May 27 '17 at 14:11