This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".


Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper modular curve of full level $N$ (over $\mathbb Q$, say), $f\colon\overline E(N)\rightarrow X(N)$ the universal generalized elliptic curve over it and by $e\colon X(N)\rightarrow E(N)$ the unit section. Further let $\omega=e^\*\Omega^1\_{E(N)/X(N)}$ and $\Omega^1=\Omega^1\_{X(N)}$. Let $Y(N)$ be the open modular curve and denote the universal elliptic curve over it still by $f$. Put $V=H^1\_{\mathrm{p}}(Y(N)\times\overline{\mathbb Q},\operatorname{Sym}^{k-2}R^1f\_\*\mathbb Z\_p)$, $W\_0 = H^0(X(N),\omega^{k-2}\otimes\Omega^1)$ and $W\_{k-1}=H^1(X(N),\omega^{2-k})$.


Faltings' $p$-adic Eichler-Shimura isomorphism, which is Thm. 6 (iii) in his paper, states that canonically
$$ \mathbb C\_p\otimes V(1)\cong \mathbb C\_p\otimes W\_0 \oplus \mathbb C\_p(k-1)\otimes W\_{k-1}. $$


It seems to be well-known that this Eichler-Shimura isomorphism "is" the $p$-adic comparison isomorphism for the modular motive ${}^N\_kW$ introduced by Scholl in his 1990 paper "Motives for Modular Forms".
In trying to make this statement precise, I obtained the following.


The motive introduced by Scholl which he calls ${}^N\_kW$ has $V$ as its $p$-adic étale realization and $W:=W\_0\oplus W\_{k-1}$ as its Hodge realization, with the $W\_i$ sitting in degree $i$, respectively.
The comparison isomorphism from $p$-adic Hodge theory gives
$$ B\_{\mathrm{HT}}\otimes V \cong B\_{\mathrm{HT}}\otimes W. $$
Taking the degree $0$ part of this one obtains an isomorphism
$$ \mathbb C\_p\otimes V \cong \mathbb C\_p\otimes W\_0 \oplus \mathbb C\_p(1-k)\otimes W\_{k-1}, $$
which differs from Faltings' isomorphism above.


What is the precise relation between Faltings' Eichler-Shimura isomorphism and the general comparison isomorphism? Since I am not familiar with the techniques used by Faltings, I was not able to prove anything about this by myself.



Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism?
The issue seems to be about notation. Your $k$ is what Faltings calls $k + 2$. And what Faltings calls $\underline{V}\_k$ is what you would call $V(k-1)$. (Possibly you confused Faltings' $\underline{V}\_k$ with his $V\_k$?) So when I write out Faltings' comparison isomorphism and then twist, I get (using your notation)
\begin{equation}
\mathbb{C}\_p \otimes V \simeq \mathbb{C}\_p \otimes W\_{k-1} \oplus \mathbb{C}\_p(1-k) \otimes W\_0, 
\end{equation}
as one wants. 


Faltings' modular forms paper seems to be a prelude (a special case even) of his proof of the general comparison isomorphism which you refer to. So it would be strange if the two comparison maps were not literally the same map. 


