# Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism?

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.

• I added a top-level tag. These are those with two-letter prefix, corresponding to arXiv categories. Each question should ideally have (at least) one top-level tag. I hope I picked the right one. If not you can change it easily via an edit. – user9072 Feb 16 '16 at 19:32

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.
• Thanks for your answer! However, I am still confused. I indeed got confused about $\underline V_k$, but if I just copy Faltings' result and write it in my notation, then I get $\mathbb C_p\otimes V(k-1)\cong \mathbb C_p(k-1)\otimes W_{k-1}\oplus\mathbb C_p\otimes W_0$. So if I twist this $1-k$ times, I obtain $\mathbb C_p\otimes V\cong\mathbb C_p\otimes W_{k-1}\oplus\mathbb C_p(1-k)\otimes W_0$, so the $W_i$ are interchanged. – Michael Fütterer Feb 18 '16 at 15:56
• Yeah, I think that's because $W_{k-1}$ is actually in degree 0, and $W_0$ is in degree $k-1$. Think about the weight 2 case. Then $W_1 = H^1(X(N), \mathcal{O}) = F^0/F^1$ is in degree 0, not degree 1. – Ari Shnidman Feb 18 '16 at 16:18
• I edited my answer to include the $W_i$'s. – Ari Shnidman Feb 18 '16 at 16:24