## de Rham structures and modular forms

In this post I want to use Scholzian techniques to look at the relationship between modular forms and the etale cohomology of modular curves. Since everyone cares about modular forms, this exercise might be a good way of getting exposure to some of the latest ideas in p-adic Hodge theory. The basic reference for this post is Scholze’s paper “p-adic Hodge theory for rigid analytic varieties”, and most of the things stated without proof below can be found there (unless it’s false, in which case I made it up).

First a little background. Let ${K}$ be an extension of ${\mathbf{Q}_{p}}$ with perfect residue field ${k}$, complete for a rank one valuation; any complete subfield of ${\mathbf{C}_{p}}$ is fine. Let ${X}$ be a smooth rigid analytic space over ${K}$. Often I really mean ${X}$ as an adic space over ${\mathrm{Spa}(K,\mathcal{O}_{K})}$, for example whenever I start talking about topologies on ${X}$, and always when I’m talking about proetale objects. The categories of qs rigid spaces and qs adic spaces locally of topologically finite type are canonically equivalent, so this isn’t too dangerous. In any case rigid spaces are more familiar.

We have the analytic site ${X_{\mathrm{an}}}$, the etale site ${X_{\mathrm{et}}}$, and the brand new spiffy proetale site ${X_{\mathrm{proet}}}$, with their structure sheaves ${\mathcal{O}_{X}}$, ${\mathcal{O}_{X_{\mathrm{et}}}}$, and ${\mathcal{O}_{X_{\mathrm{proet}}}}$. (WARNING: My ${\mathcal{O}_{X}}$ is Scholze’s ${\mathcal{O}_{X_{\mathrm{an}}}}$, and my ${\mathcal{O}_{X_{\mathrm{proet}}}}$ is Scholze’s ${\mathcal{O}_{X}}$.) Opens in the proetale site are basically maps ${U=\lim_{i\in I}U_{i}\rightarrow U_{0}\rightarrow X}$ where ${U_{0}\rightarrow X}$ is etale and ${\lim_{i\in I}U_{i}}$ is a directed inverse limit in which the morphisms ${U_{j}\rightarrow U_{i}}$ are finite etale surjective. We have natural maps of sites ${\nu:X_{\mathrm{proet}}\rightarrow X_{\mathrm{et}}}$ and ${\lambda:X_{\mathrm{et}}\rightarrow X_{\mathrm{an}}}$. The pullback ${\nu^{\ast}}$ is a mild operation: for any sheaf ${\mathcal{F}}$ on ${X_{\mathrm{et}}}$, the natural map ${\mathcal{F}\rightarrow R\nu_{\ast}\nu^{\ast}\mathcal{F}}$ is an isomorphism. On a typical proetale open as above, we have ${H^{n}(U,\nu^{\ast}\mathcal{F})=\lim_{i\rightarrow}H^{n}(U_{i},\mathcal{F})}$ for any abelian sheaf on ${X_{\mathrm{et}}}$.

Let ${\Omega_{X}^{\bullet}}$ be the de Rham complex of ${X}$ on the analytic site. Let ${(\mathcal{E},\nabla)}$ be a pair consisting of a coherent locally free ${\mathcal{O}_{X}}$-module equipped with a separated and exhaustive decreasing filtration ${\cdots\subset\mathrm{Fil}^{i+1}\mathcal{E}\subset\mathrm{Fil}^{i}\mathcal{E}\subset\cdots\subset\mathcal{E}}$ by local direct summands, and an integrable connection ${\nabla:\mathcal{E}\rightarrow\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}}$ satisfying Griffiths transversality (i.e. ${\nabla}$ carries ${\mathrm{Fil}^{i}\mathcal{E}}$ into ${\mathrm{Fil}^{i-1}\mathcal{E}\otimes\Omega_{X}^{1}}$). We get a filtration on the de Rham complex

$\displaystyle \Omega^{\bullet}\mathcal{E}=\Omega_{X}^{\bullet}\otimes\mathcal{E}:=[\mathcal{E}\rightarrow\mathcal{E}\otimes\Omega_{X}^{1}\rightarrow\mathcal{E}\otimes\Omega_{X}^{2}\rightarrow\cdots]$

by setting

$\displaystyle \mathrm{Fil}^{n}\Omega^{\bullet}\mathcal{E}:=[\mathrm{Fil}^{n}\mathcal{E}\rightarrow\mathrm{Fil}^{n-1}\mathcal{E}\otimes\Omega_{X}^{1}\rightarrow\mathrm{Fil}^{n-2}\mathcal{E}\otimes\Omega_{X}^{2}\rightarrow\cdots],$

so the associated graded pieces are given by

$\displaystyle \mathrm{gr}^{j}\Omega^{\bullet}\mathcal{E}=[\mathrm{gr}^{j}\mathcal{E}\rightarrow\mathrm{gr}^{j-1}\mathcal{E}\otimes\Omega_{X}^{1}\rightarrow\mathrm{gr}^{j-2}\mathcal{E}\otimes\Omega_{X}^{2}\rightarrow\cdots].$

Define ${H_{\mathrm{dR}}^{i}(X,\mathcal{E})=\mathbf{H}^{i}(X,\Omega^{\bullet}\mathcal{E})}$ and ${H^{i,j}(X,\mathcal{E})=\mathbf{H}^{i+j}(X,\mathrm{gr}^{j}\Omega^{\bullet}\mathcal{E})}$. It’s a good sanity check to verify that if we take ${\mathcal{E}=\mathcal{O}_{X}}$ with ${\mathrm{Fil}^{i}=\mathcal{O}_{X}}$ for ${i\leq0}$ and ${\mathrm{Fil}^{i}=0}$ for ${i\geq1}$, then

$\displaystyle \mathrm{gr}^{j}\Omega^{\bullet}\mathcal{E}=\begin{cases} 0 & \mathrm{if}\, j<0\\ \Omega_{X}^{j}[-j] & \mathrm{otherwise}.\end{cases}$

In particular ${H^{i,j}(X,\mathcal{E})=H^{i}(X,\Omega_{X}^{j})}$ in this instance. In general we have the Hodge-de Rham spectral sequence

$\displaystyle E_{1}^{i,j}=H^{j,i}(X,\mathcal{E})\Rightarrow H_{\mathrm{dR}}^{i+j}(X,\mathcal{E}).$

Note that for all intents and purposes, we can work interchangeably with ${\mathcal{E}}$‘s on either ${X_{\mathrm{an}}}$ or ${X_{\mathrm{et}}}$: writing ${\mathcal{E}_{\mathrm{et}}=\lambda^{\ast}\mathcal{E}}$ and ${\Omega^{\bullet}\mathcal{E}_{\mathrm{et}}=\lambda^{\ast}\Omega^{\bullet}\mathcal{E}}$, the association ${\mathcal{E}\rightsquigarrow\mathcal{E}_{\mathrm{et}}}$ is fully faithful and essentially surjective onto the obvious category of coherent locally free ${\mathcal{O}_{X_{\mathrm{et}}}}$-modules with suitable filtration and connection, and we have ${\mathbf{H}^{\ast}(X_{\mathrm{et}},\Omega^{\bullet}\mathcal{E}_{\mathrm{et}})=H_{\mathrm{dR}}^{\ast}(X,\mathcal{E})}$ compatibly with all structures, etc. We can also go from ${\mathcal{E}_{\mathrm{et}}}$ to ${\mathcal{E}_{\mathrm{proet}}=\nu^{\ast}\mathcal{E}_{\mathrm{et}}}$ and back again with no trouble, by the above remarks on ${\nu}$.

On the other side of the universe we have locally constant sheaves on ${X_{\mathrm{et}}}$, and their pullbacks to ${X_{\mathrm{proet}}}$. Recall that a lisse ${\mathbf{Z}_{p}}$-sheaf ${\mathbf{L}}$ on ${X_{\mathrm{et}}}$ is an inverse system ${(\mathbf{L}_{n})}$ of sheaves of ${\mathbf{Z}/p^{n}}$ modules satisfying some natural finiteness and local constancy conditions. We define ${\hat{\mathbf{L}}=\lim_{\leftarrow n}\nu^{\ast}\mathbf{L}_{n}}$. Any lisse ${\mathbf{Z}_{p}}$-sheaf gives rise to an honest sheaf of ${\hat{\mathbf{Z}}_{p}}$-modules on ${X_{\mathrm{proet}}}$, and under some reasonable conditions we have ${H^{\ast}(X_{\mathrm{et}},\mathbf{L})=H^{\ast}(X_{\mathrm{proet}},\hat{\mathbf{L}})}$, where the left side is defined by the usual inverse limit but the right side is honest-to-goodness sheaf cohomology!

Supposing the valuation on ${K}$ is discrete, there are certain canonically defined period sheaves ${\mathbb{B}_{\mathrm{dR}}^{+}}$, ${\mathbb{B}_{\mathrm{dR}}}$, ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}^{+}}$ and ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ on ${X_{\mathrm{proet}}}$. These are all sheaves of filtered rings. (As usual, the sheaves with pluses are subsheaves of the sheaves without pluses, and you remove the ${+}$ by inverting an element ${t}$.) The sheaf ${\mathbb{B}_{\mathrm{dR}}}$ is a relative version of Fontaine’s period ring ${\mathbf{B}_{\mathrm{dR}}}$: if we take ${X=\mathrm{Spa}(\mathbf{Q}_{p},\mathbf{Z}_{p})}$ and let ${(K_{i})_{i\in I}}$ be the directed system of all finite extension of ${\mathbf{Q}_{p}}$ in a fixed algebraic closure ${\overline{\mathbf{Q}_{p}}}$, then ${U=\lim_{\leftarrow i\in I}\mathrm{Spa}(K_{i},\mathcal{O}_{K_{i}})}$ is a valid proetale open of ${X}$, and ${\mathbb{B}_{\mathrm{dR}}(U)\cong\mathbf{B}_{\mathrm{dR}}}$. The sheaf ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ is something much stranger, with no good absolute analogue:

• ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ is a sheaf of ${\mathcal{O}_{X_{\mathrm{proet}}}}$-algebras, and ${\nu_{\ast}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\cong\mathcal{O}_{X_{\mathrm{et}}}}$.
• ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ comes equipped with a canonical integrable connection $\displaystyle \nabla:\mathcal{O}\mathbb{B}_{\mathrm{dR}}\rightarrow\mathcal{O}\mathbb{B}_{\mathrm{dR}}\otimes_{\mathcal{O}_{X_{\mathrm{proet}}}}\nu^{\ast}\Omega_{X_{\mathrm{et}}}^{1}=\mathcal{O}\mathbb{B}_{\mathrm{dR}}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}$
which satisfies Griffiths transversality, and the complex of sheaves $\displaystyle \Omega^{\bullet}\mathcal{O}\mathbb{B}_{\mathrm{dR}}=[\mathcal{O}\mathbb{B}_{\mathrm{dR}}\overset{\nabla}{\rightarrow}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\otimes_{\mathcal{O}_{X_{\mathrm{proet}}}}\nu^{\ast}\Omega_{X_{\mathrm{et}}}^{1}\overset{\nabla}{\rightarrow}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\otimes_{\mathcal{O}_{X_{\mathrm{proet}}}}\nu^{\ast}\Omega_{X_{\mathrm{et}}}^{2}\overset{\nabla}{\rightarrow}\cdots]$
is exact away from degree zero (and each filtered or graded piece is exact away from degree zero as well).
• The subring of horizontal sections ${\left(\mathcal{O}\mathbb{B}_{\mathrm{dR}}\right)^{\nabla=0}\subset\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ is canonically identified with ${\mathbb{B}_{\mathrm{dR}}}$.

Combining the second and third point, we get the all-important Poincaré lemma: the complex ${\Omega^{\bullet}\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ is a resolution of ${\mathbb{B}_{\mathrm{dR}}}$ as sheaves on the proetale site. Since ${\mathbb{B}_{\mathrm{dR}}}$ is a relative version of ${\mathbf{B}_{\mathrm{dR}}}$, which we all know is really the “correct” p-adic analogue of ${\mathbf{C}}$, this is pretty suggestive! It’s worth convincing yourself that there isn’t really any naive Poincaré lemma for the de Rham complex ${\Omega_{X}^{\bullet}}$ on a rigid analytic space: differential forms on rigid spaces just don’t like to get integrated without enlarging their domain, so already the de Rham complex of a closed affinoid polydisk isn’t exact. (This is one reason why rigid cohomology is a series of tubes.) Taking all of these behaviors together, it’s natural to think of ${\mathbf{\mathcal{O}\mathbb{B}_{\mathrm{dR}}}}$ as a mysterious bridge between the etale and analytic worlds.

As far as I know, Andreatta and Iovita deserve credit for realizing that Poincaré lemmas of this type exist and can be used to prove Fontaine’s comparison conjectures. It’s also interesting to note, in retrospect, the degree to which these constructions were {}“in the air”, in some cases for a surprisingly long time: for example, the ring we would now call ${\mathrm{gr}^{0}\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ and the zeroth graded piece of the Poincaré lemma both appear in a twenty-five year old paper of Hyodo, “On variation of Hodge-Tate structures”, with ${\mathrm{gr}^{0}\mathcal{O}\mathbb{B}_{\mathrm{dR}}}$ masquerading under the name “${S_{\infty}}$”.

It may be helpful to actually give the evaluations of these sheaves in a representative situation. Suppose ${U_{0}\subset X}$ is a reduced and irreducible smooth affinoid, say ${U_{0}=\mathrm{Sp}R}$ (or ${\mathrm{Spa}(R,R^{\circ})}$, it doesn’t matter). Let ${(R_{i})_{i\in I}}$ be the directed system of all finite etale extensions of ${R}$ contained in a fixed algebraic closure of ${\mathrm{Frac}(R)}$. The object

$\displaystyle U=\lim_{\leftarrow i\in I}\mathrm{Spa}(R_{i},R_{i}^{\circ})\rightarrow U_{0}\subset X$

is a perfectly good proetale open for ${X}$. Set ${R_{\infty}=\lim_{i\in I}R_{i}}$ and ${R_{\infty}^{\circ}=\lim_{i\in I}R_{i}^{\circ}}$, and let ${\widehat{R_{\infty}}}$ and ${\widehat{R_{\infty}^{\circ}}}$ be the p-adic completions of these rings. Then ${\mathcal{O}_{X_{\mathrm{proet}}}(U)=R_{\infty}}$ and ${\widehat{\mathcal{O}}_{X_{\mathrm{proet}}}(U)=\widehat{R_{\infty}}}$. We have the usual theta map ${\theta:W(R_{\infty}^{\circ\flat})[\frac{1}{p}]\rightarrow\widehat{R_{\infty}}}$ (whose kernel turns out to always be principal), and ${\mathbb{B}_{\mathrm{dR}}^{+}(U)}$ is exactly the ${\ker\theta}$-adic completion of ${W(R_{\infty}^{\circ\flat})[\frac{1}{p}]}$. On the other hand, we can extend ${\theta}$ to a natural map

$\displaystyle 1\otimes\theta:R_{\infty}\otimes_{W(k)[\frac{1}{p}]}W(R_{\infty}^{\circ\flat})[\frac{1}{p}]\rightarrow\widehat{R_{\infty}}$

via the inclusion ${R_{\infty}\subset\widehat{R_{\infty}}}$, and ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}^{+}(U)}$ is the ${\ker1\otimes\theta}$-adic completion of this tensor product; the connection is then gotten by extending the direct limit of the connctions ${R_{i}\rightarrow\Omega_{R_{i}/K}^{1}}$. This completion process really ties ${R}$ and ${W(R_{\infty}^{\circ\flat})}$ together in some inscrutable way, which is why I was using words like “strange” and “mysterious” earlier.

Definition. A lisse ${\mathbf{Z}_{p}}$-sheaf ${\mathbf{L}}$ on ${X_{\mathrm{et}}}$ is de Rham if there exists ${\mathcal{E}}$ as above such that

$\displaystyle \hat{\mathbf{L}}\otimes_{\hat{\mathbf{Z}}_{p}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\cong\mathcal{E}\otimes_{\mathcal{O}_{X}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}$

compatibly with the filtrations and connections on both sides. Such an ${\mathcal{E}}$ is unique if it exists, in which case we say ${\mathbf{L}}$ and ${\mathcal{E}}$ are associated (or that ${\mathcal{E}}$ is the associated ${\mathcal{O}_{X}}$-module of ${\mathbf{L}}$).

Note that we can recover ${\mathcal{E}_{\mathrm{et}}}$ with its filtration and connection from ${\mathbf{L}}$ via

$\displaystyle \mathcal{E}_{\mathrm{et}}\cong\nu_{\ast}\left(\hat{\mathbf{L}}\otimes_{\hat{\mathbf{Z}}_{p}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\right).$

Of course we could also take this as the definition of ${\mathcal{E}}$, for any ${\mathbf{L}}$, and then define ${\mathbf{L}}$ to be de Rham if the natural map

$\displaystyle \nu_{\ast}\left(\hat{\mathbf{L}}\otimes_{\hat{\mathbf{Z}}_{p}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\right)\otimes_{\mathcal{O}_{X_{\mathrm{et}}}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\rightarrow\hat{\mathbf{L}}\otimes_{\hat{\mathbf{Z}}_{p}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}$

is an isomorphism. This perhaps makes clearer the analogy with de Rham representations of ${\mathrm{Gal}(\overline{\mathbf{Q}_{p}}/\mathbf{Q}_{p})}$. Also, just like in the case of Galois representations, the functor from ${\mathbf{L}}$ to ${\mathcal{E}}$ is certainly not fully faithful. Now we have the following result.

Theorem (Scholze).

a. If ${X}$ is smooth and proper, and ${\mathbf{L}}$ is de Rham with associated ${\mathcal{O}_{X}}$-module ${\mathcal{E}}$, then the Hodge-de Rham spectral sequence for ${\mathcal{E}}$ degenerates, and there is a canonical isomorphism

$\displaystyle H_{\mathrm{et}}^{n}(X_{\overline{K}},\mathbf{L})\otimes_{\mathbf{Z}_{p}}\mathbf{B}_{\mathrm{dR}}\cong H_{\mathrm{dR}}^{n}(X,\mathcal{E})\otimes_{K}\mathbf{B}_{\mathrm{dR}}$

compatible with the filtrations and Galois actions. In particular, the etale cohomology ${H_{\mathrm{et}}^{n}(X_{\overline{K}},\mathbf{L})}$ is de Rham as a ${\mathrm{Gal}(\overline{K}/K)}$-representation, and we have a Hodge-Tate decomposition

$\displaystyle H_{\mathrm{et}}^{n}(X_{\overline{K}},\mathbf{L})\otimes_{\mathbf{Z}_{p}}\widehat{\overline{K}}\cong\oplus_{i+j=n}H^{i,j}(X,\mathcal{E})\otimes_{K}\widehat{\overline{K}}(-j).$

b. If ${f:V\rightarrow X}$ is the analytification of a smooth proper morphism of varieties, then ${\mathbf{L}=R^{i}f_{\ast\mathrm{et}}\mathbf{Z}_{p}}$ is de Rham, with associated ${\mathcal{O}_{X}}$-module ${\mathcal{E}=R^{i}f_{\ast}(\Omega_{V/X}^{\bullet})}$.

(In the numbering of Scholze’s paper, part a. is basically Theorem 8.4, and part b. follows from Theorems 8.8 and 9.3.)

This is already highly nontrivial in the case when ${\mathbf{L}=\mathbf{Z}_{p}}$ and ${\mathcal{E}=\mathcal{O}_{X}}$, in which case it becomes Fontaine’s de Rham comparison conjecture. The idea of the proof isn’t very difficult, and I’ll summarize it in the diagram

$\displaystyle H_{\mathrm{et}}^{n}(X_{\overline{K}},\mathbf{L})\otimes_{\mathbf{Z}_{p}}\mathbf{B}_{\mathrm{dR}}\rightarrow H^{n}(X_{\overline{K},\mathrm{proet}},\hat{\mathbf{L}}\otimes_{\hat{\mathbf{Z}}_{p}}\mathbb{B}_{\mathrm{dR}})\cong\mathbf{H}^{n}(X_{\overline{K},\mathrm{proet}},\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega^{\bullet}\mathcal{O}\mathbb{B}_{\mathrm{dR}})\leftarrow H_{\mathrm{dR}}^{n}(X,\mathcal{E})\otimes_{K}\mathbf{B}_{\mathrm{dR}}.$

The middle isomorphism here is an immediate consequence of the Poincaré lemma. The lefthand arrow is an isomorphism by a surprisingly direct calculation, using the acyclicity of period sheaves on affinoid perfectoid opens in ${X_{\mathrm{proet}}}$. The righthand arrow is also an isomorphism, by a calculation which seems (to my untutored eye) to be the least conceptual aspect of the proof. It seems tempting to reprove this step by studying the pushforward of ${\mathcal{O}\mathbb{B}_{\mathrm{dR}}^{+}}$ under the projection ${\nu_{\overline{K}}:X_{\overline{K},\mathrm{proet}}\rightarrow X_{\overline{K},\mathrm{et}}}$: this should be a sort of Fréchet completion of ${\mathbf{B}_{\mathrm{dR}}^{+}\otimes_{\overline{K}}\mathcal{O}_{X_{\overline{K},\mathrm{et}}}}$, and might be interesting in its own right.

For the purposes of this post, let me assume the following mild generalization of these results. Suppose ${Y}$ is smooth but not necessarily proper, with ${\mathbf{L}=(\mathbf{L}_{n})}$ a lisse ${\mathbf{Z}_{p}}$-sheaf on ${Y_{\mathrm{\acute{e}t}}}$, and suppose ${\mathbf{L}}$ is de Rham, with ${\mathcal{E}}$ the associated ${\mathcal{O}_{Y}}$-module with integrable connection. Say ${Y}$ is compactifiable if we can find a smooth proper rigid space ${X}$ over ${K}$ together with an open immersion ${j:Y\hookrightarrow X}$ whose complement is a normal crossings divisor ${D}$. Let us define

$\displaystyle H_{c,\mathrm{\acute{e}t}}^{\ast}(Y_{\overline{K}},\mathbf{L})=H_{\mathrm{\acute{e}t}}^{\ast}(X_{\overline{K}},j_{!}\mathbf{L})$

and

$\displaystyle H_{c,\mathrm{dR}}^{\ast}(Y,\mathcal{E})=\mathbf{H}^{\ast}(X,j_{!}\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{\bullet}(\mathrm{log}D)).$

I will ASSUME that these groups are independent of the chosen compactification, and that they satisfy the obvious analogue of part a. of the theorem above. Probably the independence is straightforward (and written somewhere?), the de Rham comparison slightly less so.

Now fix ${N\geq3}$ prime to ${p}$ and an integer ${k\geq2}$, and let ${Y=Y_{\Gamma_{1}(N)}}$ and ${X=X_{\Gamma_{1}(N)}}$ be the usual modular curves, considered as smooth rigid analytic spaces over ${\mathbf{Q}_{p}}$. We have the universal elliptic curve ${f:E\rightarrow Y}$. Let ${T_{p}E=(E[p^{n}])_{n\geq1}}$ be the Tate module sheaf of ${E}$; this is a lisse ${\mathbf{Z}_{p}}$-sheaf on ${Y_{\mathrm{et}}}$. On the other hand, let ${\mathcal{E}=R^{1}f_{\ast}(\Omega_{E/Y}^{\bullet})}$ be the first relative de Rham cohomology of ${E}$ with its natural filtration, so ${\mathrm{Fil}^{i}\mathcal{E}=0}$ for ${i\geq2}$ and ${=\mathcal{E}}$ for ${i\leq0}$, ${\mathrm{Fil}^{1}\mathcal{E}=\omega=f_{\ast}\Omega_{E/Y}^{1}}$, and ${\mathrm{gr}^{0}\mathcal{E}=\mathrm{Lie}E=\omega^{-1}}$. Note that ${\omega}$ extends canonically to an invertible sheaf on ${X}$ denoted by the same letter, and (somewhat abusively) we have ${j_{!}\omega\cong\omega(-D)}$ where ${D=X\smallsetminus Y}$ denotes the cusps. It’s not hard to see by part b. of the theorem above that ${\mathcal{E}}$ and ${\hat{T}_{p}E(-1)}$ are associated (on ${Y}$).

Theorem. There is a canonical isomorphism

$\displaystyle H_{c,\mathrm{dR}}^{1}(Y,\mathrm{sym}^{k-2}\mathcal{E})\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}\cong H_{c,\mathrm{\acute{e}t}}^{1}(Y_{\overline{\mathbf{Q}_{p}}},\mathrm{sym}^{k-2}(T_{p}E)(2-k))\otimes_{\mathbf{Z}_{p}}\mathbf{B}_{\mathrm{dR}}$

compatible with filtrations and Galois actions. Taking ${\mathrm{Fil}^{0}}$ of this isomorphism gives a canonical Galois-equivariant short exact sequence

$\displaystyle 0\rightarrow H^{0}(X,\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1})\otimes_{\mathbf{Q}_{p}}t^{1-k}\mathbf{B}_{\mathrm{dR}}^{+}\rightarrow H_{c,\mathrm{\acute{e}t}}^{1}(Y_{\overline{\mathbf{Q}_{p}}},\mathrm{sym}^{k-2}T_{p}E)\otimes_{\mathbf{Z}_{p}}t^{2-k}\mathbf{B}_{\mathrm{dR}}^{+}\rightarrow H^{1}(X,\omega^{2-k}(-D))\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}\rightarrow0$

of ${\mathbf{B}_{\mathrm{dR}}^{+}}$-modules.

The first sentence is a trivial consequence of the assumption above and the fact that the notions of being de Rham and associated are preserved under functors like ${\mathrm{sym}^{j}}$, but the second sentence requires an actual calculation. The reader who dislikes assumptions can amuse themselves by rewriting all of this in the context of quaternionic Shimura curves. Note that by the Kodaira-Spencer isomorphism (recalled below) we have

$\displaystyle H^{0}(X,\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1})=H^{0}(X,\omega^{k}(-D)),$

which by definition is the space ${S_{k}}$ of cusp forms of level ${N}$. Likewise, by Serre duality and Kodaira-Spencer again we get

$\displaystyle \begin{array}{rcl} H^{1}(X,\omega^{2-k}(-D)) & \cong & H^{0}(X,\omega^{k-2}(D)\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1})^{\ast}\\ & = & H^{0}(X,\omega^{k})^{\ast},\end{array}$

and ${H^{0}(X,\omega^{k})}$ is the space ${M_{k}}$ of all modular forms of level ${N}$. Therefore taking the zeroth graded piece of the isomorphism in the theorem, we get the Hodge-Tate decomposition

$\displaystyle H_{c,\mathrm{\acute{e}t}}^{1}(Y_{\overline{\mathbf{Q}_{p}}},\mathrm{sym}^{k-2}(T_{p}E)(2-k))\otimes_{\mathbf{Z}_{p}}\mathbf{C}_{p}\cong S_{k}\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p}(1-k)\oplus M_{k}^{\ast}\otimes_{\mathbf{Q}_{p}}\mathbf{C}_{p}.$

This result was essentially obtained by Faltings twenty-seven years in his paper “Hodge-Tate structures and modular forms”. By the way, the reason I like this normalization is that if ${f=\sum_{n\geq1}a_{f}(n)q^{n}}$ is a normalized cuspidal newform of level ${N}$ whose Hecke eigenvalues generate some finite extension ${L}$, then

$\displaystyle \left(H_{c,\mathrm{\acute{e}t}}^{1}(Y_{\overline{\mathbf{Q}}},\mathrm{sym}^{k-2}(T_{p}E)(2-k))\otimes L\right)[I_{f}]$

is exactly the Galois representation ${V_{f}}$ associated with ${f}$, where ${I_{f}}$ is the ideal in the Hecke algebra generated by the operators ${T_{\ell}-a_{f}(\ell),\ell\nmid N}$.

We need to understand the filtration and grading of the logarithmic de Rham complex

$\displaystyle \mathcal{L}_{k}^{\bullet}:=j_{!}\mathrm{sym}^{k-2}\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{\bullet}(\log D).$

The filtration vanishes in degrees ${k}$ and above. In degree ${k-1}$ we have

$\displaystyle \begin{array}{rcl} \mathrm{Fil}^{k-1}\mathcal{L}_{k}^{\bullet}[1] & = & \omega^{k-2}(-D)\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}(D)\\ & = & \omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}.\end{array}$

Next for ${1\leq i\leq k-2}$ we have

$\displaystyle \begin{array}{rcl} \mathrm{gr}^{i}\mathcal{L}_{k}^{\bullet} & = & [j_{!}\omega^{i}\otimes\mathrm{Lie}E^{k-2-i}\rightarrow j_{!}\omega^{i-1}\otimes\mathrm{Lie}E^{k-1-i}\otimes\Omega_{X}^{1}(D)]\\ & = & [\omega^{2i+2-k}(-D)\rightarrow\omega^{2i-k}\otimes\Omega_{X}^{1}]\\ & = & \omega^{2i-k}(-D)\otimes[\omega^{2}\rightarrow\Omega_{X}^{1}(D)],\end{array}$

and the indicated map ${\omega^{2}\rightarrow\Omega_{X}^{1}(D)}$ is nothing more or less than the Kodaira-Spencer isomorphism, so each of these graded pieces has trivial cohomology! Finally we have ${\mathrm{gr}^{0}\mathcal{L}_{k}^{\bullet}=\omega^{2-k}(-D)}$. Feeding all of this into the Hodge-de Rham spectral sequence, we find that the filtration on ${H_{c,\mathrm{dR}}^{1}(Y,\mathrm{sym}^{k-2}\mathcal{E})}$ satisfies

$\displaystyle \begin{array}{rcl} \mathrm{Fil}^{k} & = & 0,\\ \mathrm{Fil}^{k-1}=\cdots=\mathrm{Fil}^{1} & = & H^{0}(X,\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}),\\ \mathrm{Fil}^{0}/\mathrm{Fil}^{1} & = & H^{1}(X,\omega^{2-k}(-D)),\\ \mathrm{Fil}^{0} & = & H_{c,\mathrm{dR}}^{1}(Y,\mathrm{sym}^{k-2}\mathcal{E}).\end{array}$

This is enough to conclude the theorem.

It’s natural to ask: can we construct the map

$\displaystyle \delta:H^{0}(X,\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1})\rightarrow H_{c,\mathrm{\acute{e}t}}^{1}(Y_{\overline{\mathbf{Q}_{p}}},\mathrm{sym}^{k-2}T_{p}E(1))\otimes_{\mathbf{Z}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$

directly, as a boundary map in the cohomology of some short exact sequence? Well, the Poincare lemma is a short exact sequence in this case; tensoring it with ${\mathrm{sym}^{k-2}(\hat{T}_{p}E)(2-k)}$ and recalling the definition of association, we get a filtered short exact sequence

$\displaystyle 0\rightarrow\mathrm{sym}^{k-2}(\hat{T}_{p}E)(2-k)\otimes_{\hat{\mathbf{Z}}_{p}}\mathbb{B}_{\mathrm{dR}}\rightarrow\mathrm{sym}^{k-2}\mathcal{E}\otimes_{\mathcal{O}_{Y}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\rightarrow\mathrm{sym}^{k-2}\mathcal{E}\otimes_{\mathcal{O}_{Y}}\Omega_{Y}^{1}\otimes_{\mathcal{O}_{Y}}\mathcal{O}\mathbb{B}_{\mathrm{dR}}\rightarrow0$

of sheaves on ${Y_{\mathrm{proet}}}$. Now, ${\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}}$ is a subsheaf of ${\mathrm{Fil}^{k-1}}$ of the rightmost term here, so we get a map

$\displaystyle H^{0}(X,\omega^{k-2}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1})\rightarrow H^{0}\left(X_{\overline{K},\mathrm{proet}},\mathrm{Fil}^{k-1}(j_{!}\mathrm{sym}^{k-2}\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}\otimes_{\mathcal{O}_{X}}\mathcal{O}\mathbb{B}_{\mathrm{dR}})\right).$

Taking ${\mathrm{Fil}^{k-1}}$ of the above sequence and passing to its cohomology gives a connecting map

$\displaystyle H^{0}\left(X_{\overline{K},\mathrm{proet}},\mathrm{Fil}^{k-1}(j_{!}\mathrm{sym}^{k-2}\mathcal{E}\otimes_{\mathcal{O}_{X}}\Omega_{X}^{1}\otimes_{\mathcal{O}_{X}}\mathcal{O}\mathbb{B}_{\mathrm{dR}})\right)\rightarrow H^{1}(X_{\overline{K},\mathrm{proet}},j_{!}\mathrm{sym}^{k-2}(\hat{T}_{p}E)(1)\otimes_{\hat{\mathbf{Z}}_{p}}\mathbb{B}_{\mathrm{dR}}^{+}),$

and as above we should have

$\displaystyle H^{1}(X_{\overline{K},\mathrm{proet}},j_{!}\mathrm{sym}^{k-2}(\hat{T}_{p}E)(1)\otimes_{\hat{\mathbf{Z}}_{p}}\mathbb{B}_{\mathrm{dR}}^{+})\cong H^{1}(X_{\overline{K},\mathrm{et}},j_{!}\mathrm{sym}^{k-2}(T_{p}E)(1))\otimes_{\mathbf{Z}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}.$

Composing these three maps gives ${\delta}$.