1. Introduction

Let $p$ be a prime number, $\mathbb {A}_{\mathbb {Q}}^{\infty }$ the finite adèles of $\mathbb {Q}$, $\mathbb {A}^{\infty,p}_{\mathbb {Q}}$ the finite prime-to-$p$ adèles, and $\mathbb {Z}_p$ the ring of $p$-adic integers. Let $\mathbb {C}_p$ be the $p$-adic completion of an algebraic closure of $\mathbb {Q}_p$, and $G_{\mathbb {Q}_p}=\operatorname {Gal}(\mathbb {C}_p/\mathbb {Q}_p)$ the absolute Galois group. From now on, we fix a neat compact open subgroup $K^p\subset \operatorname {GL}_2(\mathbb {A}^{\infty,p}_{\mathbb {Q}})$. Given an open compact subgroup $K_p\subset \operatorname {GL}_2(\mathbb {Q}_p)$, we denote by $Y_{K_p}^{{\rm alg}}$ the modular curve over $\operatorname {Spec} \mathbb {Q}_p$ of level $K^pK_p\subset \operatorname {GL}_2(\mathbb {A}^{\infty }_{\mathbb {Q}})=\operatorname {GL}_2(\mathbb {A}^{\infty,p}_{\mathbb {Q}})\times \operatorname {GL}_2(\mathbb {Q}_p)$, and by $X_{K_p}^{{\rm alg}}$ its compactification by adding cusps. Let $Y_{K_p}$ and $X_{K_p}$ be the rigid analytic varieties attached to the modular curves, seen as adic spaces over $\operatorname {Spa}(\mathbb {Q}_p, \mathbb {Z}_p)$ (cf. [Reference HuberHub96]). Letting $D= X_{K_p}\backslash Y_{K_p}$ be the cusp divisor, we endow $X_{K_p}$ with the $\log$-structure defined by $D$.

Given a fine and saturated (fs) $\log$ adic space $Z$ and $?\in \{{\rm an},{\rm \acute {e}t},{\rm k\acute {e}t},{\rm pro\acute {e}t},{\rm prok\acute {e}t}\}$, we denote by $Z_{?}$ its analytic, étale, Kummer-étale, proétale and pro-Kummer-étale sites, respectively (see [Reference ScholzeSch13] and [Reference Diao, Lan, Liu and ZhuDLLZ23b]).

In [Reference FaltingsFal87], Faltings described the Hodge–Tate decomposition of the étale cohomology (with coefficients) of the modular curve $Y_{K_p }$. More precisely, let $E$ be the universal elliptic curve over $Y_{K_p}$. This admits an extension to a semi-abelian adic space $E^{sm}$ over $X_{K_p}$ (cf. [Reference Deligne and RapoportDR73]). Let $e: X_{K_p}\rightarrow E^{sm}$ be the unit section, $\omega _E= e^* \Omega ^1_{ E^{sm}/X}$ the modular sheaf and $T_p E =\varprojlim _{n} E[p^n]$ the Tate module over $Y_{K_p}$. We have the following theorem.

Theorem 1.0.1 (Faltings)

Let $k\geq 0$. There exists a Galois and Hecke equivariant isomorphism

(1)\begin{equation} H^1_{\rm \acute{e}t}(Y_{K_p,\mathbb{C}_p}, \operatorname{Sym}^k T_p E ) \otimes_{\mathbb{Q}_p}\mathbb{C}_p(1)= H^0_{\rm an}(X_{K_p,\mathbb{C}_p}, \omega_E^{k+2})\oplus H^1_{\rm an}(X_{K_p,\mathbb{C}_p}, \omega_E^{-k})(k+1) \end{equation}

called the Eichler–Shimura decomposition.

The first result of this paper is a new proof of Faltings's Eichler–Shimura decomposition using Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. Our proof is the proétale analogue of the BGG decomposition for the de Rham cohomology of Faltings and Chai [Reference Faltings and ChaiFC90, Ch. 5 and Theorem 5.5]. Let us develop the ideas behind it.

Let $X_{\infty }:= \varprojlim _{K_p} X_{K_p}$ be Scholze's perfectoid modular curve and ${\pi _{\operatorname {HT}}: X_{\infty }\rightarrow \mathbb {P}^1_{\mathbb {Q}_p}}$ the Hodge–Tate period map [Reference ScholzeSch15]. The morphism $\pi _{\operatorname {HT}}$ is $\operatorname {GL}_2(\mathbb {Q}_p)$-equivariant, where we see $\mathbb {P}^1_{\mathbb {Q}_p}$ as the left quotient of $\operatorname {GL}_2$ by the upper triangular Borel $\mathbf {B}$. Let $\pi _{K_p}: X_{\infty }\rightarrow X_{K_p}$ be the natural map. We can see $X_{\infty }$ as a pro-Kummer-étale $K_p$-torsor over $X_{K_p}$. We let $\widehat {\mathscr {O}}_{X_{K_p}}$ denote the $p$-adic completion of the structural sheaf over $X_{K_p, {\rm prok\acute {e}t}}$. Let $\mathbb {Z}_p(1)= \varprojlim _{n} \mu _{p^n}$ be the Tate twist and $\widehat {\mathscr {O}}_{X_{K_p}}(i)$ the $i$th twist of $\widehat {\mathscr {O}}_{X_{K_p}}$. By [Reference Diao, Lan, Liu and ZhuDLLZ23b, Theorem 4.6.1], $T_pE$ admits a natural extension to the pro-Kummer-étale site of $X_{K_p}$ which we denote in the same way. From now on, we fix the level $K_p$ and write $Y=Y_{K_p}$ and $X=X_{K_p}$.

The map $\pi _{\operatorname {HT}}$ is defined from the Hodge–Tate exact sequence

(2)\begin{equation} 0 \rightarrow \omega_E^{-1}\otimes_{\mathscr{O}_X} \widehat{\mathscr{O}}_X(1) \xrightarrow{\operatorname{HT}^{\vee}} T_p E \otimes_{\mathbb{Z}_p} \widehat{\mathscr{O}}_X \xrightarrow{\operatorname{HT}} \omega_E \otimes_{\mathscr{O}_X} \widehat{\mathscr{O}}_X \rightarrow 0, \end{equation}

which is the variation in families of the Hodge–Tate decomposition for elliptic curves (cf. [Reference TateTat67]). More precisely, let $\Psi : \mathbb {Z}_p^2\xrightarrow {\sim } T_pE$ be the universal trivialization over $X_{\infty }$; then (2) defines a line subbundle of $\mathscr {O}_{X_{\infty }}^{\oplus 2}$ which induces the map $\pi _{\operatorname {HT}}$.

The $\operatorname {GL}_2(\mathbb {Q}_p)$-equivariance of $\pi _{\operatorname {HT}}$ recovers (2) from a short exact sequence of $\operatorname {GL}_2$-equivariant sheaves over $\mathbb {P}^1_{\mathbb {Q}_p}$. Indeed, the presentation $\mathbb {P}^1_{\mathbb {Q}_p}= \mathbf {B} \backslash\!\operatorname {GL}_2$ induces an equivalence between algebraic $\mathbf {B}$-representations and $\operatorname {GL}_2$-equivariant vector bundles over $\mathbb {P}^1_{\mathbb {Q}_p}$. More explicitly, let $V$ be a $\mathbf {B}$-representation; then one defines the vector bundle $\mathcal {V}= \operatorname {GL}_2 \times ^{\mathbf {B}} V = \mathbf {B}\backslash (\operatorname {GL}_2\times V)$, where in the right-hand side term $\operatorname {GL}_2 \times V$, the group $\mathbf{B}$ acts diagonally. Let $\mathscr {F}$ be a $K_p$-equivariant sheaf over $\mathbb {P}^1_{\mathbb {Q}_p}$; we shall identify $\pi _{\operatorname {HT}}^*(\mathscr {F})$ with the pro-Kummer-étale sheaf over $X_{K_p}$ obtained by descent from the $K_p$-equivariant pullback over $X_{\infty }$.

Let $\mathbf {T}\subset \mathbf {B}$ be the diagonal torus and let $\kappa =(k_1,k_2) \in X^*(\mathbf {T})$ be a character. We denote by $\mathbb {Q}_p(\kappa )$ the representation defined by $\kappa$ and consider it as a $\mathbf {B}$-representation by letting the unipotent radical act trivially. We say that $\kappa =(k_1,k_2)$ is dominant if $k_1\geq k_2$. Given a dominant character $\kappa$, we let $V_{\kappa }= \operatorname {Sym}^{k_1-k_2}\operatorname {St}\otimes \det ^{k_2}$ be the irreducible representation of $\operatorname {GL}_2$ of highest weight $\kappa$, and we denote by $V_{\kappa,{\rm \acute {e}t}}$ the associated pro-Kummer-étale sheaf obtained by descent from $X_{\infty }$. Let $W=\{1,w_0\}$ be the Weyl group of $\operatorname {GL}_2$. We denote by $\mathscr {L}(\kappa )$ the $\operatorname {GL}_2$-equivariant sheaf over $\mathbb {P}^1_{\mathbb {Q}_p}$ given by $\mathbf {B}\backslash ( \operatorname {GL}_2\times w_0(\kappa ))$. The standard representation $\operatorname {St}$ has a $\mathbf {B}$-filtration

(3)\begin{equation} 0\rightarrow \mathbb{Q}_p(1,0) \rightarrow \operatorname{St} \rightarrow \mathbb{Q}_p(0,1) \rightarrow 0. \end{equation}

By construction, the pullback of (3) by $\pi _{\operatorname {HT}}$ is equal to the Hodge–Tate exact sequence (2). In particular, $\pi _{\operatorname {HT}}^*( \operatorname {St} \otimes \mathscr {O}_{\mathbb {P}^1_{\mathbb {Q}_p}})= T_p E \otimes \widehat {\mathscr {O}}_X$ and $\pi _{\operatorname {HT}}( \mathscr {L}(\kappa ))= \omega _E^{k_1-k_2}\otimes _{\mathscr {O}_X} \widehat {\mathscr {O}}_X(k_2)$. A natural question arises, namely, how to describe the pullbacks by $\pi _{\operatorname {HT}}$ of the $\operatorname {GL}_2$-equivariant vector bundles over $\mathbb {P}^1_{\mathbb {Q}_p}$. We already know that the pullbacks of those vector bundles constructed from characters of $\mathbf {T}$ are related to modular sheaves, and it remains to understand the $\operatorname {GL}_2$-equivariant sheaves constructed from non-semi-simple representations of $\mathbf {B}$.

Let $\mathscr {O}(\mathbf {B})$ be the ring of algebraic functions of $\mathbf {B}$ endowed with the right regular action; note that any algebraic representation of $\mathbf {B}$ occurs in $\mathscr {O}(\mathbf {B})$. The presentation $\mathbf {B}= \mathbf {T}\ltimes \mathbf {N}$ as a semidirect product gives an isomorphism $\mathscr {O}(\mathbf {B})= \mathscr {O}(\mathbf {T})\otimes \mathscr {O}(\mathbf {N})$, where $\mathbf {B}$ acts on the first factor by the right regular action of $\mathbf {T}$, and it acts on the second factor under the formula $(n,b)\mapsto t_{b}^{-1}nt_{b} n_b$, with $(n,b)\in \mathbf {N}\times \mathbf {B}$ and $b=(t_b,n_b)\in \mathbf {T}\ltimes \mathbf {N}$. We have the following theorem.

Theorem 1.0.2 (Theorem 4.1.2)

Let $\underline {\mathscr {O}}(\mathbf {N})$ be the $\operatorname {GL}_2$-equivariant quasi-coherent sheaf over $\mathbb {P}^1_{\mathbb {Q}_p}$ associated to $\mathscr {O}(\mathbf {N})$. Let $\mathscr {O}\!\mathbb {B}_{{\rm dR},\log }$ be the geometric de Rham period sheaf of [Reference ScholzeSch13] and [Reference Diao, Lan, Liu and ZhuDLLZ23b], and let $\mathscr {O}\!\mathbb {C}_{\log }= \operatorname {gr}^0 \mathscr {O}\!\mathbb {B}_{{\rm dR},\log }$ be the Hodge–Tate sheaf. We have a natural isomorphism of pro-Kummer-étale sheaves over $X$,

\[ \pi^{*}_{\operatorname{HT}}(\underline{\mathscr{O}}(\mathbf{N})) = \mathscr{O}\!\mathbb{C}_{\log}. \]

As a corollary one obtains the Eichler–Shimura decompositions.

Theorem 1.0.3 (Theorem 4.1.4)

Let $\kappa =(k_1,k_2)$ be a dominant character and set $\alpha = (1,-1)\in X^*(\mathbf {T})$. Let $\operatorname {BGG}(\kappa )$ be the dual BGG complex of weight $\kappa$ (Proposition 2.4.3),

\[ 0 \rightarrow V_{\kappa} \rightarrow V(\kappa) \rightarrow V(w_0(\kappa)-\alpha)\rightarrow 0. \]

We denote by $\underline {\operatorname {BGG}(\kappa )}$ the associated $\operatorname {GL}_2$-equivariant complex over $\mathbb {P}^1_{\mathbb {Q}_p}$. Then the pullback of $\underline {\operatorname {BGG}(\kappa )}$ by $\pi _{\operatorname {HT}}$ is the short exact sequence

\[ 0 \rightarrow \operatorname{Sym}^{k_1-k_2} T_pE \otimes \widehat{\mathscr{O}}_X(k_2) \rightarrow \omega_E^{k_2-k_1}\otimes \mathscr{O}\!\mathbb{C}_{\log}(k_1) \rightarrow \omega_E^{k_1-k_2+2}\otimes \mathscr{O}\!\mathbb{C}_{\log}(k_2-1)\rightarrow 0. \]

Furthermore, let $\lambda : X_{\mathbb {C}_p,{\rm prok\acute {e}t}}\rightarrow X_{ \mathbb {C}_p,{\rm an}}$ be the projection of sites. Then

\[ R\lambda_{*} (\operatorname{Sym}^k T_pE \otimes \widehat{\mathscr{O}}_X(1))= \omega_E^{-k}\otimes \mathbb{C}_p(k+1)[0] \oplus \omega_E^{k+2} [-1]. \]

Taking $H^1$-cohomology in the analytic site of $X_{\mathbb {C}_p}$, we recover the Eichler–Shimura decomposition of Theorem 1.0.1.

The proof of Theorem 1.0.2 follows from the isomorphism between Faltings extension $\operatorname {gr}^1 \mathscr {O}\!\mathbb {B}_{{\rm dR},\log }^+$ and the sheaf $T_pE \otimes \widehat {\mathscr {O}}_X \otimes \omega _E$. This isomorphism was known to Faltings, and used in his original proof of Theorem 1.0.1. Our new proof provides a more explicit definition of the Eichler–Shimura maps in terms of cocycles and can be generalized to any Shimura variety. Moreover, using this method, one easily deduces the degeneration of the spectral sequence appearing in [Reference FaltingsFal87], as well as its natural splitting using simple properties of the dual BGG resolution. It is worth mentioning that the isomorphism between the twist of the Tate module and the Faltings extension was used by Lue Pan in [Reference PanPan22] to compute the relative Sen operator of the modular curve.

The second goal of this paper is the interpolation of the Eichler–Shimura decomposition (1). The $H^0$-map of the overconvergent Eichler–Shimura maps was previously constructed by Andreatta, Iovita and Stevens in [Reference Andreatta, Iovita and StevensAIS15]. The strategy followed in this paper is close to the construction of the Eichler–Shimura map for Shimura curves in [Reference Chojecki, Hansen and JohanssonCHJ17]. Roughly speaking, we take pullbacks by $\pi _{\operatorname {HT}}$ of certain locally analytic sheaves over $\mathbb {P}^1_{\mathbb {Q}_p}$. In this way, we interpolate all the terms appearing in the Hodge–Tate exact sequence (2): we get overconvergent modular sheaves whose cohomology is the object of study in higher Coleman theory developed by Boxer and Pilloni [Reference Boxer and PilloniBP21, Reference Boxer and PilloniBP22]. The interpolation of the symmetric powers of the Tate module will be given by locally analytic principal series or locally analytic distributions as in [Reference Ash and StevensAS08]. Finally, the Hodge–Tate maps $\operatorname {HT}$ and $\operatorname {HT}^\vee$ can be put in families, obtaining the $\operatorname {dlog}$ map of [Reference Andreatta, Iovita and StevensAIS15] as a particular case.

Let us sketch the main steps of the construction. Let $n\geq 1$ be an integer and let

\[ \operatorname{Iw}_n:= \biggl( \begin{matrix} \mathbb{Z}_p^{\times} & \mathbb{Z}_p \\ p^n \mathbb{Z}_p & \mathbb{Z}_p^\times \end{matrix} \biggr) \]

be the Iwahori group modulo $p^n$. From now on, we will take $X=X_{\operatorname {Iw}_n}$. Let $\epsilon \geq \delta \geq n$ be rational numbers and $(R,R^+)$ a uniform Tate algebra over $\mathbb {Q}_p$ which we may assume to be sheafy (i.e. such that the pre-sheaf of rational functions in $\operatorname {Spa}(R,R^+)$ is an actual sheaf). Let $T=\mathbf {T}(\mathbb {Z}_p)$ denote the $\mathbb {Z}_p$-points of the diagonal torus and $\chi =(\chi _1,\chi _2): T\rightarrow R^{+,\times }$ a $\delta$-analytic character (cf. Lemma 2.2.5). We let $R\widehat {\otimes }\widehat {\mathscr {O}}_X$ be the $p$-adically complete tensor product of $R$ and the completed sheaf $\widehat {\mathscr {O}}_X$. Given a character $\lambda : \mathbb {Z}_p^\times \to R^{+,\times }$, we denote by $R(\lambda )$ the $G_{\mathbb {Q}_p}$-module induced by the composition $G_{\mathbb {Q}_p}\xrightarrow {\chi _{{\rm cyc}}} \mathbb {Z}_p^\times \xrightarrow {\lambda } R^{\times }$. Finally, we write $\widehat {\mathscr {O}}_X(\lambda ):= R(\lambda )\widehat {\otimes }\widehat {\mathscr {O}}_X$.

We begin with the construction of all the sheaves over $\mathbb {P}^1_{\mathbb {Q}_p}$: for $w\in W=\{1,w_0\}$ we define a family of overconvergent neighbourhoods $\{U_w(\epsilon )\operatorname {Iw}_n\}_{\epsilon \geq n}$ of $w\operatorname {Iw}_n$ in $\mathbb {P}^1_{\mathbb {Q}_p}$. The affinoid spaces $U_w(\epsilon )\operatorname {Iw}_n$ admit sections of the quotient map $\operatorname {GL}_2\rightarrow \mathbb {P}^1_{\mathbb {Q}_p}$. In particular, the natural $\mathbf {T}$-torsor $\mathbf {N}\backslash\!\operatorname {GL}_2\rightarrow \mathbb {P}^1_{\mathbb {Q}_p}$, where $\mathbf {N}$ is the unipotent radical of $\mathbf {B}$, has a trivialization over $U_{w}(\epsilon )\operatorname {Iw}_n$. We define a $R\widehat {\otimes } \mathscr {O}_{\mathbb {P}^1_{\mathbb {Q}_p}}$ line bundle $\mathscr {L}(\chi )$ in the analytic site of $U_w(\epsilon )\operatorname {Iw}_n$ in the same way we have defined the line bundles $\mathscr {L}(\kappa )$ for $\kappa \in X^*(\mathbf {T})$. Then we define the space of $\delta$-analytic principal series of weight $\chi$ to be the $R$-Banach space

\[ A^{\delta}_{\chi}= \Gamma(U_{w_0}(\delta) \operatorname{Iw}_n, \mathscr{L}(\chi)). \]

We define the $\delta$-analytic distributions $D^{\delta }_{\chi }$ to be the continuous dual of $A^{\delta }_{\chi }$ endowed with the weak topology. The space $A^{\delta }_{\chi }$ has a natural action of $\operatorname {Iw}_n$, so that it defines a constant $\operatorname {Iw}_n$-equivariant sheaf on $\mathbb {P}^1_{\mathbb {Q}_p}$. We let $A^{\delta }_{\chi,{\rm \acute {e}t}}$ and $D_{\chi,{\rm \acute {e}t}}^{\delta }$ denote the pro-Kummer-étale sheaves over $X$ obtained by descent from the topological $K_p$-equivariant sheaves over $X_{\infty }$.

In Proposition 2.4.4 we construct maps

\begin{gather*} R(\chi) \xrightarrow{\iota} A^{\delta}_{\chi} \text{ equivariant for the action of } \mathbf{B}(\mathbb{Z}_p)\cap \operatorname{Iw}_n,\\ A^{\delta}_{\chi}\xrightarrow{\operatorname{ev}_{w_0}} R(\chi) \text{ equivariant for the action of } w_0^{-1} \mathbf{B}(\mathbb{Z}_p)w_0 \cap \operatorname{Iw}_n, \end{gather*}

with $\iota$ being the highest weight vector, and $\operatorname {ev}_{w_0}$ the evaluation at $w_0$. We prove that these maps give rise morphisms of $\operatorname {Iw}_n$-equivariant sheaves

(4)\begin{equation} \begin{gathered} \mathscr{L}(w_0(\chi)) \rightarrow A^{\delta}_{\chi}\widehat{\otimes}_{\mathbb{Q}_p} \mathscr{O}_{\mathbb{P}^1_{\mathbb{Q}_p}} \text{ over } U_{1}(\epsilon)\operatorname{Iw}_n,\\ A^{\delta}_{\chi}\widehat{\otimes}_{\mathbb{Q}_p} \mathscr{O}_{\mathbb{P}^1_{\mathbb{Q}_p}} \rightarrow \mathscr{L}(\chi) \text{ over } U_{w_0}(\epsilon)\operatorname{Iw}_n. \end{gathered} \end{equation}

The next step is to translate all the previous constructions to the modular curve $X$. We start by defining the strict neighbourhoods of the $w$-ordinary locus $\{X_{w}(\epsilon )\}_{\epsilon >n}$; they are equal to $\pi _{\operatorname {Iw}_n}( \pi _{\operatorname {HT}}^{-1}(U_w(\epsilon )\operatorname {Iw}_n) )$. The second object we descend to $X$ are the overconvergent modular sheaves $\omega _E^{\chi }$; they are $R\widehat {\otimes } \mathscr {O}_{X}$-line bundles in the étale or analytic site of $X_{w}(\epsilon )$. We refer to [Reference Boxer and PilloniBP21] for the general construction of these sheaves. The dictionary provided by $\pi _{\operatorname {HT}}$ then gives (see Corollary 3.2.11)

(5)\begin{equation} \pi_{\operatorname{HT}}^*(\mathscr{L}(\chi))= \omega_E^{\chi}\otimes \widehat{\otimes} \widehat{\mathscr{O}}_X(\chi_2). \end{equation}

We continue with the pullback of the $\delta$-analytic principal series and distributions, seen as $\operatorname {Iw}_n$-equivariant sheaves over $\mathbb {P}^1_{\mathbb {Q}_p}$. By definition one has $\pi _{\operatorname {HT}}^*(A_{\chi }^{\delta }\widehat {\otimes } \mathscr {O}_{\mathbb {P}^1_{\mathbb {Q}_p}})= A^{\delta }_{\chi,{\rm \acute {e}t}}\widehat {\otimes } \widehat {\mathscr {O}}_X$ (and similarly for $D^{\delta }_{\chi }$). Finally, we pullback the maps (4) obtaining overconvergent Hodge–Tate maps of pro-Kummer-étale sheaves

\begin{gather*} \omega_E^{w_0(\chi)} \widehat{\otimes} \widehat{\mathscr{O}}_X(\chi_1) \xrightarrow{\operatorname{HT}^\vee} A^{\delta}_{\chi,\rm \acute{e}t}\widehat{\otimes} \widehat{\mathscr{O}}_X \text{ over } X_{1}(\epsilon),\\ A^{\delta}_{\chi,\rm \acute{e}t}\widehat{\otimes}\widehat{\mathscr{O}}_X \xrightarrow{\operatorname{HT}} \omega_E^{\chi} \widehat{\otimes} \widehat{\mathscr{O}}_X(\chi_2) \text{ over } X_{w_0}(\epsilon) \end{gather*}

(and similarly for $D^{\delta }_{\chi }$). Taking pro-Kummer-étale cohomology, one obtains the following theorem.

Theorem 1.0.4 (Theorem 4.2.2)

There are overconvergent Eichler–Shimura maps

(6) \begin{equation} 0\rightarrow H^1_{1,c}(X_{\mathbb{C}_p}, \omega_E^{w_0(\chi)})_{\epsilon}(\chi_1) \xrightarrow{ES_{A}^{\vee}} H^1_{\rm prok\acute{e}t}(X_{\mathbb{C}_p}, A^{\delta}_{\chi,\rm \acute{e}t}\widehat{\otimes} \widehat{\mathscr{O}}_X) \xrightarrow{ES_{A}} H^0_{w_0}(X_{\mathbb{C}_p}, \omega_E^{\chi+\alpha})_{\epsilon}(\chi_2-1)\rightarrow 0 \end{equation}

satisfying the following properties.

1. (i) The composition $ES_{A}\circ ES^{\vee }_{A}$ is zero.

2. (ii) Assume that $\mathcal {V}=\operatorname {Spa}(R,R^+)$ is an affinoid subspace of the weight space $\mathcal {W}_T$ of $T=\mathbf {T}(\mathbb {Z}_p)$, and let $\kappa =(k_1,k_2)\in \mathcal {V}$ be a dominant weight of $\mathbf {T}$. Let $\alpha =(1,-1)\in X^*(\mathbf {T})$ and let $\chi =\chi ^{{\rm un}}_{\mathcal {V}}$ be the universal character of $\mathcal {V}$. Then we have the following commutative diagram.

   

3. (iii) The maps of (ii) are Galois and $U_p^t$-equivariant with respect to the good normalizations of the $U_p^t$-operators. In particular, the above diagram restricts to the finite slope part with respect to the $U_p^t$-action.

4. (iv) Let $h< k_1-k_2+1$. There exists an open affinoid $\mathcal {V}'\subset \mathcal {V}$ containing $\kappa$ such that the $(\leq h)$-slope part of the restriction of (6) to $\mathcal {V}'$ is a short exact sequence of finite free $\mathbb {C}_p\widehat {\otimes }_{\mathbb {Q}_p} \mathscr {O}(\mathcal {V}')$-modules.

5. (v) Keep the hypothesis of (iv), and let $\chi$ be the universal character of $\mathcal {V}'$. Let $\widetilde {\chi }= \chi _1-\chi _2+1 : \mathbb {Z}_p^\times \rightarrow R^{+,\times }$, and $b= {d}/{dt}|_{t=1} \widetilde {\chi }(t)$. Then we have a Galois equivariant split after inverting $b$

   \[ H^1_{\rm prok\acute{e}t}(X_{\mathbb{C}_p}, A^{\delta}_{\chi,\rm \acute{e}t}\widehat{\otimes}\widehat{\mathscr{O}}_X)^{\leq h}_{b} =[H^1_{1,c}(X_{\mathbb{C}_p}, \omega_E^{w_0(\chi)})^{\leq h}_{\epsilon}(\chi_1)]_b \oplus [H^0_{w_0}(X_{\mathbb{C}_p}, \omega_E^{\chi+\alpha})^{\leq h}_{\epsilon}(\chi_2-1)]_b. \]

We finish with the compatibility of the oveconvergent Eichler–Shimura maps (6) with the Poincaré and Serre pairings. One can define a Poincaré pairing between the overconvergent proétale cohomologies

(7)\begin{equation} \langle -,- \rangle_P : H^1_{\rm pro\acute{e}t,c}(Y_{\mathbb{C}_p}, D^{\delta}_{\chi,\rm \acute{e}t}(1)) \times H^1_{\rm pro\acute{e}t}(Y_{\mathbb{C}_p}, A^{\delta}_{\chi,\rm \acute{e}t} ) \rightarrow \mathscr{O}(\mathcal{V}') \end{equation}

where the left-hand side term is the proétale cohomology with compact support. On the other hand, one also has Serre pairings between overconvergent coherent cohomologies

(8)\begin{equation} \begin{aligned} & \langle -, -\rangle_S: H^1_{w,c}(X, \omega_E^{-\chi}(-D))_{\epsilon}\times H^{0}_{w}(X,\omega_E^{\chi+\alpha})_{\epsilon} \rightarrow \mathscr{O}(\mathcal{V}'),\\ & \langle -, -\rangle_S: H^1_{w,c}(X, \omega_E^{w_0(\chi)})_{\epsilon}\times H^{0}_{w}(X,\omega_E^{-w_0(\chi)+\alpha}(-D))_{\epsilon} \rightarrow \mathscr{O}(\mathcal{V}'). \end{aligned} \end{equation}

We have the following theorem.

The outline of the paper is as follows. In § 2 we develop the overconvergent theory over the flag variety. We define the affinoid subspaces $U_w(\epsilon )\operatorname {Iw}_n$ and the sheaves $\mathscr {L}(\chi )$. We construct the $\delta$-analytic principal series $A^{\delta }_{\chi }$ and the maps (4). We recall some facts of the BGG theory for irreducible representations of $\operatorname {GL}_2$; in particular, we define the dual BGG complex $\operatorname {BGG}(\kappa )$.

Then in § 3, we translate all the previous constructions from $\mathbb {P}^1_{\mathbb {Q}_p}$ to the modular curves via $\pi _{\operatorname {HT}}$. We define the strict neighbourhoods of the $w$-ordinary locus, the overconvergent modular sheaves and the overconvergent Hodge–Tate maps. We give the good normalizations of the Hecke operators and show that the Hodge–Tate maps are compatible with the normalized $U_p$-correspondence.

Finally, in § 4, we show how to obtain the classical Eichler–Shimura decomposition from the dual BGG complex, proving Theorems 1.0.3 and 1.0.1, in the process we also prove the theorem for the cohomology with compact support. Next, we construct the overconvergent Eichler–Shimura maps and obtain Theorem 1.0.4. Finally, we show the compatibility of Poincaré and Serre duality for the overconvergent Eichler–Shimura maps, obtaining Theorem 1.0.6.