2.1 Pro-Kummer étale site

Given a finite saturated (for short ‘fs’) locally noetherian log scheme X (resp. an fs locally noetherian log adic space $\mathcal {X}$ ) we denote by $X_{\mathrm {ket}}$ , $X_{\mathrm {fket}}$ (resp. $\mathcal {X}_{\mathrm {ket}}$ , $\mathcal {X}_{\mathrm {fket}}$ ) the Kummer étale site, respectively the finite Kummer étale site (see [Reference Illusie18, Def. 2.1], [Reference Diao, Lan, Liu and Zhu15, Def. 4.1.2]). Following Scholze [Reference Scholze22], we denote by $X_{\mathrm {pke}}$ , resp. $X_{\mathrm {profket}}$ (resp. $\mathcal {X}_{\mathrm {pke}}$ , $\mathcal {X}_{\mathrm {profket}}$ ) the pro-Kummer étale site, resp. the pro-finite Kummer étale site (see [Reference Diao, Lan, Liu and Zhu15, Def. 5.1.2 & 5.1.9]) of X, respectively $\mathcal {X}$ .

As a category, it is the full subcategory of pro- $X_{\mathrm {ket}}$ , resp. pro- $X_{\mathrm {fket}}$ (resp. pro- $\mathcal {X}_{\mathrm {ket}}$ , pro- $\mathcal {X}_{\mathrm {fket}}$ ) of pro-objects that are pro-Kummer étale over X, resp. pro-finite Kummer étale over X (resp. $\mathcal {X}$ ), that is, objects that are equivalent to cofiltered systems $\displaystyle {\lim _{\leftarrow }} Z_i$ such that $Z_i\to X$ is Kummer étale, resp. finite Kummer étale, for every i and there exists an index $i_0$ such that $Z_j\to Z_i$ is finite Kummer étale and surjective for i and $j\geq i_0$ (and similarly for $\mathcal {X}$ ). For the covering families we refer to loc. cit.

We have a natural projection $\nu \colon X_{\mathrm {pke}}\to X_{\mathrm {ket}}$ (resp. $\nu \colon \mathcal {X}_{\mathrm {pke}}\to \mathcal {X}_{\mathrm {ket}}$ ) sending $U\in X_{\mathrm {ket}}$ (resp. in $\mathcal {X}_{\mathrm {ket}}$ ) to the constant inverse system defined by U. Then, by [Reference Diao, Lan, Liu and Zhu15, Prop. 5.1.6 & 5.1.7] for every sheaf of abelian groups ${\cal F}$ on $X_{\mathrm {ket}}$ (resp. in $\mathcal {X}_{\mathrm {ket}}$ ) and any quasi-compact and quasi-separated object $U={\displaystyle {\lim _{\leftarrow }}} U_j$ in $X_{\mathrm {pke}}$ (resp. in $\mathcal {X}_{\mathrm {pke}}$ ), we have natural isomorphisms of $\delta $ -functors

$$ \begin{align*}\mathrm{H}^i\bigl(U_{\mathrm{pke}},\nu^{-1}({\cal F})\bigr)\cong \lim_{\to} \mathrm{H}^i\bigl(U_{j,{\mathrm{ket}}},{\cal F}\bigr), \qquad {\cal F}\to \mathrm{R}\nu_{\ast}\nu^{-1}\bigl({\cal F}\bigr). \end{align*} $$

2.2 Sheaves on the pro-Kummer étale site

We then have the following sheaves on $\mathcal {X}_{{\mathrm {pke}}}$ defined in [Reference Diao, Lan, Liu and Zhu15, Def. 5.4.1] and in [Reference Diao, Lan, Liu and Zhu16, Def. 2.2.3] following [Reference Scholze22, Def. 6.1]:

1. i. The structure sheaf ${\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}:=\nu ^{-1}\bigl ({\cal O}_{\mathcal {X}_{\mathrm {ket}}}\bigr )$ and its subsheaf of integral elements ${\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}^+:=\nu ^{-1}\bigl ({\cal O}_{\mathcal {X}_{\mathrm {ket}}}^+\bigr )$ . It comes endowed with a morphism of sheaves of multiplicative monoids $\alpha \colon {\cal M} \to {\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}$ defined by taking $\nu ^{-1}$ of the morphism of sheaves of multiplicative monoids defining the log structure on $\mathcal {X}$ .

2. ii. The completed sheaf $\widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}^+:=\displaystyle {\lim _{\infty \leftarrow n}} {\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}^+/p^n {\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}^+$ and the completed structure sheaf $ \widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}:=\widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}^+\bigl [ \frac {1}{p}\bigr ]$ .

3. iii. Let K be a perfectoid field of characteristic $0$ with an open and bounded subring $K^+$ . Assume that $\mathcal {X}$ is defined over $\mathrm {Spa}(K,K^+)$ . Then we have the tilted integral structure sheaf $\widehat {{\cal O}}_{\mathcal {X}^{\flat }_{{\mathrm {pke}}}}^+:= \displaystyle {\lim _{\leftarrow \varphi }} {\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}^+/p {\cal O}_{\mathcal {X}_{{\mathrm {pke}}}}^+$ and the tilted structure sheaf $\widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}^{\flat }}:=\widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}^{\flat }}^+\otimes _{K^{\flat +}} K^{\flat }$ . It comes endowed with a morphism of monoids $\alpha ^{\flat } \colon {\cal M}^{\flat } \to \widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}^{\flat }}$ , where ${\cal M}^{\flat }$ is the inverse limit $\displaystyle {\lim _{\leftarrow }} {\cal M}$ indexed by ${\mathbb N}$ with transition maps given by raising to the p-th power, $\widehat {{\cal O}}_{\mathcal {X}^{\flat }}$ is identified as a sheaf of mutiplicative monoids with the inverse limit $\displaystyle {\lim _{\leftarrow }} \widehat { {\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}$ indexed by ${\mathbb N}$ with transition maps given by rasing to the p-th power and the map $\alpha ^{\flat }$ is the inverse limit of the maps $\alpha $ composed with the natural maps ${ {\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}\to \widehat { {\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}$ .

4. iv. The period sheaf $\mathbb {A}_{\mathrm {inf}}:=\mathrm {W}\bigl (\widehat {{\cal O}}_{\mathcal {X}^{\flat }_{{\mathrm {pke}}}}^+ \bigr )$ and the period map $\vartheta \colon \mathbb {A}_{\mathrm {inf}}\to \widehat {{\cal O}}_{\mathcal {X}_{{\mathrm {pke}}}}^+$ .

2.3 Log affinoid perfectoid opens

Consider a locally noetherian fs log adic space $\mathcal {X}$ over $\mathrm {Spa}({\mathbb Q}_p,{\mathbb Z}_p)$ . Following [Reference Diao, Lan, Liu and Zhu15, Def. 5.3.1 & Rmk. 5.3.2], an object $U=\lim _{i\in I} U_i$ , with $U_i=\bigl (\mathrm {Spa}(R_i,R_i^+),{\cal M}_i\bigr )$ in $\mathcal {X}_{\mathrm {pke}}$ is called log affinoid perfectoid if:

1. a. There is an initial object $0 \in I$ .

2. b. Each $U_i$ admits a global sharp finite saturated chart $P_i$ such that each transition map $U_j\to U_i$ is modeled on the Kummer chart $P_i \to P_j$ ;

3. c. $\bigl (\mathrm {Spa}(R_i,R_i^+)\bigr )_i$ is affinoid perfectoid, that is, the p-adic completion $(R,R^+)$ of $\lim _i (R_i,R_i^+)$ is a perfectoid affinoid ${\mathbb Q}_p$ -algebra;

4. d. The monoid $P = \lim _i P_i$ is n-divisible for all n.

Given a log affinoid perfectoid U as above, we denote by $\widehat {U}:=\mathrm {Spa}\bigl (R,R^+\bigr )$ the associated perfectoid affinoid space. By [Reference Diao, Lan, Liu and Zhu15, Lemma 5.3.6], it has the same underlying topological space as U (which is defined as the inverse limit of topological spaces $\displaystyle \lim _{\leftarrow \ i} \vert U_i\vert $ ). Moreover, by [Reference Diao, Lan, Liu and Zhu15, Thm. 5.4.3] and [Reference Scholze22, Thm. 6.5], we have that

$$ \begin{align*} \widehat{{\cal O}}_{\mathcal{X}_{\mathrm{pke}}}^+(U)&=R^+,\quad \widehat{{\cal O}}_{\mathcal{X}_{\mathrm{pke}}}(U)=R, \quad \widehat{{\cal O}}_{\mathcal{X}^{\flat}_{\mathrm{pke}}}^+(U)=R^{\flat+}, \\ \widehat{{\cal O}}_{\mathcal{X}^{\flat}_{\mathrm{pke}}}(U)&=R^{\flat}, \quad \mathbb{A}_{\mathrm{inf}}(U)=W\bigl(R^{\flat+}\bigr) \end{align*} $$

and the cohomology groups

$$ \begin{align*} \mathrm{H}^i\bigl(U,\widehat{{\cal O}}_{\mathcal{X}_{\mathrm{pke}}}^+\bigr) \sim 0 ,\quad \mathrm{H}^i\bigl(U,\widehat{{\cal O}}_{\mathcal{X}^{\flat}_{\mathrm{pke}}}^+\bigr) \sim 0 , \quad \mathrm{H}^i\bigl(U, \mathbb{A}_{\mathrm{inf}}\bigr) \sim 0 \quad \forall i\geq 1 \end{align*} $$

(where $\sim $ means almost $0$ ).

Thanks to [Reference Diao, Lan, Liu and Zhu15, Prop. 5.3.12 & Prop. 5.3.13], there exists a basis $\mathcal {B}$ for the site $\mathcal {X}_{\mathrm {pke}}$ given by log affinoid perfectoid objects such that for every locally constant p-torsion sheaf $\mathbb {L}$ on $\mathcal {X}_{\mathrm {ket}}$ and every $U\in \mathcal {B}$ we have $\mathrm {H}^i\bigl ( \mathcal {X}_{{\mathrm {pke}}}\vert _U,\mathbb {L}\bigr )=0$ for $i\geq 1$ . In case X is a fs log scheme over ${\mathbb Q}_p$ , there is an analogous notion of log affinoid perfectoid opens of $X_{\mathrm {pke}}$ , and it follows from the arguments in loc. cit. that there exists a basis of $X_{\mathrm {pke}}$ with the same property.

We recall that K was defined in the previous section as a perfectoid field of characteristic $0$ with an open and bounded subring $K^+$ . Assume that $\mathcal {X}$ is defined over $\mathrm {Spa}(K,K^+)$ . In this case, $\mathbb {A}_{\mathrm {inf}}$ , resp. $\widehat {{\cal O}}_{\mathcal {X}}^+$ , is a sheaf of algebras over the classical period ring $\mathrm {A}_{\mathrm {inf}}:=\mathrm {W}\bigl (K^{\flat +}\bigr )$ , resp. over $K^+$ , and given a generator $\zeta \in \mathrm {A}_{\mathrm {inf}}$ for the kernel of the canonical ring homomorphism $\mathrm {A}_{\mathrm {inf}}\to K^+$ , it follows from [Reference Scholze22, Lemma 6.3] that we have an exact sequence

$$ \begin{align*} 0 \longrightarrow \mathbb{A}_{\mathrm{inf}} \stackrel{\cdot \zeta}{\longrightarrow} \mathbb{A}_{\mathrm{inf}} \stackrel{\vartheta}{\longrightarrow} \widehat{{\cal O}}_{\mathcal{X}_{\mathrm{pke}}}^+ \longrightarrow 0. \end{align*} $$

2.4 Comparison results

Assume that $\mathcal {X}$ is a finite saturated locally noetherian log adic space over a perfectoid field $\mathrm {Spa}(K,K^+)$ with K algebraically closed of characteristic $0$ . Firstly, the main result of [Reference Diao, Lan, Liu and Zhu15], namely Theorem 6.2.1, states that if the underlying adic space to $\mathcal {X}$ is log smooth and proper and $\mathbb {L}$ is an $\mathbb {F}_p$ -local system on $\mathcal {X}_{\mathrm {ket}}$ , then the cohomology groups $\mathrm {H}^i(\mathcal {X}_{\mathrm {ket}},\mathbb {L}\big )$ are finite for all i, they vanish for $i\gg 0$ and the natural map

$$ \begin{align*} \mathrm{H}^i(\mathcal{X}_{\mathrm{ket}},\mathbb{L}) \otimes K^+/p \to \mathrm{H}^i(\mathcal{X}_{\mathrm{ket}},\mathbb{L}\otimes {\cal O}_{\mathcal{X}_{\mathrm{ket}}}^{+}/p) \end{align*} $$

is an almost isomorphism for every $i\geq 0$ . As ${\cal F}\cong \mathrm {R}\nu _{\ast }\nu ^{-1}\bigl ({\cal F}\bigr )$ for any sheaf of abelian goups, we obtain the same cohomology groups replacing $\mathcal {X}_{\mathrm {ket}}$ with $\mathcal {X}_{\mathrm {pke}}$ in the isomorphisms above. Here, we denoted $\nu \colon \mathcal {X}_{\mathrm {pke}}\longrightarrow \mathcal {X}_{\mathrm {ket}}$ the natural morphism of sites.

Second, in the case X is finite separated locally noetherian log scheme, proper and log smooth over K, we have a géométrie algébrique et géométrie analytique (GAGA) type comparison isomorphism. Let $\mathcal {X}$ be the associated log adic space over $\mathrm {Spa}(K,K^+)$ . We have a natural morphism of sites $\gamma \colon \mathcal {X}_{\mathrm {ket}} \to X_{\mathrm {ket}}$ . Let $\mathbb {L}$ be an $\mathbb {F}_p$ -local system on $X_{\mathrm {ket}}$ . Then

Proof. Let $X^o$ and $\mathcal {X}^o$ be the scheme, resp. the adic space defined by X and $\mathcal {X}$ forgetting the log structures. In this case, the morphism of sites $\gamma ^o\colon \mathcal {X}_{\mathrm {et}}^o \to X_{\mathrm {et}}^o$ induces the map $\mathrm {H}^i(X_{\mathrm {et}}^o,F\big ) \longrightarrow \mathrm {H}^i(\mathcal {X}_{\mathrm {et}}^o,\gamma ^{o,\ast }\bigl (F\bigr )\big )$ for every sheaf of torsion abelian groups F. It is an isomorphism due to [Reference Huber17, Thm. 3.2.10]. Consider the commutative diagram of sites

$$ \begin{align*}\begin{array}{ccc} \mathcal{X}_{\mathrm{et}} & \stackrel{\gamma}{\longrightarrow} & X_{\mathrm{et}} \cr \alpha \big\downarrow & & \big\downarrow \beta \cr \mathcal{X}_{\mathrm{et}}^o & \stackrel {\gamma^o}{\longrightarrow} & X_{\mathrm{et}}^o. \cr \end{array} \end{align*} $$

Using the compatibility of the Leray spectral sequences $\mathrm {H}^i(\mathcal {X}_{\mathrm {et}}^o,\mathrm {R}^j \alpha _{\ast } \gamma ^{\ast }(\mathbb {L})\big ) \Longrightarrow \mathrm {H}^{i+j}(\mathcal {X}_{\mathrm {ket}},\gamma ^{\ast }\bigl (\mathbb {L}\bigr )\big )$ and $\mathrm {H}^i(X_{\mathrm {et}}^o,\mathrm {R}^j \beta _{\ast } (\mathbb {L})\big ) \Longrightarrow \mathrm {H}^{i+j}(X_{\mathrm {ket}},\mathbb {L}\big )$ and the result of Huber, it suffices to prove that the natural morphism

$$ \begin{align*} \gamma^{o,\ast}\bigl(\mathrm{R}^j \beta_{\ast} (\mathbb{L}) \bigr) \longrightarrow \mathrm{R}^j \alpha_{\ast}\bigl( \gamma^{\ast}(\mathbb{L}\bigr) \end{align*} $$

is an isomorphism of sheaves for every j. It suffices to prove that we get an isomorphism after passing to stalks at geometric points $\zeta =\mathrm {Spa}\bigl (l,l^+\bigr ) \to \mathcal {X}^o$ as those form a conservative family by [Reference Huber17, Prop. 2.5.5]. Recall that $\zeta $ might consist of more than one point but it has a unique closed point $\zeta _0$ . Taking the stalk at $\zeta $ is equivalent to take global sections over the associated strictly local adic space $\mathcal {X}^o(\zeta )$ (see [Reference Huber17, Lemma 2.5.12]). Let $X^o(\zeta _0)$ be the spectrum of the strict Henselization of X at $\zeta _0$ . Taking the stalk at $\zeta $ of $\gamma ^{o,\ast }$ of a sheaf is equivalent to taking the sections of that sheaf over $X^o(\zeta _0)$ . We have a natural map of sites $\mathcal {X}^o(\zeta )_{\mathrm {ket}}\to X^o(\zeta _0)_{\mathrm {ket}}$ , considering on $X^o(\zeta )$ and on $ \mathcal {X}^o(\zeta )$ the log structures coming from X and $\mathcal {X}$ . We need to show that it is an equivalence. In both cases, the Kummer étale sites are the same as the finite Kummer étale sites; indeed by definition the Kummer étale topology is generated in both cases by finite Kummer étale covers and classical étale morphisms and a Kummer cover of $X^o(\zeta )$ , resp. $ \mathcal {X}^o(\zeta )$ , is still strictly local and hence does not admit any nontrivial classical étale cover (see [Reference Huber17, Lemma 2.5.6] in the adic setting). Both in the schematic and in the adic setting, the finite Kummer étale sites are equivalent to the category of finite sets with continuous action of the group $\mathrm {Hom}(\overline {M}^{\mathrm {gp}},\widehat {{\mathbb Z}}\big )$ with $\overline {M}$ the stalk of the log structure at $\zeta $ , modulo $l^{\ast }$ . See [Reference Illusie18, Ex. 4.7(a)] in the schematic case and [Reference Diao, Lan, Liu and Zhu15, Prop. 4.4.7] in the adic case. As such quotient is the same in the schematic and adic cases, the conclusion follows.

3.1 VBMS, that is, vector bundles with marked sections

We recall the main constructions of [Reference Andreatta and Iovita2] and [Reference Andreatta, Iovita and Pilloni4]. Let $\mathcal {X}$ denote an adic analytic space over $\mathrm {Spa}({\mathbb Q}_p, {\mathbb Z}_p)$ and let $({\cal E}, {\cal E}^+)$ denote a pair consisting of a locally free ${\cal O}_{\mathcal {X}}$ -module ${\cal E}$ of rank $2$ and a subsheaf ${\cal E}^+$ of ${\cal E}$ which is a locally free ${\cal O}_{\mathcal {X}}^+$ -module of rank $2$ such that ${\cal E}={\cal E}^+\otimes _{{\cal O}_{\mathcal {X}}^+}{\cal O}_{\mathcal {X}}$ . Let ${\cal I}\subset {\cal O}_{\mathcal {X}}^+$ be an invertible ideal such that ${\cal I}$ gives the topology on ${\cal O}_{\mathcal {X}}^+$ , and let $r\ge 0$ be an integer such that $ {\cal I} \subset p^r {\cal O}_{\mathcal {X}}^+$ .

We suppose that there is a section $s\in \mathrm {H}^0(\mathcal {X}, {\cal E}^+/{\cal I}{\cal E}^+)$ such that the submodule $\bigl ({\cal O}_{\mathcal {X}}^+/{\cal I}\bigr ) s$ is a direct summand of ${\cal E}^+/{\cal I}{\cal E}^+$ . We have the following.

Theorem 3.1 [Reference Andreatta, Iovita and Pilloni4].

a) The functor attaching to every adic space $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ such that $t^{\ast }({\cal I})$ is an invertible ideal in ${\cal O}_{{\cal Z}}^+$ , the set (group in fact):

$$ \begin{align*}{\mathbb V}({\cal E}, {\cal E}^+)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X} \bigr):=\mathrm{Hom}_{{\cal O}_{{\cal Z}}^+}\bigl(\gamma^{\ast}({\cal E}^+), {\cal O}_{{\cal Z}}^+ \bigr)=\mathrm{H}^0\bigl({\cal Z}, \gamma^{\ast}({\cal E}^+)^{\vee} \bigr), \end{align*} $$

is represented by the adic vector bundle ${\mathbb V}({\cal E},{\cal E}^+):=\mathrm {Spa}_{\mathcal {X}}\bigl (\mathrm {Sym}({\cal E}), \mathrm {Sym}({\cal E}^+) \bigr )\rightarrow \mathcal {X}$ .

b) The subfunctor of ${\mathbb V}({\cal E}, {\cal E}^+)$ denoted ${\mathbb V}_0({\cal E}^+, s)$ which associates to every adic space $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ as above, the set:

$$ \begin{align*}{\mathbb V}_0({\cal E}^+,s)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr):=\Bigl\{h\in{\mathbb V}({\cal E},{\cal E}^+)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr) \ \vert \ h\bigl(\mathrm{mod}\ \gamma^{\ast}({\cal I}) \bigr)(\gamma^{\ast}(s))=1\Bigr\}, \end{align*} $$

is represented by the open adic subspace of ${\mathbb V}({\cal E},{\cal E}^+)$ , also denoted ${\mathbb V}_0({\cal E}^+,s)$ , consisting of the points x such that $\vert \tilde {s}-1\vert _x\le \vert \alpha \vert _x$ , where $\tilde {s}$ is a (local) lift of s to ${\cal E}^+$ and $\alpha $ is a (local) generator of ${\cal I}$ at x.

c) Suppose that we have sections s and $t\in \mathrm {H}^0(\mathcal {X}, {\cal E}^+/{\cal I}{\cal E}^+)$ which form an $\bigl ({\cal O}_{\mathcal {X}}^+/{\cal I}\bigr )$ -basis of ${\cal E}^+/{\cal I}{\cal E}^+$ . Then, the subfunctor ${\mathbb V}_0({\cal E}^+,s,t)$ of ${\mathbb V}_0({\cal E}^+, s)$ which associates to every adic space $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ , the set:

$$ \begin{align*}{\mathbb V}_0({\cal E}^+,s,t)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr):=\Bigl\{h\in{\mathbb V}_0({\cal E}^+,s)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr) \ \vert \ h\bigl(\mathrm{mod}\ \gamma^{\ast}({\cal I}) \bigr)(\gamma^{\ast}(t))=0\Bigr\}, \end{align*} $$

is represented by the open adic subspace ${\mathbb V}_0({\cal E}^+,s, t)$ of ${\mathbb V}_0({\cal E}^+, s)$ consisting of the points x such that $\vert \tilde {t}\vert _x\le \vert \alpha \vert _x$ for a (any) lift $\tilde {t}$ of t to ${\cal E}^+$ and $\alpha $ a (local) generator of ${\cal I}$ at x.

Proof. The proof is local on $\mathcal {X}$ . Assume that $U\subset \mathcal {X}$ is an affinoid open $U=\mathrm {Spa}(R,R^+)$ such that ${\cal I}\vert _U$ is principal generated by $\alpha \in R^+$ and ${\cal E}^+\vert _U$ is free with basis $f_0$ , $f_1$ with $f_0(\mathrm {mod}\ \alpha )=s\vert _U$ . Then $f_1(\mathrm {mod}\ \alpha ) $ generates $\Bigl (\bigl ({\cal E}^+/{\cal I}{\cal E}^+\bigr )/s\bigl ({\cal O}_{\mathcal {X}}^+/{\cal I} \bigr )\Bigr )\vert _U$ and we assume in case (c) that $f_1(\mathrm {mod}\ \alpha )=t\vert _U$ . Then by [Reference Andreatta and Iovita2, §2] we have ${\mathbb V}({\cal E},{\cal E}^+)\vert _U=\mathrm {Spa}\bigl (R\langle X,Y\rangle , R^+\langle X,Y\rangle \bigr )$ and

$$ \begin{align*} {\mathbb V}_0({\cal E}^+,s)\vert_U=\mathrm{Spa}\bigl(R\langle X,Y\rangle\langle\frac{X-1}{\alpha}\rangle , R^+\langle X,Y\rangle\langle \frac{X-1}{\alpha}\rangle \bigr)=\mathrm{Spa}\bigl(R\langle Z,Y\rangle, R^+\langle Z,Y\rangle \bigr), \end{align*} $$

where $\displaystyle X=1+\alpha Z$ giving also the map to ${\mathbb V}({\cal E},{\cal E}^+)\vert _U$ . Similarly,

$$ \begin{align*} {\mathbb V}_0({\cal E}^+,s,t)\vert_U=\mathrm{Spa}\bigl(R\langle Z,W\rangle, R^+\langle Z,W\rangle \bigr) \end{align*} $$

with $\displaystyle Y=\alpha W$ . We have the tautological map over ${\mathbb V}({\cal E},{\cal E}^+)\vert _U$ given by

$$ \begin{align*} {\cal E}^+\otimes_{R^+} R^+\langle X,Y\rangle \to R^+\langle X,Y\rangle,\qquad f_0\mapsto X, f_1\mapsto Y\end{align*} $$

from which we deduce similarly the tautological maps over ${\mathbb V}_0({\cal E}^+,s)\vert _U$ and ${\mathbb V}_0({\cal E}^+,s,t)\vert _U$ providing the claimed representability and concluding the proof.

3.2 Dual VBMS

In this article, we’ll need a variant of the construction in Section §3.1 which we now present. Suppose that $\mathcal {X}$ , ${\cal I}$ , $({\cal E}, {\cal E}^+)$ are as in Section §3.1. Moreover, we assume that there is an exact sequence of locally free ${\cal O}_{\mathcal {X}}^+/{\cal I}$ -modules

$$ \begin{align*} 0\longrightarrow \mathcal{Q}\longrightarrow {\cal E}^+/{\cal I}{\cal E}^+\longrightarrow {\cal F}\longrightarrow 0 \end{align*} $$

and a section $s\in \mathrm {H}^0\bigl (\mathcal {X}, {\cal F} \bigr )$ such that $\bigl ({\cal O}_{\mathcal {X}}^+/{\cal I} \bigr )s={\cal F}$ . We have:

Theorem 3.2. The subfunctor ${\mathbb V}_0^D({\cal E}^+, \mathcal {Q}, s)$ of ${\mathbb V}({\cal E},{\cal E}^+)$ , defined by associating to every adic space $t\colon {\cal Z}\rightarrow \mathcal {X}$ as in Section §3.1 the set

$$ \begin{align*}{\mathbb V}_0^D({\cal E}^+,\mathcal{Q}, s)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr):= \end{align*} $$

$$ \begin{align*}:=\!\Bigl\{\!h\in{\mathbb V}({\cal E},{\cal E}^+)\bigl(\gamma\colon {\cal Z}\rightarrow \mathcal{X}\bigr) \, |\, h\bigl(\mathrm{mod}\ \gamma^{\ast}({\cal I}) \bigr)\bigl(\gamma^{\ast}(\mathcal{Q})\bigr)=0\ \mathrm{ and }\ h\bigl(\mathrm{mod}\ \gamma^{\ast}({\cal I}) \bigr)(\gamma^{\ast}(s))=1\!\Bigr\} \end{align*} $$

is represented by the the open adic subspace of ${\mathbb V}({\cal E}, {\cal E}^+)$ denoted ${\mathbb V}_0^D({\cal E}^+, \mathcal {Q},s)$ and consisting of the points x such that $\vert q\vert _x\le \vert \alpha \vert _x$ and $\vert \tilde {s}-1\vert _x\le \vert \alpha \vert _x$ , where q is a (local) lift to ${\cal E}^+$ of a local generator of $\mathcal {Q}$ at x, $\alpha $ is a (local) generator of ${\cal I}$ at x and $\tilde {s}$ is a (local) lift of s to ${\cal E}^+$ .

3.3 The sheaves ${\mathbb W}_k$ and ${\mathbb W}^D_k$

Let the hypothesis be as in Section §3.1 and §3.2. We assume that we have a morphism of adic spaces $\mathcal {X}\to {\cal W}$ where let us recall that ${\cal W}$ is the adic weight space for $\mathrm {\mathbf {GL}}_{2,/{\mathbb Q}}$ . For every adic space ${\cal Z}$ , the morphisms ${\cal Z}\to {\cal W}$ classify continuous homomorphisms ${\mathbb Z}_p^{\ast } \to \Gamma \bigl ({\cal Z},{\cal O}_{{\cal Z}}\bigr )$ . We denote by $k^{\mathrm {univ}}\colon {\mathbb Z}_p^{\ast } \to \Gamma \bigl (\mathcal {X},{\cal O}_{\mathcal {X}}\bigr )$ the continuous homomorphism defined by $\mathcal {X}\to {\cal W}$ . We assume that $k^{\mathrm {univ}}$ satisfies the following analyticity assumption: There exists a section $u_{\mathrm {univ}}$ of ${\cal O}_{\mathcal {X}}$ such that $\vert u_{\mathrm {univ}} \vert _x < \vert p^{\frac {1}{p-1} -r} \vert _x$ for every $x\in \mathcal {X}$ and

$$ \begin{align*} k^{\mathrm{univ}}(t)=\exp u_{\mathrm{univ}} \log(t), \quad \forall t\in 1+ p^r {\mathbb Z}_p. \end{align*} $$

We recall that the integer $r\ge 0$ is such that ${\cal I}\subset p^r {\cal O}_{\mathcal {X}}^+$ . Let us denote by ${\cal T}$ the adic torus representing the functor which associates to an adic space $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ the group

$$ \begin{align*} {\cal T}(\gamma\colon {\cal Z}\rightarrow \mathcal{X}):=1+\gamma^{\ast}({\cal I}). \end{align*} $$

Then $k^{\mathrm {univ}}$ defines a character $k^{\mathrm {univ}}\colon {\cal T} \to \mathbb {G}_m$ , that is, a morphism of adic spaces and group functors, using the fomula above.

We have natural actions of ${\cal T}$ on ${\mathbb V}_0({\cal E}^+, s)$ , ${\mathbb V}_0({\cal E}^+, s,t)$ and ${\mathbb V}_0^D\bigl ({\cal E}^+,\mathcal {Q}, s \bigr )$ defined on $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ points by: $u\ast h:=uh$ and $u\ast h':=uh'$ , where $u\in {\cal T}(\gamma \colon {\cal Z}\rightarrow \mathcal {X})$ , $h\in {\mathbb V}_0({\cal E}^+, s)(\gamma \colon {\cal Z}\rightarrow \mathcal {X})$ , resp. $h\in {\mathbb V}_0({\cal E}^+, s,t)(\gamma \colon {\cal Z}\rightarrow \mathcal {X})$ , resp. $h'\in {\mathbb V}_0^D\bigl ( {\cal E}^+,\mathcal {Q}, s\bigr )(\gamma \colon {\cal Z}\rightarrow \mathcal {X})$ . Let us denote by $f\colon {\mathbb V}_0({\cal E}^+,s)\longrightarrow \mathcal {X}$ , $g\colon {\mathbb V}_0({\cal E}^+,s,t)\longrightarrow \mathcal {X}$ and by $f^D:{\mathbb V}_0^D\bigl ({\cal E}^+,\mathcal {Q}, s\bigr )\longrightarrow \mathcal {X}$ the structural morphisms.

Definition 3.3. We denote

$$ \begin{align*} {\mathbb W}_{k^{\mathrm{univ}}}({\cal E}^+, s):=f_{\ast}\bigl({\cal O}^+_{{\mathbb V}_0({\cal E}^+,s) } \bigr)[k^{\mathrm{univ}}],\ \ {\mathbb W}_{k^{\mathrm{univ}}}({\cal E}^+, s,t):=g_{\ast}\bigl({\cal O}_{{\mathbb V}_0({\cal E}^+,s, t) }^+\bigr)[k^{\mathrm{univ}}] \end{align*} $$

and

$$ \begin{align*}{\mathbb W}^D_{k^{\mathrm{univ}}}\bigl({\cal E}^+,\mathcal{Q}, s \bigr):=f^D_{\ast}\Bigl({\cal O}_{{\mathbb V}_0^D\bigl({\cal E}^+,\mathcal{Q}, s \bigr)}^+ \Bigr)[k^{\mathrm{univ}}], \end{align*} $$

where if ${\cal G}$ is an ${\cal O}_{\mathcal {X}}^+$ -module on $\mathcal {X}$ with an action by the torus ${\cal T}:=1+{\cal I}$ , we denote ${\cal G}[k^{\mathrm {univ}}]$ the subsheaf of ${\cal G}$ of sections x such that $u\ast x=k^{\mathrm {univ}}(u)x$ for all corresponding sections u of ${\cal T}$ .

3.3.1 Local descriptions of the sheaves ${\mathbb W}_k$ and ${\mathbb W}_k^D$

We assume all notations and assumptions of the proof of Theorem 3.1 and of Section §3.3. Let $U=\mathrm {Spa}(R,R^+)\subset \mathcal {X}$ be an affinoid open such that ${\cal E}^+\vert _U=f_0{\cal O}_U^++f_1{\cal O}_U^+$ , ${\cal I}\vert _U=\alpha {\cal O}_U^+$ and such that $f_0(\mathrm {mod}\ \alpha )=s\vert _U$ and $f_1(\mathrm {mod}\ \alpha ) $ generates $\Bigl (\bigl ({\cal E}^+/{\cal I}{\cal E}^+\bigr )/s\bigl ({\cal O}_{\mathcal {X}}^+/{\cal I} \bigr )\Bigr )\vert _U$ (respectively $f_1(\mathrm {mod}\ \alpha )=t\vert _U$ ).

In view of the next section, we also consider the dual situation, that is, recall that s is a global marked section of ${\cal E}^+$ modulo ${\cal I}$ and denote ${\cal F}:=({\cal E}^+/{\cal I}{\cal E}^+)/\bigl (s({\cal O}_{\mathcal {X}}^+/{\cal I})\bigr )$ . By dualizing, we obtain the exact sequence

$$ \begin{align*} 0\longrightarrow \mathcal{Q}\longrightarrow ({\cal E}^+)^{\vee}/{\cal I}({\cal E}^+)^{\vee} \longrightarrow ({\cal O}_{\mathcal{X}}^+/{\cal I})s^{\vee}\longrightarrow 0, \end{align*} $$

where $\mathcal {Q}:={\cal F}^{\vee }$ , which defines data for ${\mathbb V}_0^D$ . Then $({\cal E}^+)^{\vee }\vert _U=f_0^{\vee }{\cal O}_{U}^++ f_1^{\vee }{\cal O}_{U}^+$ , $\mathcal {Q}\vert _U=f_1^{\vee }\bigl ({\cal O}_U^+/\alpha {\cal O}_U^+\bigr )$ and $f_0^{\vee }(\mathrm {mod}\ \alpha )=s^{\vee }\vert _U$ . Therefore, as in [Reference Andreatta and Iovita2, Lemma 2.4], one proves that

$$ \begin{align*} {\mathbb V}_0({\cal E}^+,s)\vert_U=\mathrm{Spa}\bigl(R\langle X,Y\rangle\langle\frac{X-1}{\alpha}\rangle , R^+\langle X,Y\rangle\langle \frac{X-1}{\alpha}\rangle \bigr)=\mathrm{Spa}\bigl(R\langle Z,Y\rangle, R^+\langle Z,Y\rangle \bigr), \end{align*} $$

where $\displaystyle X=1+\alpha Z$ and

$$ \begin{align*}{\mathbb V}_0({\cal E}^+,s,t)\vert_U=\mathrm{Spa}\bigl(R\langle Z,W\rangle, R^+\langle Z,W\rangle\bigr) \end{align*} $$

with $Y=\alpha W$ . Similarly, we have

$$ \begin{align*} {\mathbb V}_0^D\bigl(({\cal E}^+)^{\vee},\mathcal{Q}, s^{\vee}\bigr)|_U=\mathrm{Spa}\Bigl(R\langle A,B\rangle\langle \frac{A-1}{\alpha}, \frac{B}{\alpha} \rangle, R^+\langle A,B\rangle\langle \frac{A-1}{\alpha}, \frac{B}{\alpha} \rangle \Bigr)= \end{align*} $$

$$ \begin{align*}=\mathrm{Spa}\bigl(R\langle C, D\rangle, R^+\langle C, D\rangle \bigr), \end{align*} $$

where $A=1+\alpha C$ and $B=\alpha D$ .

Therefore, we have

$$ \begin{align*}{\mathbb W}_{k^{\mathrm{univ}}}({\cal E}^+, s)(U)=R^+\langle Z,Y\rangle[k^{\mathrm{univ}}]=R^+\langle \frac{Y}{1+\alpha Z}\rangle k^{\mathrm{univ}}(1+\alpha Z), \end{align*} $$

$$ \begin{align*}{\mathbb W}_{k^{\mathrm{univ}}}({\cal E}^+, s,t)(U)=R^+\langle Z,W\rangle[k^{\mathrm{univ}}]=R^+\langle \frac{W}{1+\alpha Z}\rangle k^{\mathrm{univ}}(1+\alpha Z) \end{align*} $$

and

$$ \begin{align*}{\mathbb W}^D_{k^{\mathrm{univ}}}\bigl(({\cal E}^+)^{\vee},\mathcal{Q}, s^{\vee} \bigr)(U)=R^+\langle C,D\rangle[k^{\mathrm{univ}}]=R^+\langle \frac{D}{1+\alpha C}\rangle k^{\mathrm{univ}}(1+\alpha C). \end{align*} $$

3.4 The duality

Suppose that $\mathcal {X}$ , ${\cal I}$ , $ ({\cal E}, {\cal E}^+)$ , s are as in Section §3.1, and let us denote by

$$ \begin{align*} \iota \colon {\cal F}:=s\bigl({\cal O}_{\mathcal{X}}^+/{\cal I}\bigr)\hookrightarrow {\cal E}^+/{\cal I}{\cal E}^+.\\[-9pt] \end{align*} $$

Let $\bigl ({\cal E}', ({\cal E}^+)^{\vee }\bigr )$ denote the pair where $({\cal E}^+)^{\vee }$ is the ${\cal O}_{\mathcal {X}}^+$ -dual of ${\cal E}^+$ and ${\cal E}':=({\cal E}^+)^{\vee }\otimes _{{\cal O}_{\mathcal {X}}^+}{\cal O}_{\mathcal {X}}$ . Moreover, let us consider $\mathcal {Q}:=\mathrm {Ker}\bigl (({\cal E}^+/{\cal I}{\cal E}^+)^{\vee }\stackrel {\iota ^{\vee }}{\longrightarrow }{\cal F}^{\vee } \bigr )$ , where ${\cal F}^{\vee }$ is the ${\cal O}_{\mathcal {X}}^+/{\cal I}$ -dual of ${\cal F}$ . As explained in Theorems 3.1 and 3.2, we have adic spaces ${\mathbb V}_0({\cal E}^+,s)$ and ${\mathbb V}_0^D\bigl (({\cal E}^+)^{\vee }, \mathcal {Q}, s^{\vee }\bigr )$ over $\mathcal {X}$ . Consider the morphism of adic spaces

$$ \begin{align*} \langle \ , \ \rangle\colon {\mathbb V}_0({\cal E}^+, s)\times_{\mathcal{X}}{\mathbb V}_0^D\bigl(({\cal E}^+)^{\vee}, \mathcal{Q}, s^{\vee}\bigr)\longrightarrow {\mathbb V}_0({\cal O}_{\mathcal{X}}^+, 1),\\[-9pt] \end{align*} $$

defined on points as follows. Fix a morphism of adic spaces $\gamma \colon {\cal Z}\rightarrow \mathcal {X}$ . Consider $h\in {\mathbb V}_0({\cal E}^+,s)\bigl (\gamma \colon {\cal Z}\rightarrow \mathcal {X}\bigr )$ and $h'\in {\mathbb V}_0^D\bigl (({\cal E}^+)^{\vee }, \mathcal {Q}, s^{\vee }\bigr )\bigl (\gamma \colon {\cal Z}\rightarrow \mathcal {X}\bigr )$ . By definition, this is equivalent to giving morphisms of ${\cal O}_{{\cal Z}}^+$ -modules $h\colon \gamma ^{\ast }({\cal E}^+)\rightarrow {\cal O}_{{\cal Z}}^+$ with $h(\mathrm {mod}\ {\cal I})(\gamma ^{\ast }(s))=1$ and $h'\colon \gamma ^{\ast }({\cal E}^+)^{\vee }\rightarrow {\cal O}_{{\cal Z}}^+$ with $h'(\mathrm {mod}\ \gamma ^{\ast }({\cal I}))(\mathcal {Q})=0$ and $h'(\mathrm {mod}\ \gamma ^{\ast }({\cal I}))(\gamma ^{\ast }(s^{\vee }))=1$ . Then $h'(h)\in \mathrm {H}^0({\cal Z}, {\cal O}_{{\cal Z}}^+)$ and $h'(h)(\mathrm {mod}\ \gamma ^{\ast }({\cal I}))=1$ , that is, $h'(h)\in {\mathbb V}_0({\cal O}_{\mathcal {X}}^+, 1)\bigl (\gamma \colon {\cal Z}\rightarrow \mathcal {X} \bigr )$ . We define

$$ \begin{align*} \langle h, h'\rangle:=h'(h)\in {\mathbb V}_0({\cal O}_{\mathcal{X}}^+, 1)\bigl(\gamma\colon{\cal Z}\rightarrow \mathcal{X} \bigr). \end{align*} $$

Notice that ${\mathbb V}_0({\cal O}_{\mathcal {X}}^+, 1)$ is the affine one-dimensional space $\mathbb {A}^1_{\mathcal {X}}$ over $\mathcal {X}$ , with standard coordinate T. The torus ${\cal T} \times {\cal T}$ acts componentwisely on ${\mathbb V}_0({\cal E}^+, s)\times _{\mathcal {X}}{\mathbb V}_0^D\bigl (({\cal E}^+)^{\vee }, \mathcal {Q}, s^{\vee }\bigr )$ . Given sections $(h,h')$ of the latter and $(u,u')$ of ${\cal T}\times {\cal T}$ , we have $\langle u \ast h, u' \ast h'\rangle =(u \cdot u') \ast \langle h, h'\rangle $ .

Proof. The statement is local on $\mathcal {X}$ . We use the explicit coordinates of §3.3.1. Then, $T=1+\alpha V$ and $T^{k^{\mathrm {univ}}}$ is the section $ k^{\mathrm {univ}}(1+\alpha V)$ . By loc. cit., we have ${\mathbb W}_{k^{\mathrm {univ}}}({\cal O}_{\mathcal {X}}^+, 1)(U)= T^{k^{\mathrm {univ}}} \cdot R^+$ .

As $\langle \ , \ \rangle ^{\ast }\bigl ( T \bigr )=X\otimes A + Y \otimes B=\big (1+\alpha Z )\big ) \cdot \bigl (1+\alpha C \bigr )+ \alpha Y \otimes D$ , we conclude that

(1) $$ \begin{align} \langle \ , \ \rangle^{\ast}\bigl( T^{k^{\mathrm{univ}}} \bigr)=k^{\mathrm{univ}}(1+\alpha Z) k^{\mathrm{univ}}(1+\alpha C) \cdot k^{\mathrm{univ}}\left(1+\alpha \frac{Y}{1+\alpha Z} \otimes \frac{D}{1+\alpha C}\right). \end{align} $$

This concludes the proof.

Definition 3.5. Write ${\mathbb W}_{k^{\mathrm {univ}}}({\cal E}^+, s)^{\vee }$ for the ${\cal O}_{\mathcal {X}}^+$ -dual of ${\mathbb W}_{k^{\mathrm {univ}}}({\cal E}^+, s)$ . Define the map of ${\cal O}_{\mathcal {X}}^+$ -modules

$$ \begin{align*} \xi_{k^{\mathrm{univ}}}\colon {\mathbb W}_{k^{\mathrm{univ}}}({\cal E}^+, s)^{\vee} \longrightarrow {\mathbb W}^D_{k^{\mathrm{univ}}}\bigl(({\cal E}^+)^{\vee},\mathcal{Q}, s^{\vee} \bigr),\quad \gamma \mapsto (\gamma \otimes 1) \bigl(\langle \ , \ \rangle^{\ast}\bigl(T^{k^{\mathrm{univ}}}\bigr)\bigr). \end{align*} $$

3.4.1 Local descriptions of the duality between ${\mathbb W}_k$ and ${\mathbb W}_k^D$

We put ourselves in the setting of §3.3.1 and compute explicitly the pairing $\xi _{k^{\mathrm {univ}}}$ on the affinoid U in terms of the local coordinates of loc. cit. As $k^{\mathrm {univ}}$ is supposed to be r-analytic on $\mathcal {X}$ , we can write $k^{\mathrm {univ}}(t)=\exp u_{\mathrm {univ}} \log (t)$ for every $t\in 1+p^r {\mathbb Z}_p$ . We then claim that

$$ \begin{align*} \xi_{k^{\mathrm{univ}}}\left( k^{\mathrm{ univ}}(1+\alpha Z) \bigl(\frac{Y}{1+\alpha Z} \bigr)^n\right)^{\vee}=\alpha^n \left( \begin{matrix}u_{\mathrm{univ}} \cr n \cr \end{matrix}\right) k^{\mathrm{univ}}(1+\alpha C) \bigl(\frac{D}{1+\alpha C}\bigr)^n , \end{align*} $$

where $\displaystyle \left ( \begin {matrix}u_{\mathrm {univ}} \cr n \cr \end {matrix}\right ) :=\frac { u_{\mathrm {univ}} (u_{\mathrm {univ}}-1) \cdots (u_{\mathrm {univ}}-n+1)}{n!}$ if $n\ge 1$ and $\displaystyle \left ( \begin {matrix}u_{\mathrm {univ}} \cr 0 \cr \end {matrix}\right )=1$ .

First of all, one computes that, if we write $f(X):=\exp u_{\mathrm {univ}} \log (1+X)=\sum _{n=0}^{\infty } a_n X^n$ as a formal power series in X, then $\displaystyle a_n= \left ( \begin {matrix}u_{\mathrm {univ}} \cr n \cr \end {matrix}\right )$ . Using equation (1), we deduce that

$$ \begin{align*} \frac{\langle \ , \ \rangle^{\ast}\bigl( T^{k^{\mathrm{univ}}} \bigr)}{k^{\mathrm{univ}}(1+\alpha Z) k^{\mathrm{univ}}(1+\alpha C)}= \sum_ {n=0}^{\infty} \alpha^n \left( \begin{matrix}u_{\mathrm{univ}} \cr n \cr \end{matrix}\right) \left(\frac{Y}{1+\alpha Z} \right)^n\otimes \left(\frac{D}{1+\alpha C} \right)^n, \end{align*} $$

and the claim follows.

3.5 An example: locally analytic functions and distributions

We consider in this article weights defined as follows. Let ${\cal W}$ denote the weight space seen as an adic space.

Let $k\colon {\mathbb Z}_p^{\ast }\longrightarrow B^{\ast }$ be a weight as in definition 3.6, and suppose it is an r-analytic character. Let $T={\mathbb Z}_p\oplus {\mathbb Z}_p$ ; denote by $f_0=(1,0)$ and $f_1=(0,1)\in T$ the standard ${\mathbb Z}_p$ -basis. Denote by $T^{\vee }$ the ${\mathbb Z}_p$ -dual of T, and let $T_0^{\vee } \subset T^{\vee }$ be the subset of elements ${\mathbb Z}_p^{\ast } e_0 \times {\mathbb Z}_p e_1 $ with $e_0=f_0^{\vee }$ and $e_1=f_1^{\vee }$ , the ${\mathbb Z}_p$ -basis of $T^{\vee }$ dual to $(f_0, f_1)$ . Then $T_0^{\vee }$ is a profinite set with an action of the Iwahori subgroup $\mathrm {Iw}_1\subset \mathrm {\mathbf {GL}}_2({\mathbb Z}_p)$ . Following [Reference Barrera and Gao7, Def. 3.1], we set:

By [Reference Barrera and Gao7, Lemma 3.1], the action of $\mathrm {Iw}_1$ on $T_0^{\vee }$ induces actions of $\mathrm {Iw}_1$ on $A_k^o(T_0^{\vee })[n]$ , $D_k^o(T_0^{\vee })[n]$ , $A_k(T_0^{\vee })[n]$ and $D_k(T_0^{\vee })[n]$ . Moreover, [Reference Barrera and Gao7, Def. 3.3 & Prop. 3.3] the B module $D_k^o(T_0^{\vee })[n]$ admits a decreasing filtration $\mathrm {Fil}^{\bullet } D_k^o(T_0^{\vee })[n]$ of B-modules, stable under the action of $\mathrm {Iw}_1$ , such that the graded pieces are finite and $D_k^o(T_0^{\vee })[n]$ is the inverse limit $\displaystyle {\lim _{\infty \leftarrow m}} D_k^o(T_0^{\vee })[n]/\mathrm {Fil}^m D_k^o(T_0^{\vee })[n]$ .

3.5.1 An alternative description

For later purposes, we end this section by describing $A_k(T_0^{\vee })[n]$ and $D_k(T_0^{\vee })[n]$ using the formalism of VBMS. For every $\lambda =0,\ldots , p^n-1$ denote by ${\mathbb W}_k(T,s,t+\lambda s ,p^n)$ , or simply ${\mathbb W}_k(T,s,t+\lambda s)$ if the power of p we are working with is clear from the context, the sections ${\mathbb W}_{k}(T, s,t+\lambda s)(U)$ over the adic space $U=\mathrm {Spa}(B[1/p],B)$ associated to the rank $2$ , free ${\cal O}_U^+$ -module $T\otimes {\cal O}^+_U$ and the two sections $s=f_0\otimes 1$ and $t+\lambda s=f_1\otimes 1+\lambda (f_0\otimes 1)$ modulo ${\cal I}:=p^n {\cal O}_U^+$ . Let $\mathrm {Iw}_n$ be the subgroup of matrices $M=\left (\begin {matrix}\alpha & \beta \cr \gamma & \delta \end {matrix} \right )\in \mathrm {GL}_2({\mathbb Z}_p)$ such that $\gamma \equiv 0$ modulo $p^n$ . Then:

Proposition 3.8. There is an $\mathrm {Iw}_n$ -equivariant isomorphism of B-modules

$$ \begin{align*} \nu_n\colon \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s) \longrightarrow A_k^o(T_0^{\vee})[n]. \end{align*} $$

Then, taking duals with respect to the weak topology on B, we get a decomposition into a direct sum and a $\mathrm {Iw}_n$ -equivariant isomorphism

$$ \begin{align*} \nu_n^{\vee}\colon D_k^o(T_0^{\vee})[n]\cong \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s)^{\vee}. \end{align*} $$

Proof. We describe the isomorphism explicitly. First of all, notice that $A_k^o(T_0^{\vee })[n]$ decomposes as a direct sum

$$ \begin{align*}A_k^o(T_0^{\vee})[n]=\oplus_{\lambda=0}^{p^n-1} A_{k,\lambda}^o(T_0^{\vee})\\[-9pt] \end{align*} $$

according to residue classes: We say that $f\in A_k^o(T_0^{\vee })[n]$ lies in $A_{k,\lambda }^o(T_0^{\vee })$ if an only if $f(1,w)$ is zero if $w\not \in \lambda + p^n {\mathbb Z}_p$ . In particular,

$$ \begin{align*} A_{k,\lambda}^o(T_0^{\vee})=B\langle w_{\lambda} \rangle \cdot u^k,\\[-9pt] \end{align*} $$

where $\sum _{m=0}^{\infty } a_m w_{\lambda }^m \cdot u^k\colon T_0^{\vee } \to B$ sends $u e_0 + v e_1 \mapsto k(u) \cdot \sum _m a_m \bigl (\frac {v/u -\lambda }{p^n}\bigr )^m $ if $v/u\in \lambda + p^n{\mathbb Z}_p$ and to $0$ otherwise. The standard left action of $\mathrm {Iw}_n$ on T is described as follows: Given $M=\left (\begin {matrix}\alpha & \beta \cr \gamma & \delta \end {matrix} \right )\in \mathrm {Iw}_n$ , we have $M (f_0) =\alpha f_0 + \gamma f_1$ , $M(f_1)=\beta f_0 + \delta f_1$ . This induces a right action given by $e_0 \cdot M =\alpha e_0 + \beta e_1$ , $e_1 \cdot M=\gamma e_0 + \delta e_1$ . We finally obtain the left action of $\mathrm {Iw}_n$ on $A_k^o(T_0^{\vee })[n]$ . Explicitly, as $(u e_0 + v e_1)\cdot M= (\alpha u +\gamma v) e_0 + (\beta u+ \delta v) e_1$ , then

$$ \begin{align*} M(w_0^m \cdot u^k ) = k(\alpha + \gamma w_0) \left(\frac{\beta + \delta w_0 }{\alpha +\gamma w_0}\right)^m \cdot u^k.\\[-9pt] \end{align*} $$

As $w_{\lambda }^m \cdot u^k = \left (\begin {matrix} 1 & -\lambda \cr 0 & 1 \end {matrix}\right ) (w_0^m \cdot u^k) $ we get the sought for action of $\mathrm {Iw}_n$ on $\oplus _{\lambda =0}^{p^n-1} A_{k,\lambda }^o(T_0^{\vee })$ .

On the other hand, consider the subfunctors $\amalg _{\lambda =0}^{p^n-1} {\mathbb V}_0(T,s,t-\lambda s) \to {\mathbb V}(T)$ over the adic space $U=\mathrm {Spa}(B[1/p],B)$ . The action of $\mathrm {Iw}_n$ on T restricts to an action on this subfunctors and induces an action on $\oplus _{\lambda =0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s)$ . Explicitly,

$$ \begin{align*} {\mathbb W}_{k}(T, s,t-\lambda s)=B\langle \frac{W_{\lambda}}{1+p^n Z}\rangle k(1+p^n Z)\\[-9pt] \end{align*} $$

according to §3.3.1. It contains the B-submodule of the space of integral functions $B\langle X,Y\rangle $ of ${\mathbb V}(T\otimes {\cal O}_U^+)$ , where $X=1+p^n Z$ and $Y= p^n W_{\lambda }+\lambda X$ . Recall that we have a universal map $T \to B\langle X,Y\rangle $ defined by sending $m f_0 + r f_1\mapsto m X + r Y$ . The left action of $\mathrm {Iw}_n$ on T defines by universality an action on $B\langle X,Y\rangle $ : If $M=\left (\begin {matrix} \alpha & \beta \cr \gamma & \delta \cr \end {matrix}\right )$ , then $M (f_0) =\alpha f_0 + \gamma f_1$ , $M(f_1)=\beta f_0 + \delta f_1$ and $M(X)=\alpha X + \gamma Y$ , $M(Y)=\beta X + \delta Y$ . Denote by $\mathrm {Iw}_n^1\subset \mathrm {Iw}_n$ the subgroup of matrices with $\alpha =1 $ modulo $p^n$ . We then get an action of $\mathrm {Iw}_n^1$ on the integral functions on $\amalg _{\lambda =0}^{p^n-1} {\mathbb V}_0(T,s,t-\lambda s)$ , and hence on $\oplus _{\lambda } {\mathbb W}_k(T,s,t-\lambda s )$ , determined on the variables Z and $W_{\lambda }$ ’s by the formulas

$$ \begin{align*}M(Z)=\frac{(\alpha X-1)}{p^n} + \gamma W_0,\quad M(W_0)=\frac{\beta}{p^n} X + \delta W_0, \quad \left(\begin{matrix} 1 &- \lambda \cr 0 & 1 \end{matrix}\right)(W_0)=W_{\lambda}. \end{align*} $$

Notice that if $M=\left (\begin {matrix} \alpha & 0 \cr 0 & \delta \cr \end {matrix}\right ) \in \mathrm {Iw}_n$ , then $M\bigl (k(1+p^n Z)\bigr )=k(\alpha ) k(1+p^n Z)$ and $M(W_0)=\delta W_0$ giving explicitly the action of diagonal matrices on each ${\mathbb W}_{k}(T, s,t-\lambda s)$ . For every $\lambda =0,\ldots , p^n-1$ define the map

$$ \begin{align*} \nu_{\lambda}\colon {\mathbb W}_{k}(T, s,t-\lambda s )\longrightarrow A_{k,\lambda}^o(T_0^{\vee})[n] , \quad k(1+p^n Z) \sum_i a_i \left(\frac{W_{\lambda}}{1+p^nZ}\right)^i\mapsto \sum_i a_i w_{\lambda}^i \cdot u^k. \end{align*} $$

It is clearly an isomorphism of B-modules. We are left to show that

$$ \begin{align*} \nu_n:=\sum_{\lambda=0}^{p^n-1} \nu_{\lambda}\colon \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s)\to \oplus_{\lambda=0}^{p^n-1} A_{k,\lambda}^o(T_0^{\vee})[n]=A_k^o(T_0^{\vee})[n] \end{align*} $$

is $\mathrm {Iw}_n$ -equivariant.

Consider the B-linear map $\xi \colon B\langle X,Y\rangle \to A(T^{\vee })$ , where $A(T^{\vee })$ is the ring of analytic functions from $T^{\vee }$ to B, sending $f(X,Y)=\sum _{h,m} a_{h,m} X^h Y^m$ to the function $\xi \bigl (f(X,Y)\bigr )\colon T^{\vee } \to B$ , $u e_0 + v e_1 \to \sum _{h,m} a_{h,m} u^hv^m$ . This map is $\mathrm {Iw}_n$ -equivariant. Indeed, given $M\in \mathrm {Iw}_n$ such that $M (f_0) =\alpha f_0 + \gamma f_1$ , $M(f_1)=\beta f_0 + \delta f_1$ then $M (e_0) =\alpha e_0 + \beta e_1$ , $M(e_1)=\gamma e_0 + \delta e_1$ so that $M(u e_0 + v e_1)= (u \alpha + v \gamma ) e_0 + (u\beta + \delta v) e_1$ . Hence, $M\bigl (\xi (f(X,Y)) \bigr )=\xi \bigl (f\bigl (M(X),M(Y)\bigr )\bigr )$ . As $\nu _n$ is determined by $\xi $ using that $X=1+p^nZ$ and $Y= p^n W_{\lambda }+\lambda X$ , this implies that $\nu _n$ is $\mathrm {Iw}_n$ -equivariant as well.

Note that we have a $\mathrm {Iw}_n$ -equivariant map of functors, and hence of representing objects, $\amalg _{\lambda \in {\mathbb Z}/p^n{\mathbb Z}}{\mathbb V}_0(T,s,t-\lambda s) \to {\mathbb V}_0(T,s)$ . This provides a $\mathrm {Iw}_n$ -equivariant map

$$ \begin{align*} {\mathbb W}_{k}(T, s) \to \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s). \end{align*} $$

Let $Q_n\subset T^{\vee }/p^n T^{\vee }$ be the $({\mathbb Z}/p^n{\mathbb Z})$ -dual of the quotient $\bigl (T/p^n T\bigr )/ ({\mathbb Z}/p^n{\mathbb Z}) s$ . The duality between

$$ \begin{align*} \zeta_k\colon {\mathbb W}_{k}(T, s)^{\vee} \longrightarrow {\mathbb W}_{k}^D(T^{\vee}, s^{\vee},Q_n) \end{align*} $$

of Definition 3.5 composed with the $\mathrm {Iw}_n$ -equivariant isomorphism

$$ \begin{align*} \nu_n^{\vee}\colon D_k^o(T_0^{\vee})[n] \cong \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s)^{\vee} \end{align*} $$

of Proposition 3.8 give the following.

Corollary 3.9. We have a $\mathrm {Iw}_n$ -equivariant, B-linear map

$$ \begin{align*} D_k^o(T_0^{\vee})[n] \cong \oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-\lambda s)^{\vee}\longrightarrow {\mathbb W}_{k}^D\bigl(T^{\vee}, s^{\vee},Q_n\bigr). \end{align*} $$

3.5.2 The $U_p$ operator

For $\rho =0,\ldots ,p-1$ , let $\pi _{\rho }\colon T \to T$ be the map defined by $\left (\begin {matrix} 1 & \rho \cr 0 & p \cr \end {matrix}\right ) $ , that is, $f_0\mapsto f_0$ , $f_1\mapsto p f_1+ \rho f_0$ . It defines the map $\pi _{\rho }^{\vee }\colon T^{\vee } \to T^{\vee }$ that sends $u e_0+ v e_1 \mapsto u e_0+(pv+\rho ) e_1$ . In particular, taking $(\pi _{\rho }^{\vee })^{\ast }$ it induces a map $A_{k}^o(T_0^{\vee })[n+1] \to A_{k}^o(T_0^{\vee })[n]$ that is $0$ on $A_{k,\lambda }^o(T_0^{\vee })[n+1] $ for $\lambda \not \equiv \rho $ modulo p and it induces a map $A_{k,\lambda }^o(T_0^{\vee })[n+1] \to A_{k,\lambda _0}^o(T_0^{\vee })[n] $ if $\lambda =\rho + p \lambda _0$ , with $\lambda _0\in \{ 0,\ldots , p^n-1\}$ . Taking the sum over the $\rho $ ’s, we get a map $\pi _n=\sum _{\rho =0}^{p-1} (\pi _{\rho }^{\vee })^{\ast }$ , where

$$ \begin{align*} \pi_n \colon A_{k}^o(T_0^{\vee})[n+1]&=\oplus_{\rho=0}^{p-1} \oplus_{\lambda_0=0}^{p^n-1} A_{k,\rho+p\lambda_0}^o(T_0^{\vee})[n+1] \to \oplus_{\lambda_0\in{\mathbb Z}/p^n {\mathbb Z}} A_{k,\lambda_0}^o(T_0^{\vee})[n]\\&=A_{k}^o(T_0^{\vee})[n]. \end{align*} $$

Notice that $\pi _{\rho }$ defines by functoriality a map ${\mathbb V}_0(T,s,t-\lambda s,p^n) \to {\mathbb V}_0(T,s,t-(\rho +p\lambda ) s,p^{n+1})$ (we have added the dependence on the power of p in the definition of ${\mathbb V}_0$ to avoid confusion). This gives a map $\mu _{\rho }\colon {\mathbb W}_{k}(T, s,t-(\rho +p\lambda ) s,p^{n+1})\to {\mathbb W}_k( s, t-\lambda s,p^{n}) $ and, summing over all $\rho $ ’s,

$$ \begin{align*} \mu_n=\sum_{\rho=0}^{p-1}\mu_{\rho}\colon\oplus_{\rho=0}^{p-1}\oplus_{\lambda=0}^{p^n-1} {\mathbb W}_{k}(T, s,t-(\rho+p\lambda) s,p^{n+1})\to \oplus_{\lambda\in {\mathbb Z}/p^n{\mathbb Z}} {\mathbb W}_{k}(T, s,t-\lambda s,p^{n}). \end{align*} $$

Lemma 3.10. With the notation of Proposition 3.8, we have $\pi _n \circ \nu _{n+1}=\nu _n \circ \mu _n$ and similarly taking strong duals $ \nu _{n+1}^{\vee } \circ \pi _n^{\vee }=\mu _n^{\vee } \circ \nu _n^{\vee }$ .

Proof. This is an explicit computation using the notation of the proof of Proposition 3.8 and follows from the fact that $ (\pi _{\rho }^{\vee })^{\ast }$ sends $w_{\rho +p\lambda _0} \mapsto w_{\lambda _0}$ and $\mu _{\lambda }$ sends $W_{\rho +p\lambda }\mapsto W_{\lambda }$ .

4.1 On the pro-Kummer étale topology of modular curves

For every $s\in {\mathbb N}$ , we write $\displaystyle {\lim _{\infty \leftarrow r}} \mathcal {X}(p^r,N)$ (for $r\geq s$ ) for the pro-finite Kummer étale cover of $\mathcal {X}_0(p^s,N)$ defined by the Kummer étale covers $\mathcal {X}(p^r,N)\to \mathcal {X}_0(p^s,N)$ , for $r\geq s$ . The following lemma provides a basis for the pro-Kumemr étale topology of $\mathcal {X}_0(p^s,N)$ :

4.2 Standard opens

We start by defining certain opens of ${\mathbb P}^1:={\mathbb P}^1_{{\mathbb Q}_p}$ , namely let for every $n\ge 1 U_0, U_{\infty }, U_{\infty }^{(n)}, U_0^{(n)}\subset {\mathbb P}^1$ be defined as follows. Let T denote a parameter at $0$ on ${\mathbb P}^1$ , then

1. a) $\displaystyle U_{\infty }=\{x\in {\mathbb P}^1 \ | \ \Vert \frac {1}{T}\Vert _x\le 1\}$ , $U_0=\{x\in {\mathbb P}^1 \ | \ \Vert T-\lambda \Vert _x\le 1, \mbox { for some }\lambda \in \{0,1,\ldots ,p-1\}\}$ ,

2. b) $\displaystyle U_{\infty }^{(n)}={\mathbb P}^1\bigl ( \frac {1}{p^nT} \bigr )=\{x\in {\mathbb P}^1\ | \ \Vert \frac {1}{T}\Vert _x\le \Vert p^n\Vert _x\}$ ,

3. c) $U_0^{(n)}=\cup _{\lambda } U_{0,\lambda }^{(n)}$ with $\lambda =\lambda _0+\lambda _1p+\ldots +\lambda _{n-1}p^{n-1}$ , where $\lambda _0,\lambda _1,\ldots ,\lambda _{n-1}\in \{0,1,\ldots ,p-1\}$ and we have $\displaystyle U_{0,\lambda }^{(n)}:={\mathbb P}^1\bigl ( \frac {T-\lambda }{p^n} \bigr )$ .

We remark that for every $n\ge 1$ we have ${\mathbb P}^1({\mathbb Q}_p, {\mathbb Z}_p)\subset U_{\infty }^{(n)}\cup U_0^{(n)}$ , and moreover, the family $\{U_{\infty }^{(n)}\cup U_0^{(n)}\}_{n\ge 1}$ is a fundamental system of open neighbourhoods of ${\mathbb P}^1({\mathbb Q}_p,{\mathbb Z}_p)$ in ${\mathbb P}^1$ .

We recall from [Reference Scholze23, Thm. 3.3.18] that for every rational open subset $U'\subset U_0$ or $U'\subset U_{\infty }$ the inverse image ${\cal U}':=\pi _{\mathrm {HT}}^{-1}\bigl (U'\bigr )$ is an affinoid perfectoid open subspace of $\mathcal {X}(p^{\infty },N)$ . In particular, it is quasi-compact so that it is also the inverse image of an open ${\cal U}^{\prime }_r$ via $\pi _r$ of $ \mathcal {X}(p^r,N)$ for some r using the homeomorphism $\vert \mathcal {X}(p^{\infty },N) \vert \cong \displaystyle {\lim _{\infty \leftarrow n}} \vert \mathcal {X}(p^r,N) \vert $ . As the transition maps in the inverse limit are finite and surjective, ${\cal U}^{\prime }_r$ is in fact the image of ${\cal U}'$ via $\pi _r$ for r large enough. If $U'$ is further invariant for the action of the Iwahori subgroup $\mathrm {Iw}_s\subset \mathrm {\mathbf {GL}}_2({\mathbb Z}_p)$ of matrices which are upper triangular modulo $p^s$ , then also ${\cal U}'$ is $\mathrm {Iw}_s$ -invariant as $\pi _{\mathrm {HT}}$ is $\mathrm { Iw}_s$ -equivariant and ${\cal U}'$ is the inverse image of a unique open ${\cal U}^{\prime }_{0,s}$ of $ \mathcal {X}_0(p^s,N)$ . Indeed, given ${\cal U}^{\prime }_r \subset \mathcal {X}(p^r,N)$ for some $r\geq s$ such that its inverse image gives ${\cal U}'$ , then ${\cal U}^{\prime }_r$ is $\mathrm { Iw}_s$ -invariant. As the morphism $\mathcal {X}(p^r,N) \to \mathcal {X}_0(p^s,N) $ is finite Kummer étale and Galois with group $G_s$ equal to the image of $\mathrm {Iw}_s\subset \mathrm {\mathbf {GL}}_2({\mathbb Z}_p)\to \mathrm {\mathbf {GL}}_2({\mathbb Z}/p^r{\mathbb Z})$ then ${\cal U}^{\prime }_{0,s}:={\cal U}^{\prime }_r/G_s$ is an open of $ \mathcal {X}_0(p^s,N) $ with the required properties. Furthermore, ${\cal U}^{\prime }_{0,s}$ defines the open $({\cal U}^{\prime }_r)_{r\geq s}$ for the pro-Kummer étale site of $ \mathcal {X}_0(p^s,N)$ , and hence of $\mathcal {X}$ , given by ${\cal U}^{\prime }_r:={\cal U}^{\prime }_{0,s} \times _{ \mathcal {X}_0(p^s,N)} \mathcal {X}(p^r,N)$ . By construction, ${\cal U}'\sim \displaystyle {\lim _{\infty \leftarrow r}} {\cal U}^{\prime }_r$ .

In particular, for every $n\ge 1$ , we consider the open rational subspaces $U_{\infty }^{(n)}$ of $U_{\infty }$ and $U_0^{(n)}$ of $U_0$ defined above. We remark that $U_{\infty }^{(n)}$ is invariant under the left action of the subgroup $\mathrm {Iw}_n$ . Then we denote by $\mathcal {X}(p^{\infty },N)_0^{(n)}:=\pi _{\mathrm {HT}}^{-1}\bigl (U_0^{(n)} \bigr ) $ and $\mathcal {X}(p^{\infty },N)_{\infty }^{(n)}:=\pi _{\mathrm {HT}}^{-1}\bigl (U_{\infty }^{(n)} \bigr ) $ and recall that they define affinoid perfectoid open subspaces of $ \mathcal {X}(p^{\infty },N)$ .

As explained above, they also define opens for the pro-Kummer étale site of $ \mathcal {X}_0(p^n,N)$ and of $\mathcal {X}$ respectively. Namely, for $n\ge 1$ , $\mathcal {X}(p^{\infty }, N)_{\infty }^{(n)}$ , being invariant under $\mathrm {Iw}_n$ , descends to an affinoid open denoted $\mathcal {X}_0(p^m, N)_{\infty }^{(n)}$ of $\mathcal {X}_0(p^m, N)$ , for all $m\ge n$ . We also have variants $\mathcal {X}_1(p^m, N)_{\infty }^{(n)}$ , resp. $\mathcal {X}(p^m, N)_{\infty }^{(n)}$ , if we descend to $\mathcal {X}_1(p^m, N)$ , resp. $\mathcal {X}(p^m, N)$ .

In contrast, as $U_0^{(n)}$ is invariant with respect to $\mathrm {Iw}_1$ , the open $\mathcal {X}(p^{\infty }, N)_0^{(n)}$ descends to an open affinoid denoted $\mathcal {X}_0(p^m, N)_0^{(n)}$ of $\mathcal {X}_0(p^m, N)$ , for all $m\ge 1$ . In this case, we consider the variant $\mathcal {X}(p^m, N)_0^{(n)}$ open of $\mathcal {X}(p^m, N)$ .

A reason to introduce the open subsets $\mathcal {X}_0(p^r,N)_{\infty }^{(n)}$ for $r\ge n$ and $\mathcal {X}_0(p^r,N)_0^{(n)}$ , for any $r \geq 1$ , is that they behave nicely under the $U_p$ -correspondence, as we will explain below.

4.3 On the Hodge–Tate period map

Over $\mathcal {X}(p^{\infty },N)$ , the sheaf $T_p(E)$ admits a universal trivialization

$$ \begin{align*} T_p(E)={\mathbb Z}_p a \oplus {\mathbb Z}_p b. \end{align*} $$

The map $\mathrm {dlog}$ defines a surjective map onto the sheaf of invariant differentials of E

$$ \begin{align*} T_p E^{\vee} \otimes_{{\mathbb Z}_p} {\cal O}_{\mathcal{X}(p^{\infty},N)} \longrightarrow \omega_E \end{align*} $$

which is used to define the map $\pi _{\mathrm {HT}}$ : For every log affinoid perfectoid open $W\,{=}\,\mathrm {Spa}(R,R^+)$ of $\mathcal {X}(p^{\infty },N)$ such that the universal elliptic curve extends to a (generalized) elliptic curve over $\mbox {Spec} (R^+)$ and $\omega _E$ is generated as $R^+$ -module by one element that we denote $\Omega _W$ , we write $\mathrm {dlog}(a^{\vee })= \alpha \Omega _W$ , $\mathrm {dlog}(b^{\vee })= \beta \Omega _W$ with $\alpha $ , $\beta \in R$ generating the whole ring R. Then $\pi _{\mathrm {HT}}\vert _W\colon W \to \mathbb {P}^1$ is defined in homogeneous coordinates by $[\alpha ;\beta ]$ . Namely, let $W_{\infty }\subset W$ be the rational open defined by $W(1/\alpha )$ , and let $W_0\subset W$ be the rational open defined by $W(1/\beta )$ . Then $\pi _{\mathrm {HT}}\vert _{W_0}\colon W_0\to U_0$ sends the standard coordinate T on the standard affinoid neighbourhood $U_0=\mathbb {A}^1$ of $0$ to $\alpha /\beta $ and $\pi _{\mathrm {HT}}\vert _{W_{\infty }}\colon W_{\infty }\to U_{\infty }$ sends the standard coordinate T on the standard affinoid neighbourhood $U_{\infty }:=\mathbb {A}^1$ of $\infty $ to $\beta /\alpha $ .

For every $n\in {\mathbb N}$ , we can refine such morphism to a morphism on $\mathcal {X}_0(p^n,N)_{\mathrm {pke}}$ as follows. Considering the smooth formal model $\mathfrak {X}$ of the modular curve X over ${\cal O}_{{\mathbb C}_p}$ given by moduli theory and the universal generalised elliptic curve E over $\mathfrak {X}$ , the invariant differentials of E relative to $\mathfrak {X}$ and the fact that $\mathcal {X}$ is the adic generic fibre of $\mathfrak {X}$ , give an invertible ${\cal O}_{\mathcal {X}}^+$ -module $\omega _E^+$ on $\mathcal {X}$ . Pulling back via the projection map $\mathcal {X}_0(p^n,N) \to \mathcal {X} $ , we get a ${\cal O}_{\mathcal {X}_0(p^n,N)_{\mathrm {pke}}}^+$ -module for the pro-Kummer étale topology that we still abusively denote $\omega _E^+$ and passing to p-adic completions we finally obtain an invertible $\widehat {{\cal O}}_{\mathcal {X}_0(p^n,N)}^+$ -module $ \widehat { \omega }_E^+$ . Here, for simplicity, we write $\widehat {{\cal O}}_{\mathcal {X}_0(p^n,N)}^+$ for $\widehat {{\cal O}}_{\mathcal {X}_0(p^n,N)_{\mathrm {pke}}}^+$ .

Consider the map $\mathrm {dlog}$ for the basis of $\mathcal {X}_0(p^n,N)_{\mathrm {pke}}$ given in Lemma 4.1. Let a modulo $p^n$ be a generator in $T_ p(E)/ p^n T_p(E)$ of the level subgroup of order $p^n$ . For every log affinoid perfectoid open U as in loc. cit., write $\widehat {U}=\mathrm {Spa}(R,R^+)$ . Then $T_p(E)^{\vee }$ is constant on U; we have a universal generalized elliptic curve E over $\mbox {Spec}(R^+)$ and $\widehat {{\cal O}}_{\mathcal {X}_0(p^n,N)}^+(U)=R^+$ . We then have the map $\mathrm {dlog}\colon T_p(E)^{\vee }(U) \otimes _{{\mathbb Z}_p} R^+ \longrightarrow \widehat { \omega }_E^+(U)$ . Gluing, we obtain a map of sheaves on $\mathcal {X}_0(p^n,N)_{\mathrm {pke}}$ :

(2) $$ \begin{align} \mathrm{dlog}\colon T_p(E)^{\vee} \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)}^+ \longrightarrow \widehat{ \omega}_E^+. \end{align} $$

From Proposition 4.5, we get an integral version of the Hodge–Tate exact sequence:

$$ \begin{align*} 0 \rightarrow \bigl(\widehat{\omega}_E^{\mathrm{mod}}\bigr)^{-1} \longrightarrow T_p(E)^{\vee} \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}}^+ \longrightarrow \widehat{\omega}_E^{\mathrm{mod}} \rightarrow 0. \end{align*} $$

This will be useful to compute the cohomology $\mathrm {H}^1\bigl (\mathcal {X}_0(p^n,N)_{\infty ,{\mathrm {pke}}}^{(m)}, \widehat {{\cal O}}_{\mathcal {X}_0(p^n,N)_{\infty }^{(n)}} \bigr )$ . In fact, tensoring the exact sequence with $\widehat {\omega }_E^{\mathrm {mod,2}}$ and taking the long exact sequence in cohomology we obtain a map

(3) $$ \begin{align} \mathrm{H}^0\bigl(\mathcal{X}_0(p^n,N)_{\infty,{\mathrm{pke}}}^{(m)}, \widehat{\omega}_E^{\mathrm{mod,2}}\bigr) \longrightarrow \mathrm{H}^1\bigl(\mathcal{X}_0(p^n,N)_{\infty,{\mathrm{pke}}}^{(m)}, \widehat{{\cal O}}_{\mathcal{X}_1(p^n,N)_{\infty}^{(m)}}^+ \bigr) \end{align} $$

and similarly for $\mathcal {X}_0(p^n,N)_0^{(m)}$ , or their covers $\mathcal {X}_1(p^n,N)_{\infty }^{(m)}$ , $\mathcal {X}(p^n,N)_0^{(m)}$ .

4.4 The sheaf $\omega _E^k$

Take r, m and $n \in {\mathbb N}$ as in Proposition 4.5. The map $\mathrm {dlog}$ provides $\omega _E^{\mathrm {mod}}/p^r \omega _E^{\mathrm {mod}} $ with a marked section s over $\mathcal {X}_1(p^n,N)_{\infty }^{(m)}$ as the image of the tautological generator of $C_r^{\vee }$ .

Similarly, recall that we have a decomposition $\mathcal {X}(p^n,N)_0^{(m)}=\amalg _{\lambda \in {\mathbb Z}/p^n{\mathbb Z}} \mathcal {X}(p^n,N)_{0,\lambda }^{(m)}$ , where over $\mathcal {X}(p^n,N)_{0,\lambda }^{(m)}$ , using the trivialization $T_p(E)/p^n T_p(E)=({\mathbb Z}/p^n{\mathbb Z}) a \oplus ({\mathbb Z}/p^n{\mathbb Z}) b$ , we have $\mathrm {dlog} (a^{\vee })=\lambda \mathrm {dlog} (b^{\vee })$ . In particular, the canonical subgroup $C_r$ is generated by $b+\lambda a$ and $\omega _E^{\mathrm {mod}}/p^r \omega _E^{\mathrm {mod}} $ acquires a marked section $s:=\mathrm { dlog} (b^{\vee })$ .

Suppose we are given an h-analytic character $k\colon {\mathbb Z}_p^{\ast } \to B^{\ast }$ , for $h\leq r$ , as in Definition 3.6. Using the formalism of VBMS from §3.3 for $\omega _E^{\mathrm {mod}}$ and the section s modulo $p^r$ , we get the invertible ${\cal O}_{\mathcal {X}_1(p^n,N)_{\infty }^{(m)}}^+\widehat {\otimes } B$ -module, resp. ${\cal O}_{\mathcal {X}(p^n,N)_0^{(m)}}\widehat {\otimes } B$ -module

$$ \begin{align*} \omega_E^k:={\mathbb W}_{k}(\omega_E^{\mathrm{mod}}, s). \end{align*} $$

Since $\omega _E^{\mathrm {mod}}$ and the section s modulo $p^r$ are stable under the action of the automorphism group $\Delta _n:=({\mathbb Z}/p^n{\mathbb Z})^{\ast }$ of $j_n\colon \mathcal {X}_1(p^n,N)_{\infty }^{(m)}\to \mathcal {X}_0(p^n,N)_{\infty }^{(m)}$ , then $({\mathbb Z}/p^n{\mathbb Z})^{\ast }$ acts on $j_{n,\ast }\bigl (\omega _E^k[1/p]\bigr )$ and, taking the subsheaf of $j_{n,\ast }\bigl (\omega _E^k[1/p]\bigr )$ on which ${\mathbb Z}_p^{\ast }$ acts via the character k, then $j_{n,\ast }\bigl (\omega _E^k[1/p]\bigr )$ descends to an invertible ${\cal O}_{\mathcal {X}_0(p^n,N)_{\infty }^{(m)}}\widehat {\otimes } B[1/p]$ -module that we denote $\omega _E^k[1/p] $ .

Similarly, the Galois group of $j^{\prime }_n\colon \mathcal {X}(p^n,N)_0^{(m)}\to \mathcal {X}_0(p^n,N)_0^{(m)}$ , which is identified with the standard Borel subgroup of $\mathrm {GL}_2({\mathbb Z}/p^n{\mathbb Z})$ , acts compatibly on $(\omega _E^{\mathrm {mod}},s)$ and hence it acts on $j^{\prime }_{n,\ast }\bigl (\omega _E^k[1/p]\bigr )$ . In this case we let $\omega _E^k[1/p]$ be the invertible ${\cal O}_{\mathcal {X}_0(p^n,N)_0^{(m)}}\widehat {\otimes } B[1/p]$ -module defined as the subsheaf of $j^{\prime }_{n,\ast }\bigl (\omega _E^k[1/p]\bigr )$ on which the standard Borel subroup of $\mathrm {GL}_2({\mathbb Z}_p)$ acts via the projection onto the lower right entry ${\mathbb Z}_p^{\ast }$ composed with the character k.

4.5 The $U_p$ -correspondence

Given the modular curve $\mathcal {X}_0(p^s,N)$ for $s\geq 1$ , we have correspondences $T_{\ell }$ , for $\ell $ not dividing $p N$ , and $U_p$ . They are defined by the analytification $\mathcal {X}_0(p^s,N, \ell )$ , resp. $\mathcal {X}_0(p^s,N,p)$ , of the modular curve $X_0(p^s,N,\ell )$ , resp. $X_0(p^s,N,p) $ , classifying, at least away from the cusps, subgroups D of order $\ell $ of the universal elliptic curve E, resp. subgroups of order p of E complementary to the p-torsion $H_1$ of the cyclic subgroup $H_s$ of order $p^s$ defined by the level structure. We have two maps $q_1,q_2\colon ,\mathcal {X}_0(p^s,N, \ell ) \to \mathcal {X}_0(p^s,N)$ defined by the analytification of the maps $q_1,q_2\colon X_0(p^s,N,\ell )\to X_0(p^s,N) $ , where $q_1$ sends the universal object $(E,H_s,\Psi _N,D)$ to $(E,H_s,\Psi _N)$ (the forgetful map) and $q_2$ sends the universal object $(E,H_s,\Psi _N,D)$ to $(E/D, H_s',\Psi _N')$ , where $H_s'$ is the image of $H_s$ via the isogeny $E\to E/D$ and $\Psi _N'$ is $\Psi _N$ composed with this isogeny.

These maps induce morphisms of sites $\mathcal {X}_0(p^s,N,\ell )_{\mathrm {pke}} \to \mathcal {X}_0(p^s,N)_{\mathrm {pke}} $ , resp. $\mathcal {X}_0(p^s, N,p )_{\mathrm {pke}} \to \mathcal {X}_0(p^s,N)_{\mathrm {pke}} $ . As $q_1, q_2$ are finite Kummer étale, the following follows from the discussion in [Reference Andreatta, Iovita and Stevens5, Cor. 2.6] or [Reference Diao, Lan, Liu and Zhu15, Prop. 4.5.2].

The maps $q_1$ and $q_2$ induce maps of perfectoid spaces. Take $\ell $ prime to N but possibly equal to p. The fibre product $\mathcal {X}(p^{\infty },N,\ell ):=\mathcal {X}(p^{\infty },N) \times _{\mathcal {X}_0(p^s,N)}^{q_i} \mathcal {X}_0(p^s,N,\ell )$ exists and is independent of the choice of maps $q_1$ or $q_2$ . We then have the two projections $q_1,q_2\colon \mathcal {X}(p^{\infty },N,\ell ) \to \mathcal {X}(p^s,N)$ . Notice that $\mathcal {X}(p^{\infty },N,p )$ splits completely as

$$ \begin{align*} \mathcal{X}(p^{\infty},N,p )=\amalg_{\lambda=0,\ldots,p-1} \mathcal{X}(p^{\infty},N), \end{align*} $$

where over the copy labeled by $\lambda =0,\ldots ,p-1$ the isogeny $E\to E':=E/D$ produces the map of ${\mathbb Z}_p$ -modules

$$ \begin{align*} u_{\lambda} \colon T_p E&={\mathbb Z}_p a \oplus {\mathbb Z}_p b \to T_p(E'):=T_p(E_{\lambda}):={\mathbb Z}_p a' \oplus {\mathbb Z}_p b', \\ a'&=a, b'=\frac{b+\lambda a}{p}\in T_p E \otimes {\mathbb Q}_p. \end{align*} $$

Using this description, the maps $q_1$ and $q_2$ restricted to the component labeled $\lambda $ define maps $q_{1,\lambda }$ and $q_{2,\lambda }$ , where $q_{1,\lambda }$ is the identity map and $q_{2,\lambda }\colon \mathcal {X}(p^{\infty },N) \to \mathcal {X}(p^{\infty },N) $ is an isomorphism such that the pullback of $T_p E$ is $T_p(E_{\lambda })$ .

We consider the maps $t_1,t_2\colon \amalg _{\lambda =0,\ldots ,p-1} {\mathbb P}^1 \to {\mathbb P}^1$ where on the component labeled by $\lambda $ the map $t_{1,\lambda }$ induced by $t_1$ is the identity while the map $t_2$ is the isomorphism $t_{2,\lambda }\colon {\mathbb P}^1 \to {\mathbb P}^1$ defined on points by $[\alpha ,\beta ]\mapsto [\alpha - \lambda \beta ,p \beta ]$ . We then have the following diagram:

$$ \begin{align*}\begin{matrix} \mathcal{X}(p^{\infty},N) & \stackrel{q_2}{\longleftarrow} & \mathcal{X}(p^{\infty},N,p )= \amalg_{\lambda=0,\ldots,p-1} \mathcal{X}(p^{\infty},N) & \stackrel{q_1}{\longrightarrow} & \mathcal{X}(p^{\infty},N) \cr \pi_{\mathrm{HT}} \big\downarrow & & \amalg_{\lambda=0,\ldots,p-1} \pi_{\mathrm{HT}} \big\downarrow & & \pi_{\mathrm{HT}} \big\downarrow\\ {\mathbb P}^1 & \stackrel{t_2}{\longleftarrow} & \amalg_{\lambda=0,\ldots,p-1} {\mathbb P}^1 & \stackrel{t_1}{\longrightarrow} & {\mathbb P}^1.\cr \end{matrix} \end{align*} $$

In fact, the squares are commutative. This follows from the functoriality of $\mathrm {dlog}$ with respect to isogenies and by computing $u_{\lambda }^{\vee }$ : The map $u_{\lambda }$ sends $a\mapsto a'$ and $b\mapsto p b'-\lambda a'$ so that on the dual basis $u_{\lambda }^{\vee }$ sends $(a')^{\vee }\mapsto a^{\vee }-\lambda b^{\vee }$ and $(b')^{\vee }\mapsto p b^{\vee }$ .

We conclude this section with a lemma on the dynamic of the operators $t_{2,\lambda }$ . For $\lambda =0,\ldots ,p-1$ we write $t_{\lambda }$ in place of $t_{2,\lambda }$ in the next lemma.

4.6 Étale sheaves

Let $H\subset \mathrm {\mathbf {GL}}_2({\mathbb Z}_p)$ be a finite index subgroup. In this section, we recall the tensor functor from the category of profinite H-representations to the category of sheaves on the pro-Kummer étale site of the modular curve $X(H,N)$ defined by H or, equivalently, of the associated adic space $\mathcal {X}(H,N)$ . We work with the latter.

We fix log geometric points $\zeta _i$ , one for every connected component $Z_i$ of $\mathcal {X}(H,N)$ . Due to [Reference Diao, Lan, Liu and Zhu15, Prop. 5.1.12], the sites $Z_{i,{\mathrm {fket}}}$ are Galois categories with underlying profinite group, the Kummer étale fundamental group $\pi _1^{\mathrm {ket}}(Z_i,\zeta _i)$ . In particular, the pro-Kummer finite étale cover $\bigl (\mathcal {X}(p^r,N)\bigr )_{r}$ of $\mathcal {X}(H,N)$ for r big enough, restricted to each $Z_i$ defines a homomorphism

$$ \begin{align*} \pi_1(Z_i,\zeta_i)\to \displaystyle{\lim_{\infty \leftarrow r}} \mathrm{Aut}\bigl(\mathcal{X}(p^r,N)/\mathcal{X}(H,N) \bigr)=H. \end{align*} $$

Given a finite representation $L_n$ of H, we view it as a representation of $\pi _1(Z_i,\zeta _i)$ , for every i, and hence as a local system on each $Z_{i,{\mathrm {ket}}}$ and, hence, a local system $\mathbb {L}_n$ on $\mathcal {X}(H,N)_{\mathrm {ket}}$ . In fact, $\mathbb {L}_n$ is a sheaf on $\mathcal {X}(H,N)_{\mathrm {ket}}$ such that there exists a finite Kummer étale cover $\mathcal {X}(p^r,N) \to \mathcal {X}(H,N)$ , for $r\gg 0$ , on which the sheaf $\mathbb {L}_n$ is constant.

Given a profinite representation L of H, that is, an inverse limit $L=\displaystyle {\lim _{\infty \leftarrow n}} L_n$ of finite representations $L_n$ for $n\in {\mathbb N}$ , we let $\mathbb {L}$ be the inverse limit $\mathbb {L}=\displaystyle {\lim _{\infty \leftarrow n}} \mathbb {L}_n$ . It is a sheaf on $\mathcal {X}(H,N)_{\mathrm {ket}}$ . Notice that using the scheme $X(H,N)$ one gets, as mentioned before, a sheaf on $X(H,N)_{\mathrm {ket}}$ , that we will denote $\mathbb {L}$ . We have the following GAGA type of results:

Theorem 4.10. For every $i\in {\mathbb N}$ , the maps

$$ \begin{align*} \mathrm{H}^i\bigl(X(H,N)_{\mathrm{pke}},\mathbb{L} \bigr)\longrightarrow \mathrm{H}^i\bigl(\mathcal{X}(H,N)_{\mathrm{pke}},\mathbb{L} \bigr) \end{align*} $$

are isomorphisms. Analogously, the natural map

$$ \begin{align*} \mathrm{H}^i\bigl(\mathcal{X}(H,N)_{\mathrm{pke}},\mathbb{L} \bigr)\widehat{\otimes} {\cal O}_{{\mathbb C}_p} \longrightarrow \mathrm{H}^i\bigl(\mathcal{X}(H,N)_{\mathrm{pke}},\mathbb{L}\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}(H,N)}^+\bigr) \end{align*} $$

is an almost isomorphism.

Proof. The result for each $\mathbb {L}_n$ follows from the discussion in §2.4 and from Proposition 2.1.

Consider the natural map $\displaystyle {\lim _{\leftarrow }}\colon \mathrm {Sh}^{\mathbb N}(Z) \to \mathrm {Sh}(Z)$ from inverse systems of sheaves on $Z=X(H,N)_{\mathrm {pke}}$ , and $Z=\mathcal {X}(H,N)_{\mathrm {pke}}$ , respectively. Using the existence of bases of log affinoid perfectoid opens with the properties recalled in §2.3, it follows from [Reference Scholze22, lemma 3.18] that we have $\mathrm {R}^i \displaystyle {\lim _{\leftarrow }} (\mathbb {L})=0$ , both in the algebraic and in the adic setting, and $\mathrm {R}^i \displaystyle {\lim _{\leftarrow }} \bigl ( \mathbb {L}\widehat {\otimes } \widehat {{\cal O}}_{\mathcal {X}(H,N)}^+\bigr )=0$ for $i\geq 1$ . Hence, $\mathrm {H}^j\bigl (Z,\mathbb {L}\bigr )$ and $\mathrm {H}^j\bigl (\mathcal {X}(H,N)_{\mathrm {pke}},\mathbb {L}\widehat {\otimes } \widehat {{\cal O}}_{\mathcal {X}(H,N)}^+\bigr )$ coincide with the derived functors $\mathrm {H}^j\bigl (Z,(\mathbb {L}_n)_{n\in {\mathbb N}}\bigr )$ , resp. $ \mathrm {H}^j\bigl (\mathcal {X}(H,N)_{\mathrm {pke}}, (\mathbb {L}_n \otimes {\cal O}_{\mathcal {X}(H,N)_{\mathrm {pke}}}^+)_{n\in {\mathbb N}}\bigr ) $ of $\displaystyle {\lim _{\leftarrow }} \mathrm {H}^0(Z,\_) $ introduced by [Reference Jannsen19] on the inverse system $\mathrm {Sh}^{\mathbb N}(Z)$ . Due to [Reference Jannsen19, Prop. 1.6], these cohomology groups sit in exact sequences

$$ \begin{align*} 0 \longrightarrow \displaystyle{\lim_{\leftarrow}}^{(1)} \mathrm{H}^{j-1}\bigl(Z,\mathbb{L}_n\bigr) \longrightarrow \mathrm{H}^j\bigl(Z,(\mathbb{L}_n)_{n\in {\mathbb N}}\bigr) \longrightarrow \displaystyle{\lim_{\leftarrow}}\mathrm{H}^{j}\bigl(Z,\mathbb{L}_n\bigr) \longrightarrow 0 \end{align*} $$

and similarly for the inverse system $(\mathbb {L}_n \otimes {\cal O}_{\mathcal {X}(H,N)_{\mathrm {pke}}}^+)_{n\in {\mathbb N}}$ . The maps in the theorem are compatible with these sequences, and the claims follow from the finite case, that is, for the sheaves $\mathbb {L}_n$ .

Consider an s-analytic character $k\colon {\mathbb Z}_p^{\times }\longrightarrow B^{\ast }$ as in Definition 3.6. Consider the ${\mathbb Q}_p$ -module $D_k(T_0^{\vee })[n]$ , for $n\geq s$ , with action of the Iwahori subroup $\mathrm {Iw}_1$ , defined in §3.7. As $D_k(T_0^{\vee })[n]=\bigl (D_k^o(T_0^{\vee })[n]\bigr )[1/p]$ and $D_k^o(T_0^{\vee })[n]$ admits a $\mathrm {Iw}_1$ -equivariant filtration with finite graded pieces, we get an associated sheaf ${\mathbb D}_k(T_0^{\vee })[n]$ on the pro-Kummer étale site of $\mathcal {X}_0(p,N)$ . Then:

Proposition 4.13. For every $i\in {\mathbb N}$ , we have isomorphisms

$$ \begin{align*} \mathrm{H}^i\bigl(\Gamma_0(p)\cap \Gamma(N),D_k(T_0^{\vee})[n] \bigr)\widehat{\otimes} {\mathbb C}_p \cong \mathrm{H}^i\bigl(\mathcal{X}_0(p,N)_{\mathrm{pke}},{\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p,N)}\bigr). \end{align*} $$

For every $s\geq 1$ , we have actions of Hecke operators on $\mathrm {H}^i\bigl (\mathcal {X}_0(p^s,N)_{\mathrm {pke}},{\mathbb D}_k(T_0^{\vee })[n]\widehat {\otimes } \widehat {{\cal O}}_{\mathcal {X}_0(p^s,N)}\bigr )$ as follows. Let $\ell $ be a prime integer not dividing N, and let $q_1,q_2\colon X_0(p,N,\ell )\to X_0(p,N)$ be as in §4.5. For $\ell \neq p$ , we have natural isomorphisms $q_1^{\ast }\bigl ({\mathbb D}_k^o(T_0^{\vee })[n]\bigr ) \to q_2^{\ast }\bigl ({\mathbb D}_k^o(T_0^{\vee })[n]\bigr )$ inducing a map

$$ \begin{align*} \mathcal{T}_{\ell}\colon q_1^{\ast}\bigl({\mathbb D}_k^o(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}^+\bigr) \to q_2^{\ast}\bigl({\mathbb D}_k^o(T_0^{\vee})[n]\bigr)\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N,\ell)}^+. \end{align*} $$

Inverting p, taking $q_{2,\ast }$ and using the trace map $\mathrm {Tr}\colon q_{2,\ast } q_2^{\ast } \to \mathrm {Id}$ of Lemma 4.7, we get a map

$$ \begin{align*} q_{2,\ast} q_1^{\ast}\bigl({\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr) \to {\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}. \end{align*} $$

For $\ell =p$ , it follows by taking the dual of the $\mathrm {Iw}_s$ -equivariant map $\pi _n$ of §3.5.2 that we have a map $\mathcal {U}_p\colon q_1^{\ast }\bigl ({\mathbb D}_k^o(T_0^{\vee })[n]\bigr ) \to q_2^{\ast }\bigl ({\mathbb D}_k^o(T_0^{\vee })[n+1]\bigr )$ and, hence, a map

$$ \begin{align*} \mathcal{U}_p\colon q_1^{\ast}\bigl({\mathbb D}_k^o(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}^+\bigr) \to q_2^{\ast}\bigl({\mathbb D}_k^o(T_0^{\vee})[n+1]\bigr)\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N,p)}^+. \end{align*} $$

Inverting p, taking $q_{2,\ast }$ and using the trace map $\mathrm {Tr}\colon q_{2,\ast } q_2^{\ast } \to \mathrm {Id}$ , we get a map

$$ \begin{align*} q_{2,\ast} q_1^{\ast}\bigl({\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr) \to {\mathbb D}_k(T_0^{\vee})[n+1]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}. \end{align*} $$

We have a restriction map $D_k(T_0^{\vee })[n+1]\to D_k(T_0^{\vee })[n] $ which is $\mathrm {Iw}_s$ -equivariant and defines a map ${\mathbb D}_k(T_0^{\vee })[n+1] \to {\mathbb D}_k(T_0^{\vee })[n] $ . We finally get a morphism

$$ \begin{align*} q_{2,\ast} q_1^{\ast}\bigl({\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr) \to {\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}. \end{align*} $$

Taking cohomology groups over $\mathcal {X}_0(p^s,N)_{\mathrm {pke}}$ and using the map

$$ \begin{align*} \mathrm{H}^i\bigl(\mathcal{X}_0(p^s,N)_{\mathrm{pke}},{\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr) \to \mathrm{H}^i\bigl(\mathcal{X}_0(p^s,N,\ell)_{\mathrm{pke}},q_1^{\ast}\bigl({\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr) \bigr), \end{align*} $$

we get Hecke operators

(4) $$ \begin{align} T_{\ell}, U_p^{\mathrm{naive}}&\colon \mathrm{H}^i\bigl(\mathcal{X}_0(p^s,N)_{\mathrm{pke}},{\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr)\nonumber\\ &\quad \to \mathrm{H}^i\bigl(\mathcal{X}_0(p^s,N)_{\mathrm{pke}},{\mathbb D}_k(T_0^{\vee})[n]\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^s,N)}\bigr). \end{align} $$

(Here we introduce the unnormalized operator $U_p^{\mathrm {naive}}$ which is $p $ times the standard operator $U_p$ as it preserves integral structures, a fact that will be of crucial importance in section §5.)

4.7 A comparison result on $\mathcal {X}_0(p^n,N)_{\infty }^{(m)}$

Consider an r-analytic weight $k\colon {\mathbb Z}_p^{\ast } \to B^{\ast }$ as in Definition 3.6 and integers $ n\geq m$ as in Proposition 4.5, and define the sheaf

$$ \begin{align*} \mathfrak{D}_{k,\infty}^{o,(m)}[n]:={\mathbb D}_k^o(T_0^{\vee})[n]\vert_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}} \widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}}^+. \end{align*} $$

The aim of this section is to prove that for m large enough it admits a decreasing filtration $\mathrm {Fil}^h \mathfrak {D}_{k,\infty }^{o,(m)}[n]$ for $h\geq -1$ with the following properties.

Proof. Recall the surjective map

$$ \begin{align*} \mathrm{dlog}\colon T_p(E)^{\vee} \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}}^+ \longrightarrow \omega_E^{\mathrm{mod}}\widehat{\otimes} \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}}^+\\[-9pt] \end{align*} $$

from equation (2) and Proposition 4.5. It is defined for every m large enough and $\omega _E^{\mathrm {mod}} $ is an invertible ${\cal O}_{\mathcal {X}_0(p^n,N)_{\infty }^{(m)}}^+$ -module. We also assume that over $\mathcal {X}_0(p^n,N)_{\infty }^{(m)}$ we have a canonical subgroup $C_r$ of order $p^r$ . Consider the natural projection $j_n\colon \mathcal {X}(p^n,N)_{\infty }^{(m)} \to \mathcal {X}_0(p^n,N)_{\infty }^{(m)}$ ; then $j_n$ is Kummer étale and Galois with group $\Delta _n$ the subgroup of $\mathrm {\mathbf {GL}}_2({\mathbb Z}/p^n{\mathbb Z})$ of upper triangular matrices. In particular, to define a sheaf on $\mathcal {X}_0(p^n,N)_{\infty ,{\mathrm {pke}}}^{(m)}$ is equivalent to define a sheaf on $\mathcal {X}(p^n,N)_{\infty ,{\mathrm {pke}}}^{(m)}$ with an action of $\Delta _n$ compatible with the action on $\mathcal {X}(p^n,N)_{\infty }^{(m)}$ . In order to define $\mathrm {Fil}^h \mathfrak {D}_{k,\infty }^{o,(m)}[n]$ , we define a $\Delta _n$ -equivariant filtration on $j_n^{\ast } \bigl ( \mathfrak {D}_{k,\infty }^{o,(m)}[n]\bigr )$ .

Over $ \mathcal {X}(p^n,N)_{\infty }^{(m)}$ , we have a trivialization

$$ \begin{align*} T_p(E)/p^n T_p(E)=({\mathbb Z}/p^n{\mathbb Z}) a \oplus ({\mathbb Z}/p^n{\mathbb Z})b\\[-9pt] \end{align*} $$

such that $\mathrm {dlog}(a^{\vee })$ is a basis for $\omega _E^{\mathrm {mod}}/ p^r \omega _E^{\mathrm {mod}}$ as ${\cal O}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+$ -module. Dualizing, we get an injective map

$$ \begin{align*} (\mathrm{dlog})^{\vee}\colon \omega_E^{\mathrm{mod,-1}} \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+ \longrightarrow T_p(E) \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+\end{align*} $$

with quotient $\mathcal {Q}$ isomorphic to $\omega _E^{\mathrm {mod}}\otimes _{{\cal O}_{\mathcal {X}_0(p,N)_{\infty }^{(m)}}^+} \widehat {{\cal O}}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+$ and such that modulo $p^r$ the section a generates $\omega _E^{\mathrm {mod,-1}}/p^n \omega _E^{\mathrm {mod,-1}}$ as ${\cal O}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+$ -module. Then $(\mathrm { dlog})^{\vee }$ induces, for every $\lambda \in {\mathbb Z}/p^n{\mathbb Z}$ , an affine map

$$ \begin{align*} \rho_{\lambda}\colon {\mathbb V}_0\bigl(T_p(E) , a,b-\lambda a\bigr) \times_{{\mathbb Z}_p} \mathcal{X}(p^n,N)_{\infty}^{(m)} \longrightarrow {\mathbb V}_0\bigl(\omega_E^{\mathrm{mod,-1}},a\bigr) \times_{\mathcal{X}_0(p,N)_{\infty}^{(m)}} \mathcal{X}(p^n,N)_{\infty}^{(m)} \end{align*} $$

on the pro-Kummer étale site of $\mathcal {X}(p^n,N)_{\infty }^{(m)}$ . Here and below, the formalism f VBMS is applied with respect to the ideal ${\cal I}$ generated by $p^r$ . Notice that ${\mathbb V}_0\bigl (T_p(E) , a,b-\lambda a\bigr ) \times _{{\mathbb Z}_p} \mathcal {X}(p^n,N)_{\infty }^{(m)}\cong {\mathbb V}_0\bigl (T_p(E)\otimes _{{\mathbb Z}_p} \widehat {{\cal O}}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+,a,b-\lambda b\bigr )$ is a principal homogeneous space under the formal vector group ${\mathbb V}'\bigl (\mathcal {Q}\bigr ) \subset {\mathbb V}\bigl (\mathcal {Q}\bigr )$ classifying sections of $\mathcal {Q}^{\vee }$ which are zero modulo $p^r$ . We have the invertible ${\cal O}_{\mathcal {X}_0(p,N)_{\infty }^{(m)}}^+\widehat {\otimes } B$ -module $\omega ^{-k}_E := {\mathbb W}_k(\omega _E^{\mathrm {mod,-1}}, a\big )$ . Set

$$ \begin{align*} \mathfrak{W}_{k,\infty,\lambda}^{(m)}:= {\mathbb W}_k\bigl(T_p(E),a,b-\lambda a\bigr) \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+. \end{align*} $$

Applying the formalism of VBMS, we get the map of sheaves on the pro-Kummer étale site of $\mathcal {X}(p^n,N)_{\infty }^{(m)}$

$$ \begin{align*} \rho_{\lambda}^k\colon \omega^{-k}_E \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+ \longrightarrow \mathfrak{W}_{k,\infty,\lambda}^{(m)}. \end{align*} $$

Using that ${\mathbb V}_0\bigl (T_p(E)\otimes _{{\mathbb Z}_p} \widehat {{\cal O}}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+,a,b +\lambda a\bigr )$ is a principal homogeneous space under ${\mathbb V}'\bigl (\mathcal {Q}\bigr )$ , we obtain a ${\mathbb V}'\bigl (\mathcal {Q}\bigr )$ -stable increasing filtration $\mathrm { Fil}_h \mathfrak {W}_{k,\infty ,\lambda }^{(m)}$ for $h\geq 0$ with

(5) $$ \begin{align} \mathrm{Fil}_0 \mathfrak{W}_{k,\infty,\lambda}^{(m)}&=\omega^{-k}_E \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+,\nonumber\\ \mathrm{Gr}_h \mathfrak{W}_{k,\infty}^{(m)}&\cong \omega^{-k+2h}_E \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+. \end{align} $$

See [Reference Andreatta and Iovita2] and [Reference Andreatta, Iovita and Stevens6, Prop. 5.2]. Taking $\widehat {{\cal O}}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+$ -duals, we get a sheaf and a decreasing filtration

$$ \begin{align*} \mathfrak{W}_{k,\infty,\lambda}^{(m),\vee}=W_k\bigl(T_p(E),a,b-\lambda b\bigr)^{\vee} \otimes_{{\mathbb Z}_p} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m}}^+, \qquad \mathrm{Fil}^h \mathfrak{W}_{k,\infty,\lambda}^{(m),\vee}, h\geq -1 \end{align*} $$

on the pro-Kummer étale site of $\mathcal {X}(p^n,N)_{\infty }^{(m)}$ . Here, $\mathrm {Fil}^h \mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee }$ consists of those sections of $\mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee }$ which are zero on $\mathrm {Fil}_h \mathfrak {W}_{k,\infty ,\lambda }^{(m)}$ (where we set $\mathrm {Fil}_{-1}\mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee }=0$ so that $\mathrm {Fil}^{-1} \mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee })= \mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee }$ ). Then

(6) $$ \begin{align} \mathrm{Gr}^h \mathfrak{W}_{k,\infty,\lambda}^{(m),\vee}\cong \omega^{k-2h-2}_E \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+. \end{align} $$

Due to Proposition 3.8, we have a $\Delta _n$ -equivariant isomorphism

$$ \begin{align*} j_n^{\ast} \bigl( \mathfrak{D}_{k,\infty}^{o,(m)}[n]\bigr)\cong \oplus_{\lambda\in{\mathbb Z}/p^n{\mathbb Z}} \mathfrak{W}_{k,\infty,\lambda}^{(m),\vee}\\[-9pt] \end{align*} $$

and we set $\mathrm {Fil}^h j_n^{\ast } \bigl ( \mathfrak {D}_{k,\infty }^{o,(m)}[n]\bigr )$ to be the filtration corresponding to $\oplus _{\lambda \in {\mathbb Z}/p^n{\mathbb Z}} \mathrm {Fil}^h\mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee } $ . The map $\oplus _ \lambda \rho _{\lambda }^k$ defines a map $\rho ^k\colon \omega ^{-k}_E \otimes _{{\cal O}_{\mathcal {X}_0(p,N)_{\infty }^{(m)}}^+} \widehat {{\cal O}}_{\mathcal {X}(p^n,N)_{\infty }^{(m)}}^+ \longrightarrow \oplus _{\lambda } \mathfrak {W}_{k,\infty ,\lambda }^{(m)} $ and, hence, a map

$$ \begin{align*} \nu\colon \mathrm{Gr}^{-1} j_n^{\ast}\bigl(\mathfrak{D}_{k,\infty}^{o,(m)}[n]\bigr) \to \omega^{k}_E \otimes_{{\cal O}_{\mathcal{X}_0(p,N)_{\infty}^{(m)}}^+} \widehat{{\cal O}}_{\mathcal{X}(p^n,N)_{\infty}^{(m)}}^+. \end{align*} $$

By construction, (i) and (ii) hold over $\mathcal {X}(p^n,N)_{\infty }^{(m)}$ , and to prove those claims we need to show that the given filtration and the map $\nu $ are $\Delta _n$ -equivariant. Also, given a sheaf $\mathcal {F}$ on $\mathcal {X}_0(p,N)_{\infty }^{(m)}$ , we have a spectral sequence

$$ \begin{align*} \mathrm{H}^i(\Delta_n, \mathrm{H}^j\bigl(\mathcal{X}(p^n,N)_{\infty,{\mathrm{pke}}}^{(m)},j_n^{\ast}(\mathcal{F})\bigr) \Rightarrow \mathrm{H}^{i+j}\bigl(\mathcal{X}_0(p^n,N)_{\infty,{\mathrm{pke}}}^{(m)},\mathcal{F}\bigr). \end{align*} $$

Since the cohomology groups $\mathrm {H}^j\bigl (\Delta _n,\mathcal {G}\bigr )$ are annihilated by the order of $\Delta _n$ for $j\geq 1$ , to conclude (iv)–(v) it suffices to prove those claims for $\mathcal {X}(p^n,N)_{\infty }^{(m)}$ .

The rest of the proof is a computation using the log affinoid perfectoid cover by opens U of the adic space $\mathcal {X}_0(p^n,N)_{\infty ,{\mathrm {pke}}}^{(m)}$ defined by trivializing the full Tate module $\prod _{\ell } T_{\ell }(E)$ . It is Galois over $\mathcal {X}_0(p^n,N)_{\infty ,{\mathrm {pke}}}^{(m)}$ with group $G_U$ . Let $\widehat {U}:=\mathrm {Spa}(R,R^+)$ be the associated affinoid perfectoid space as in §2.3. Write $T_p(E)^{\vee }(U)\otimes R^+=e_0 R^+\oplus e_1R^+$ with $e_0$ mapping to a generator of $\omega _E^{\mathrm {mod}}$ and $e_1$ in the kernel of $d\log $ generating $\omega _E^{\mathrm {mod,-1}}$ . Recall from Proposition 3.8 that $\mathfrak {W}_{k,\infty ,\lambda }^{(m),\vee }(U)$ is the dual of $\mathfrak {W}_{k,\infty ,\lambda }^{(m)}(U)\cong R^+\otimes B\langle \frac {W_{\lambda }}{1+p^r Z} \rangle \cdot k(1+p^r Z)$ with increasing filtration defined by $\mathrm {Fil}^h = \oplus _{i=0}^h R^+\otimes B \bigl (\frac {W_{\lambda }}{1+p^r Z}\bigr )^i \cdot k(1+p^r Z) $ .

For every $\sigma \in G_U$ , we then have $\sigma (e_1)=e_1$ and $\sigma (e_0)=e_0+\xi (\sigma ) e_1$ . Then, $\sigma (W_{\lambda })=W_{\lambda } +\frac {\xi (\sigma )}{p^n} (1+p^r Z) $ and $\sigma (Z)= Z$ . If $\xi (\sigma )=\alpha + p^r \beta $ then $\sigma (W_{\lambda })=W_{\lambda +\alpha } + \beta (1+p^r Z) $ . Thus, the increasing filtration on $\oplus _{\lambda \in {\mathbb Z}/p^n{\mathbb Z}}\mathfrak {W}_{k,\infty ,\lambda }^{(m)}$ and the diagonal embedding of $R^+\otimes B\to \oplus _{\lambda \in {\mathbb Z}/p^n{\mathbb Z}}\mathrm {Fil}^0 \mathfrak {W}_{k,\infty ,\lambda }^{(m)}$ are both stable for the action of $G_U$ . This concludes the proof of (i) and (ii).

We pass to claims (iii)–(v). As $\omega ^{k+2}_E$ is a locally free ${\cal O}_{\mathcal {X}_0(p,N)_{\infty }^{(m)}}^+\widehat {\otimes }B$ -module, we are left to show that

$$ \begin{align*} \mathrm{H}^1\bigl(\mathcal{X}_0(p^n,N)_{\infty,{\mathrm{pke}}}^{(m)}, \widehat{{\cal O}}_{\mathcal{X}_0(p^n,N)_{\infty}^{(m)}} \bigr)\cong \mathrm{H}^0\bigl(\mathcal{X}_0(p^n,N)_{\infty}^{(m)},\omega^{2}_E\bigr)[1/p]. \end{align*} $$

So to prove claim (iii), we are left to show that that the map (3) is an isomorphism. The étale cohomology of the structure sheaf on U is trivial as recalled in §2.3. Thanks to equation (6), the sheaves $ \mathfrak {D}_{k,\infty }^{o,(m)}[n] /\mathrm {Fil}^h \mathfrak {D}_{k,\infty }^{o,(m)}[n] \vert _U$ are extensions of the structure sheaf $\widehat {{\cal O}}_U^+$ so that also their cohomology over U is trivial. Then the cohomology groups in equation (3) and those of $ \mathfrak {D}_{k,\infty }^{o,(m)}[n] /\mathrm {Fil}^h \mathfrak {D}_{k,\infty }^{o,(m)}[n]$ appearing in (iv) and (v) coincide with the continuous cohomology of $G_U$ of the sections of the relevant sheaves over U. This reduces the proof of claims (iii), (iv) and (v) to a Galois cohomology computation for which we refer to [Reference Andreatta, Iovita and Stevens6, Thm. 5.4].