Proof. i) Let $x: \operatorname {\mbox {Spa }}(F)\to X$ be a classical point, that is, $F/\breve {\mathbb Q}_p$ is a finite extension. The pull-back of $\mathbb {L}_\eta $ by x gives a continuous n-dimensional ${\mathbb Q}_p$ -representation of $\operatorname {\mathrm {Gal}}(\bar F/F)$ ; the corresponding pro-étale $\mathrm {GL}_n({\mathbb Q}_p)$ -torsor over $\operatorname {\mbox {Spa }}(F)$ is given by the specialization of

$$\begin{align*}\mathrm{Sht}_\infty\to \mathrm{Sht}_K\xrightarrow{\pi_{\mathrm{GM}}} \mathrm{Gr}_{\mathrm{GL}_n,\leq \mu} \end{align*}$$

over the resulting $y:=\pi _{\mathrm {GM}}\circ a^\times \circ x: \mathrm {Spa}(F)\to \mathrm {Gr}_{\mathrm {GL}_n,\leq \mu }$ ; here $\pi _{\mathrm {GM}}$ is the period morphism. Then y lies in the admissible locus $\mathrm {Gr}^{\mathrm {adm}}_{\mathrm {GL}_n,\leq \mu }$ of $\mathrm {Gr}_{\mathrm {GL}_n,\leq \mu }$ . Suppose y factors through the “Schubert cell” $\mathrm {Gr}_{\mathrm {GL}_n,\lambda }$ for some $\lambda \leq \mu $ . Then, by the definition of the admissible locus, y also lies in $\mathrm {Gr}^{\mathrm { adm}}_{\mathrm {GL}_n,\lambda }$ and hence we also have $b\in B(\mathrm {GL}_n, \lambda ^{-1})$ (see, e.g., [Reference Caraiani and Scholze3, Prop. 3.5.3]). Then the Galois representation given by specializing $\mathbb {L}_\eta $ , as above, is crystalline and in particular de Rham of Hodge-Tate weight $\lambda ^{-1}$ , and with corresponding Frobenius isocrystal given by b. (This is essentially a corollary of “weakly admissible $\Rightarrow $ admissible” in the sense of Fontaine. See, for example, [Reference Gleason11, Prop. 1.4, Prop. 1.11].Footnote ³ ) The claim that $\mathbb {L}$ is de Rham (in fact, of constant Hodge-Tate weights given by $\lambda ^{-1}$ ), now follows from the rigidity theorem of [Reference Liu and Zhu17].

ii) We now consider the triple $(\mathrm {D}_{\mathrm {dR}}(\mathbb {L} ), \nabla , \mathrm {Fil}^{\bullet })$ associated to the de Rham local system $\mathbb {L} $ by the p-adic Riemann-Hilbert correspondence of [Reference Liu and Zhu17]:

Denote by $(\mathcal {V} , f , \mathbb {L} )$ the data, as in §2.2.4, which correspond to the vector space shtuka $( \mathscr {V},\phi _{ \mathscr {V}})$ over X obtained from $a $ . Here, $\mathcal {V} $ is a vector bundle of rank n over the relative Fargues-Fontaine curveFootnote ⁴ $X_{FF, X^{\Diamond }}$ and $f $ is a meromorphic isomorphism from $\mathcal {V}^b$ to $\mathcal {V} $ which has pole bounded by $\mu $ at the universal untilt. We claim that f provides an isomorphism

(2.3) $$ \begin{align} V\otimes_{\underline{{\mathbb Q}_p}} \mathcal{O} {\mathbb B}_{\mathrm{dR}}\xrightarrow{\ \sim\ } \mathbb {L}\otimes_{\underline{ {\mathbb{Z}}_p}}\mathcal{O} {\mathbb B}_{\mathrm{dR}} \end{align} $$

of sheaves of $\mathcal {O} {\mathbb B}_{\mathrm {dR}}$ -modules on $X_{\text {pro\'et}}$ (see [Reference Scholze24, Def. 6.8], [Reference Scholze25] for the definition of ${\mathbb B}_{\mathrm {dR}}$ , $\mathcal {O} {\mathbb B}_{\mathrm {dR}}$ ). Consider $ S=\operatorname {\mbox {Spa }}(R, R^+)\to X^{\Diamond }$ over $\mathrm{Spd }\ (\breve {\mathbb Q}_p)$ given by the tilt of the perfectoid affinoid completion $\widehat {\mathcal {W}}=S^\sharp =\operatorname {\mbox {Spa }}(R^\sharp , R^{\sharp +})$ of a pro-étale $\mathcal {W}\to X$ , as in [Reference Scholze24, Def. 4.3]. The pull-back of $\mathcal {V}$ to S is, by construction [Reference Scholze and Weinstein29, §22.3],

$$\begin{align*}\mathcal {V}_S=\mathbb {L}\otimes_{\underline{ {\mathbb{Z}}_p}}\mathcal{O}_{X_{FF, S}}. \end{align*}$$

On the other hand, by definition,

$$\begin{align*}\mathcal {V}^b_S=(\breve V\otimes_{\breve{\mathbb Q}_p}\mathcal{O}_{\mathcal {Y}_{(0,\infty)}(S)})^{\phi=1}, \end{align*}$$

where $\phi $ acts on $\breve V=V\otimes _{{\mathbb Q}_p}\breve {\mathbb Q}_p$ by $b\otimes \mathrm {Frob}$ . It follows that the completion $\widehat {\mathcal {V}^b_S}$ of $\mathcal {V}^b_S$ along the divisor of $X_{FF, S}$ given by the untilt $S^\sharp $ is

$$\begin{align*}\widehat{\mathcal {V}^b_S}=V\otimes_{{\mathbb Q}_p}\widehat{\mathcal{O}}_{\mathcal {Y}_{(0,\infty)}(S), S^\sharp}=V\otimes_{{\mathbb Q}_p}{\mathbb B}^+_{\mathrm{dR}}(R^\sharp, R^{\sharp+}). \end{align*}$$

The similar completion of $\mathcal {V}_S$ is

$$\begin{align*}\widehat{\mathcal {V}_S}=\mathbb {L}\otimes_{ {\mathbb{Z}}_p} {\mathbb B}_{\mathrm{dR}}^+(R^\sharp, R^{\sharp+}). \end{align*}$$

Hence, by completing $f_S: \mathcal {V}^b_S\dashrightarrow \mathcal {V}_S$ , we obtain an isomorphism

$$\begin{align*}V\otimes_{{\mathbb Q}_p} {\mathbb B}_{\mathrm{dR}}(R^\sharp, R^{+\sharp})\xrightarrow{\ \sim\ } \mathbb {L}\otimes_{ {\mathbb{Z}}_p} {\mathbb B}_{\mathrm{dR}}(R^\sharp, R^{\sharp+}). \end{align*}$$

By [Reference Scholze24, Thm. 6.5], ${\mathbb B}_{\mathrm {dR}}(R^\sharp , R^{+\sharp })={\mathbb B}_{\mathrm {dR}}(\mathcal {W})$ , the sections over $\mathcal {W}\to X$ of the sheaf ${\mathbb B}_{\mathrm {dR}}$ on $X_{\mathrm {pro\acute {e}t}}$ . We can see that, for variable $\mathcal {W}\to X$ , the above isomorphisms give an isomorphism of sheaves of ${\mathbb B}_{\mathrm {dR}}$ -modules over $X_{\mathrm {pro\acute {e}t}}$ . (The required patching can be checked after passing to a cover for the v-topology and it follows since f is defined over $X^{\Diamond }$ .) We can now apply $-\otimes _{{\mathbb B}_{\mathrm {dR}}}\mathcal {O}{\mathbb B}_{\mathrm {dR}}$ and obtain the desired isomorphism (2.3) of sheaves.

By the definition in [Reference Liu and Zhu17, §3.2], we have

$$\begin{align*}\mathrm{D}_{\mathrm{dR}}(\mathbb {L})=\mathrm{R}^0\nu_*(\mathbb {L}\otimes_{\underline{ {\mathbb{Z}}_p}} \mathcal{O} {\mathbb B}_{\mathrm{ dR}})\simeq \mathrm{R}^0\nu_*(V\otimes_{\underline{{\mathbb Q}_p}} \mathcal{O} {\mathbb B}_{\mathrm{dR}}) =V\otimes_{{\mathbb Q}_p} \mathrm{R}^0\nu_*(\mathcal{O} {\mathbb B}_{\mathrm{dR}}), \end{align*}$$

which is $V\otimes _{{\mathbb Q}_p}\mathcal {O}_{X}$ , with its trivial connection. Here, $\nu : X_{\text {pro\'et}}\to X_{\text {\'et}}$ is the natural morphism of sites as in loc. cit. and we have used (2.3). This shows part (ii).

iii) The shtuka $ \mathscr {V}(\mathbb {L})$ associated to the de Rham local system $\mathbb {L}$ in [Reference Pappas and Rapoport21, §2.6.1] is constructed, by the “gluing” procedure described in loc. cit. Prop. 2.5.1, from $\mathbb {L}$ and a ${\mathbb B}_{\mathrm {dR}}^+$ -submodule sheaf $\mathbb {M}_0$ of $\mathbb {M}\otimes _{{\mathbb B}^+_{\mathrm {dR}}}{\mathbb B}_{\mathrm {dR}}$ , where $\mathbb {M}=\mathbb {L}\otimes _{\underline {{\mathbb {Z}}}_p}{\mathbb B}^+_{\mathrm {dR}}$ . Furthermore, by [Reference Pappas and Rapoport21, Prop. 2.6.3], $\mathbb {M}_0$ is given by the construction of [Reference Scholze24, Prop. 7.9] as

$$\begin{align*}\mathbb {M}_0=(\mathrm{D}_{\mathrm{dR}}(\mathbb {L})\otimes_{\mathcal{O}_X}\mathcal{O}{\mathbb B}^+_{\mathrm{dR}})^{\nabla=0}. \end{align*}$$

Since $\mathrm {D}_{\mathrm {dR}}(\mathbb {L})=V\otimes _{{\mathbb Q}_p}\mathcal {O}_{X}$ by the above, $\mathbb {M}_0=V\otimes _{\underline {\mathbb Q}_p}{\mathbb B}^+_{\mathrm {dR}}$ . Note that, by [Reference Liu and Zhu17, Thm. 3.9 (iv)], we see that the filtration $\mathrm {Fil}^{\bullet }$ on the bundle $\mathrm {D}_{\mathrm {dR}}(\mathbb {L})=V\otimes _{{\mathbb Q}_p}\mathcal {O}_{X}$ (which satisfies the Griffiths transversality condition) also gives, by the construction of [Reference Scholze24, Prop. 7.9], the ${\mathbb B}^+_{\mathrm {dR}}$ -submodule $\mathbb {M}=\mathbb {L}\otimes _{\underline { {\mathbb {Z}}_p}} {\mathbb B}_{\mathrm {dR}}^+$ of $\mathbb {M}_0\otimes _{{\mathbb B}_{\mathrm { dR}}^+}{\mathbb B}_{\mathrm {dR}}$ . Part (iii) now follows from the above, the construction of the shtuka $( \mathscr {V},\phi _{ \mathscr {V}})$ from the data $(\mathcal {V},f,\mathbb {L})$ which is implicit in the proof of the equivalence of §2.2.4, and the construction of the shtuka $ \mathscr {V}(\mathbb {L})$ in [Reference Pappas and Rapoport21, §2.6.1].

Proof. The proof of Theorem 3.7 uses the following statement which shows that the p-adic Borel extension property alone implies this form of p-adic Brody hyperbolicity, for reasonably general diamonds. Recall, Y is a smooth irreducible rational curve over F.

Proposition 3.9. Let $\mathcal {D}$ be a locally spatial diamond that is separated over $\mathrm{Spd }\ (F)$ . Suppose that for every finite extension L of F, every morphism of diamonds $a^\times : ({\mathbb B}^\times _L)^{\Diamond } \rightarrow \mathcal {D}$ over $\mathrm{Spd }\ (F)$ extends to a (necessarily unique) morphism $a : {\mathbb B}_L^{\Diamond } \rightarrow \mathcal {D}$ over $\mathrm{Spd }\ (F).$ Then for every finite extension L of F, every morphism $f : Y_L^{\Diamond } \rightarrow \mathcal {D}$ over $\mathrm{Spd }\ (F)$ is constant, that is, f factors through a L-rational point

$$\begin{align*}Y_L^{\Diamond} \rightarrow \mathrm{Spd }\ (L) \rightarrow \mathcal{D}\end{align*}$$

over $\mathrm{Spd }\ (F).$

Proof. We start with a useful lemma.

Lemma 3.10. Let $g: S\to \mathrm{Spec }\ (F)$ be an algebraic variety and $\alpha : S^{\Diamond }\to \mathcal {D}$ a morphism of diamonds over $\mathrm{Spd }\ (F)$ .

a) Suppose there is a variety T with a finite faithfully flat morphism $\pi : T\to S$ such that $\alpha \circ \pi ^{\Diamond }: T^{\Diamond } \to \mathcal {D}$ factors through a morphism $s: \mathrm{Spd }\ (F)\to \mathcal {D}$ over $\mathrm{Spd }\ (F)$ , that is, there exists such s with $\alpha \circ \pi ^{\Diamond }=s\circ (g\circ \pi )^{\Diamond }$ . Then $\alpha $ factors through $s: \mathrm{Spd }\ (F)\to \mathcal {D}$ .

b) Suppose there exists a finite field extension $F'$ of F such that

$$\begin{align*}\alpha': (S\times_FF')^{\Diamond}\to \mathcal {D}\times_{\mathrm{Spd }\ (F)}\mathrm{Spd }\ (F')\to \mathcal {D} \end{align*}$$

factors through a morphism $s': \mathrm{Spd }\ (F')\to \mathcal {D}$ over $\mathrm{Spd }\ (F)$ , that is, there exists such $s'$ with $\alpha '=s'\circ (g')^{\Diamond }$ . Then $\alpha $ factors through a morphism $s: \mathrm{Spd }\ (F)\to \mathcal {D}$ over $\mathrm{Spd }\ (F)$ , that is, there exists such s with $\alpha =s\circ g^{\Diamond }$ .

Proof. a) We have

$$\begin{align*}(s\circ g^{\Diamond})\circ \pi^{\Diamond}=s\circ (g\circ \pi)^{\Diamond}=\alpha\circ \pi^{\Diamond}. \end{align*}$$

By [Reference Scholze and Weinstein29, Prop. 10.3.7], $|T^{\Diamond }|=|T|$ and $|S^{\Diamond }|=|S|$ , where $|T|$ and $|S|$ are the topological spaces underlying the corresponding analytic adic spaces. It follows that $|\pi ^{\Diamond }|: |T^{\Diamond }|\to |S^{\Diamond }|$ is surjective. We can now see, using [Reference Scholze and Weinstein29, Lem. 17.4.9], that $\pi ^{\Diamond }: T^{\Diamond }\to S^{\Diamond }$ is v-surjective, hence this implies $s\circ g^{\Diamond }=\alpha $ , as desired.

b) By replacing $F'$ by its Galois hull over F we can assume that $F'/F$ is finite Galois with group $\Gamma $ ; then $\mathrm{Spd }\ (F)=\mathrm{Spd }\ (F')/\underline \Gamma $ . We observe that, for all $\gamma \in \Gamma $ , we have $s'\circ \gamma ^{\Diamond }=s'$ . Indeed, $(g')^{\Diamond }: (S\times _FF')^{\Diamond }\to \mathrm{Spd }\ (F')$ is v-surjective and we have

$$\begin{align*}s'\circ\gamma^{\Diamond}\circ (g')^{\Diamond}=s' \circ (g'\circ\gamma)^{\Diamond}=s'\circ (g')^{\Diamond}. \end{align*}$$

Denote by $q: \mathrm{Spec }\ (F')\to \mathrm{Spec }\ (F)$ the natural morphism. By finite étale descent, $s'=s\circ q^{\Diamond }$ , for some $s: \mathrm{Spd }\ (F)\to \mathcal {D}$ and we have $\alpha =s\circ g^{\Diamond }$ .

We now continue with the proof of the proposition. Note that if $\mathrm{Spd }\ (L)\to \mathcal {D}$ is a classical point of the diamond $\mathcal {D}$ over $\mathrm{Spd }\ (F)$ , then the image of $|\mathrm{Spd }\ (L)|$ in $|\mathcal {D}|$ is a closed point of the topological space $|\mathcal {D}|$ . In fact, since $\mathrm{Spd }\ (L)\to \mathrm{Spd }\ (F)$ is proper and $\mathcal {D}$ is a separated locally spatial diamond over $\mathrm{Spd }\ (F)$ , the morphism $\mathrm{Spd }\ (L)\to \mathcal {D}$ is proper and $\mathrm{Spd }\ (L)\to \mathcal {D}\times _{\mathrm{Spd }\ (F)}\mathrm{Spd }\ (L)$ is a closed immersion of locally spatial diamonds, see [Reference Scholze and Weinstein29, 17.4].

We can see, using Lemma 3.10 (b), that it is enough to show that there is a finite extension $L'/L$ such that $Y_{L'}^{\Diamond }\to Y_L^{\Diamond }\to \mathcal {D}$ is constant, that is, factors through a point $\mathrm{Spd }\ (L')\to \mathcal {D}$ . This allows us to assume that $Y_L$ contains an L-rational point. Further, replacing $\mathcal {D}$ by $\mathcal {D}\times _{\mathrm{Spd }\ (F)} \mathrm{Spd }\ (L)$ , we may assume also that $L = F.$ It is then also enough to show the statement assuming $Y\subset {\mathbb {A}}^1_F$ .

Now, let $f :Y_F^{\Diamond } \rightarrow \mathcal {D}$ be a morphism of diamonds over $\mathrm{Spd }\ (F).$

We first claim that the underlying map of topological spaces $|f| : |Y_F^{\Diamond }| \rightarrow |\mathcal {D}|$ is constant. Since the classical points of $|Y_F^{\Diamond }| = |Y_F^{\mathrm {an}}|$ are topologically dense in $|Y_F^{\mathrm {an}}|$ , and since, by the above, classical points are closed in $|\mathcal {D}|$ it suffices to prove that all the classical points of $|Y_F^{\mathrm {an}}|$ have the same image in $|\mathcal {D}|.$

Suppose for a contradiction, that there exist two classical points $x \neq y$ of $|Y_F^{\mathrm {an}}|$ , such that $|f|(x) \neq |f|(y)$ in $|\mathcal {D}|.$ Replacing $L=F$ by a finite extension if necessary, we may assume that both x, y are L-rational points. Further, replacing again $\mathcal {D}$ by $\mathcal {D}\times _{\mathrm{Spd }\ (F)} \mathrm{Spd }\ (L)$ , we may assume again that $L = F.$

Using our assumption at balls around the punctures, that is, around the complement of Y in $\mathbb {A}^1_F$ , we extend $f: Y^{\Diamond }\to \mathcal {D}$ to a morphism $\hat {f} : (\mathbb {A}^1_F)^{\Diamond } \rightarrow \mathcal {D}$ over $\mathrm{Spd }\ (F)$ . Let $g : {\mathbb B}^\times _{F} \rightarrow (\mathbb {A}^1_F)^{\mathrm {an}}$ be an analytic function on ${\mathbb B}_{F}^\times $ with an essential singularity at $0$ . Then, the big Picard theorem of p-adic analysis (see for instance [Reference Cherry4, Theorem 2.1]) implies that there are sequences of classical points $(x_n)_{n \geq 1} \subseteq g^{-1}(x)$ and $(y_n)_{n \geq 1} \subseteq g^{-1}(y)$ in ${\mathbb B}_{F}^\times $ with $0 = \lim _n x_n = \lim _n y_n.$ Set $a^\times : ({\mathbb B}_F^\times )^{\Diamond } \rightarrow \mathcal {D}$ to be the composite morphism $a^\times = \hat {f} \circ g^{\Diamond }.$ Since $|a^\times |(x_n) = |f|(x)$ and $|a^\times |(y_n) = |f|(y)$ , $|a^\times |$ cannot extend continuously to the disk $|{\mathbb B}_F^{\Diamond }| = |{\mathbb B}_F|$ . This contradicts our assumption that an extension a exists. This therefore proves the claim that the underlying map of topological spaces $|f| : |Y_F^{\Diamond }| \rightarrow |\mathcal {D}|$ is constant.

By picking an F-rational point of Y and considering its image in $\mathcal {D}$ , we thus have that $|f|$ factors through the image $|x|(|\mathrm{Spd }\ (F)|)$ of an F-rational point $x : \mathrm{Spd }\ (F) \rightarrow \mathcal {D}.$ Since, as above, $x : \mathrm{Spd }\ (F) \rightarrow \mathcal {D}$ is a closed immersion, in particular an injection of locally spatial diamonds, [Reference Scholze26, Prop. 12.15] now implies that f factors through $x.$

The proof of Theorem 3.7 now follows by combining Corollary 3.2 with Proposition 3.9.

Proof. For simplicity, we set

$$\begin{align*}U^\times = {\mathbb B}^{\times s} \times {\mathbb B}^t,\qquad U = {\mathbb B}^{ s} \times {\mathbb B}^t={\mathbb B}^{s+t}. \end{align*}$$

We first show the uniqueness of the extension: Note that by [Reference Scholze and Weinstein29, Prop. 10.3.7], the topological spaces $|U^{\times {\Diamond }} |$ , resp. $|U^{{\Diamond }} |$ , of the diamonds $U^{\times {\Diamond }} $ , resp. $U^{{\Diamond }} $ , are naturally homeomorphic with the underlying topological spaces $|U^\times |$ and $|U |$ of the analytic adic spaces $U^\times $ and $U $ , respectively. We can easily see that $|U^\times |=\cup _{n\geq 1}|\mathbb {A}^s_n\times {\mathbb B}^t|$ is an open and dense subset of $|U|$ . Hence, the same is true for $|U^{\times {\Diamond }} |$ in $|U^{{\Diamond }}|$ . The uniqueness of the extension $a $ of $a ^\times $ in the statement of the theorem follows from this density combined with [Reference Scholze and Weinstein29, Lem. 17.4.1] (or [Reference Scholze26, Prop. 12.15]), since ${\mathrm {Sht}}_{(G,b,\mu , K)}$ is separated over $\mathrm{Spd }\ (\breve E)$ .

As a preliminary reduction, we observe that it is enough to prove that the base change $a^\times _{ F'}$ of $a^\times $ to some finite extension $F'$ of F extends to

$$\begin{align*}a (F'): U^{\Diamond} \times_{\mathrm{Spd }\ (F)}\mathrm{Spd }\ (F')\to {\mathrm{Sht}}_{(G,b,\mu, K)}\times_{\mathrm{Spd }\ (\breve E)}\mathrm{Spd }\ (F') \end{align*}$$

over $\mathrm{Spd }\ (F')$ : Indeed, by replacing $F'$ by its Galois hull over F, we can assume that $F'/F$ is Galois with finite Galois group $\Gamma $ . Then by uniqueness as above, we have $\gamma \cdot a (F')=a (F')$ , for all $\gamma \in \Gamma $ , since this property is true for $a^\times _{ F'}$ . Since $\mathrm{Spd }\ (F)=\mathrm{Spd }\ (F')/\underline {\Gamma }$ , we deduce the existence of $a $ that extends $a^\times $ by étale descent ([Reference Scholze26, Lem. 9.2]).

A) Reduction to the case of $\mathbf {\mathrm {GL}_n}$ : We now reduce the proof to the case that the group is $\mathrm {GL}_n$ : Since $\mathcal {G}$ is smooth and affine over $ {\mathbb {Z}}_p$ , there exists a closed immersion of group schemes $\rho : \mathcal {G}\hookrightarrow \mathrm {GL}_n$ over $ {\mathbb {Z}}_p$ ; set $G'=\mathrm {GL}_n$ , $b' =\rho (b)$ , $\mu '=\rho \circ \mu $ , and take $K'=\mathrm {GL}_n( {\mathbb {Z}}_p)$ . Then $(G', [b'], \{\mu '\})$ gives local shtuka data with reflex field $E'={\mathbb Q}_p\subset E$ , and $\rho $ induces a morphism of diamonds

$$\begin{align*}\rho_*: {\mathrm{Sht}}_{(G,b,\mu, K)}\to {\mathrm{Sht}}_{(G',b',\mu', K')}\times_{\mathrm{Spd }\ (\breve {\mathbb Q}_p)}\mathrm{Spd }\ (\breve E) \end{align*}$$

over $\mathrm{Spd }\ (\breve E)$ ; this morphism is a closed immersion. (This follows from the argument in the proof of [Reference Gleason11, Prop. 2.25]. Alternatively, it can be deduced from the étaleness of the period morphism together with the fact that

$$\begin{align*}\rho^{\mathrm{Gr}}_*: \mathrm{Gr}_{G,\mathrm{Spd }\ ( E),\leq\mu}\hookrightarrow \mathrm{Gr}_{\mathrm{GL}_n, \mathrm{Spd }\ ({\mathbb Q}_p), \leq\mu'}\times_{\mathrm{Spd }\ ({\mathbb Q}_p)}\mathrm{Spd }\ (E) \end{align*}$$

is a closed immersion, for example, see [Reference Scholze and Weinstein29, Lem. 19.1.5, Prop. 19.2.3, Lect. 20]).

Consider the compositions

$$\begin{align*}\rho_*\circ a^\times : U^{\times{\Diamond}} \to {\mathrm{Sht}}_{(G',b',\mu', K')}\times_{\mathrm{Spd }\ (\breve {\mathbb Q}_p)}\mathrm{Spd }\ (\breve E), \end{align*}$$

$$\begin{align*}c^\times: U^{\times{\Diamond}} \xrightarrow{ \rho_*\circ a^\times } {\mathrm{Sht}}_{(G',b',\mu', K')}\times_{\mathrm{Spd }\ (\breve {\mathbb Q}_p)}\mathrm{Spd }\ (\breve E)\to {\mathrm{Sht}}_{(G',b',\mu', K')}. \end{align*}$$

Assuming that the conclusion of Theorem 3.1 holds for $\mathrm {GL}_n$ , the morphism $c^\times $ extends to

$$\begin{align*}c: U^{\Diamond} \to {\mathrm{Sht}}_{(G',b',\mu', K')} \end{align*}$$

over $\mathrm{Spd }\ (\breve {\mathbb Q}_p)$ . This gives

$$\begin{align*}a': U^{{\Diamond}} \xrightarrow{ \ } {\mathrm{Sht}}_{(G',b',\mu', K')}\times_{\mathrm{Spd }\ (\breve {\mathbb Q}_p)}\mathrm{Spd }\ (\breve E) \end{align*}$$

over $\mathrm{Spd }\ (\breve E)$ such that $a' =\rho _*\circ a^\times $ . Recall that $|U^{\times {\Diamond }} |$ is open and dense in $|U^{\Diamond } |$ and $\rho _*$ is a closed immersion. Since $ a'(|U^{\times {\Diamond }} |)=(\rho _*\circ a^\times )(|U^{\times {\Diamond }} |)\subset \rho _*(|{\mathrm {Sht}}_{(G,b,\mu , K)}|)$ , we see that $a'(|U^{{\Diamond }} |)\subset \rho _*(|{\mathrm {Sht}}_{(G,b,\mu , K)}|)$ . Hence, by [Reference Scholze and Weinstein29, Lem. 17.4.1 (3)] (or [Reference Scholze26, Prop. 12.15]), $a'$ factors through $\rho _*$ , that is, there is

$$\begin{align*}a : U^{\Diamond} \to {\mathrm{Sht}}_{(G,b,\mu, K)} \end{align*}$$

with $\rho _*\circ a =a'$ . We can now easily see that $a $ extends $a^\times $ . Hence, this completes our reduction of the proof to the case $G=\mathrm {GL}_n$ .

B) The case of $\mathbf {\mathrm {GL}_n}$ : From here and on, we assume that $G=\mathrm {GL}_n$ , $\mathcal {G}=(\mathrm {GL}_n)_{ {\mathbb {Z}}_p}$ ; then the reflex field is ${\mathbb Q}_p$ . Set $V={\mathbb Q}_p^n$ so that $G=\mathrm {GL}_n=\mathrm {GL}(V)$ .

B1) Extending the shtuka: The pull-back of the universal $\underline {\mathrm {GL}_n( {\mathbb {Z}}_p)}$ -torsor $\mathbb {P}^{\mathrm { univ}}$ to $ U^{\times {\Diamond }}$ gives a pro-étale $\underline {\mathrm {GL}_n( {\mathbb {Z}}_p)}$ -torsor

$$\begin{align*}\mathbb {P}_{U^{\times {\Diamond}}}=(a^\times)^*\mathbb {P}^{\mathrm{univ}} \end{align*}$$

over the diamond $U^{\times {\Diamond }}$ . As in §2.4, $\mathbb {P}_{U^{\times {\Diamond }}}$ is canonically isomorphic to the “torsor of trivializations” of a pro-étale $\underline { {\mathbb {Z}}_p}$ -local system $\mathbb {L}^\times $ over the diamond $U^{\times {\Diamond }}$ . This uniquely corresponds to an étale $ {\mathbb {Z}}_p$ -local system over $U^\times $ which we will also denote by $\mathbb {L}^\times $ .

By Prop. 2.1 (ii), the local system $\mathbb {L}^\times $ over $U^\times $ , is horizontal de Rham. We now claim that $\mathbb {L}^\times $ extends to a de Rham $ {\mathbb {Z}}_p$ -local system $\mathbb {L} $ over $U $ , with the same Hodge-Tate weights. We first note the following general lemma.

We shall now apply the argument in the proof of [Reference Oswal, Shankar and Zhu20, Thm 5.7] (which uses crucially the p-adic log Riemann-Hilbert correspondence of [Reference Diao, Lan, Liu and Zhu9]) to obtain the required extension $\mathbb {L}$ of $\mathbb {L}^\times $ .

The proof of [Reference Oswal, Shankar and Zhu20, Thm 5.7] is written for the case where the boundary divisor Z is smooth. However, we can apply the argument of loc. cit. to first see that $\mathbb {L}^\times $ extends to a de Rham étale $ {\mathbb {Z}}_p$ -local system $\mathbb {L}_1^\times $ over $U_1^\times := {({\mathbb B}^\times )}^{s-1} \times {\mathbb B}^{t+1}$ . Since the Kummer étale extensions to $U_{\text {k\'et}}$ (see [Reference Diao, Lan, Liu and Zhu8, Cor. 6.3.4]) of $\mathbb {L}_1^\times $ and $\mathbb {L}^\times $ agree, $(D_{\text {dR,}\log }(\mathbb {L}_1^\times ),\nabla _{\mathbb {L}_1^\times })$ agrees with $(D_{\text {dR}, \log }(\mathbb {L}^\times ),\nabla _{\mathbb {L}^\times })$ . We may thus iterate the above argument, inducting on s, to obtain the desired extension $\mathbb {L}$ over U. Moreover, it follows from Lemma 3.13 that $\mathbb {L} $ is horizontal de Rham, with $(\mathrm {D}_{\mathrm {dR}}(\mathbb {L} ), \nabla )$ being the (trivial) canonical extension of the trivial pair $(\mathrm {D}_{\mathrm {dR}}(\mathbb {L}^\times ), \nabla ^\times )$ .

The de Rham local system $\mathbb {L} $ over $U $ gives, by the construction in [Reference Pappas and Rapoport21, §2.6], a shtuka $ \mathscr {V}(\mathbb {L} )$ over $U^{\Diamond }$ ; this has one leg bounded by $\mu $ and by Proposition 2.1 (iii) restricts to the shtuka over $U^{\times {\Diamond }} $ obtained from $a^\times $ .

B2) Constructing a framing: We will now show that the shtuka $ \mathscr {V}(\mathbb {L} )$ over $U^{\Diamond }$ constructed above, admits a framing by b, after possibly enlarging the finite extension $F/\breve E$ . We first show:

Proof. We set $\mathbb {T}=\mathbb {T}_F=\operatorname {\mbox {Spa }}(F\langle t, t^{-1}\rangle , \mathcal {O}_F\langle t, t^{-1}\rangle )$ . Note that we can cover $U={\mathbb B}^{s+t}$ by a finite union of open rational subdomains $W\subset U$ , each isomorphic to $\mathbb {T}^{s+t}$ . For example, ${\mathbb B}$ is covered by $|x|=1$ and $|x-1|=1$ . Using this, we see that it is enough to show that all the restrictions $\mathbb {L}_{|W}$ satisfy the conclusion of the claim, that is, for each such $W\simeq \mathbb {T}^{s+t}\subset U$ , there is a finite $F'/F$ such that the base change $\mathbb {L}_{|W_{F'}}$ is crystalline at all classical points of $W_{F'}$ . Note that

$$\begin{align*}W\simeq \mathbb {T}^{s+t}=\mathbb {T}^{s+t}_F=\operatorname{\mbox{Spa }}(R[p^{-1}], R) \end{align*}$$

where $R=\mathcal {O}_F\langle t^{\pm 1}_1,\ldots , t^{\pm 1}_{s+t}\rangle $ satisfies the assumptions of [Reference Shimizu30, Set-up 3.1] and the assumption (BR) (“bonne reduction”) of [Reference Shimizu30, 3.3], see also [Reference Brinon2].

We will now apply results of Shimizu [Reference Shimizu30]. Note that the ring $\mathrm {B}^\nabla _{\mathrm {dR}}(R)={{\mathbb B}}_{\mathrm { dR}}(A,A^+)$ agrees with the sections of the pro-étale sheaf ${\mathbb B}_{\mathrm {dR}}$ of [Reference Scholze24] on the corresponding pro-étale cover of $\operatorname {\mbox {Spa }}(R[p^{-1}], R)$ (see [Reference Shimizu30, §8.2]). Also, by [Reference Shimizu30, Lem. 8.7] the notion of “de Rham local system” of [Reference Scholze24], [Reference Liu and Zhu17], used above, agrees with the notion used in [Reference Shimizu30]. Similarly, by the above, the representation $\mathcal {G}_{R} \to \mathrm {GL}(\Lambda )\subset \mathrm {GL}(V)$ ( $\Lambda \subset V$ is a $ {\mathbb {Z}}_p$ -lattice) which corresponds to $\mathbb {L}_{|W}$ is horizontally de Rham, also in the sense of [Reference Shimizu30]. Hence, since by Proposition 2.1 (i), $\mathbb {L}_{|W}$ has some crystalline fibers, the p-adic monodromy result [Reference Shimizu30, Thm. 7.4], implies that there is a finite extension $F'/F$ , such that the pull-back $\mathbb {L}_{|W_{F'}}$ of $\mathbb {L}$ to $W_{F'}\simeq \mathbb {T}^{s+t}_{F'}$ is “horizontal crystalline,” that is, the representation $\mathcal {G}_{R_{F'}} \to \mathrm {GL}(\Lambda )\subset \mathrm {GL}(V)$ which corresponds to $\mathbb {L}_{|W_{F'}}$ is horizontal crystalline in the sense of loc. cit.. This implies, in particular, that $\mathbb {L}_{|W_{F'}}$ is crystalline at all classical points of $W_{F'}$ .

By the “preliminary reduction” in the beginning of the proof, it is enough to show the extension of $a^\times $ exists after we base change to $F'$ . Hence, we can replace $F'$ by F and assume that $F'=F$ in the conclusion of Lemma 3.14; we can also assume that F is Galois over $\breve {\mathbb Q}_p=W(k)[1/p]$ . Recall, $U={\mathbb B}^{s+t}={\mathbb B}_F^{s+t}$ ; this is the generic fiber of the smooth p-adic formal scheme

$$\begin{align*}\mathfrak{U}=\mathrm{Spf}(\mathcal{O}_F\langle x_1,\ldots, x_{s+t}\rangle) \end{align*}$$

over $\mathcal {O}_F$ . Lemma 3.14 with $F'=F$ implies that the local system $\mathbb {L}$ over U satisfies the “pointwise crystallinity criterion” of Guo-Yang [Reference Guo and Yang12, Thm. 7.2] and, hence, is crystalline (with respect to $\mathfrak {U}$ ). Recall that, as in [Reference Guo and Yang12, Def. 4.4], this means that there is a locally free Frobenius isocrystal $(\mathcal {E},\phi _{\mathcal {E}})\in \mathrm {Isoc}^\phi (\mathfrak {U}_{k, {\mathrm {crys}}})$ , together with a Frobenius equivariant isomorphism of ${\mathbb B}_{\mathrm {crys}}$ -vector bundles

$$\begin{align*}\vartheta: {\mathbb B}_{\mathrm{crys}}(\mathcal {E})\xrightarrow{\sim} {\mathbb B}_{\mathrm{crys}}\otimes_{ {\mathbb{Z}}_p} \mathbb {L} \end{align*}$$

over $U_{\mathrm {pro\acute {e}t}}$ . Here, the notations are as in loc. cit.. In particular, $\mathrm {Isoc}^\phi (\mathfrak {U}_{k, {\mathrm {crys}}})$ is the category of Frobenius isocrystals over the crystalline site $(\mathfrak {U}_k/W(k))_{\mathrm {crys}}$ , where $\mathfrak {U}_k$ is the special fiber $\mathrm{Spec }\ (k[x_1,\ldots ,x_{s+t}])=\mathbb {A}^{s+t}_k$ . Moreover, in the above, we can take $(\mathcal {E},\phi _{\mathcal {E}})$ to be the value $\mathcal {E}_{\mathrm {crys}, \mathbb {L}}$ of the crystalline Riemann-Hilbert functor of [Reference Guo and Yang12, Thm. 5.20] at the local system $\mathbb {L}$ , see [Reference Guo and Yang12, Thm. 5.20 (v)] and its proof.

We next claim that $\mathcal {E}_{\mathrm {crys}, \mathbb {L}}$ is “constant,” that is, obtained by base change from the Frobenius isocrystal over $\mathrm{Spec }\ (k)$ which is given by b.

Note that, in our set-up, we can consider

$$\begin{align*}\mathfrak{U}_0=\operatorname{\mbox{Spf }}(W(k)\langle x_1,\ldots, x_{s+t}\rangle) \end{align*}$$

with its standard Frobenius lift $\phi _0$ taking $x_i$ to $x_i^p$ . Then, by a classical result (see, e.g., [Reference Guo and Yang12, Prop. 2.11]) the Frobenius isocrystal $\mathcal {E}_{\mathrm {crys}, \mathbb {L}}$ is functorially uniquely determined by a triple $(\mathcal {V}_0, \nabla _0, \phi _{\mathcal {V}_0})$ , where

• • $\mathcal {V}_0$ is a vector bundle of rank n over $U_0=\mathfrak {U}_{0,\eta }={\mathbb B}^{s+t}_{W(k)[1/p]}$ ,

• • $\nabla _0$ is a flat connection on $\mathcal {V}_0$ ,

• • $\phi _{\mathcal {V}_0}$ is a $\phi _0$ -linear Frobenius endomorphism $\phi _{\mathcal {V}_0}$ whose linearization $\phi ^\#_{\mathcal {V}_0}: \phi ^*_0\mathcal {V}_0\xrightarrow {\ } \mathcal {V}_0$ is a horizontal isomorphism.

To prove our claim, it is enough to show that the triple $(\mathcal {V}_0, \nabla _0, \phi _{\mathcal {V}_0})$ is constant, that is, isomorphic to $(\breve V\otimes _{\breve {\mathbb Q}_p}\mathcal {O}_{U_0}, 1\otimes d, b\otimes \phi _0)$ .

The vector bundle $\mathcal {V}_0$ over $U_0$ is trivial, by a result of Lütkebohmert [Reference Lütkebohmert18]. After choosing a basis of $\mathcal {V}_0$ we can write the connection $\nabla _0$ via an $n\times n$ matrix $\Omega =(\omega _{ij})$ of $1$ -forms $\omega _{ij}\in \Omega ^1_{U_0}(U_0)$ , while $\phi _{\mathcal {V}_0}=C\cdot \phi _0 $ for an $n\times n$ matrix $C=(c_{ij})$ , $c_{ij}\in \mathcal {O}_{U_0}(U_0)=\breve {\mathbb Q}_p\langle x_1,\ldots , x_{s+t}\rangle $ . By the construction of the crystalline Riemann-Hilbert functor $ \mathcal {E}_{\mathrm {crys}, -}$ and [Reference Guo and Yang12, Thm. 4.12 (iii)], the base change of $(\mathcal {V}_0, \nabla _0)$ from $\breve {\mathbb Q}_p$ to F is isomorphic to the pair $(\mathbb {D}_{\mathrm {dR}}(\mathbb {L}), \nabla )$ over U which is attached to $\mathbb {L}$ by the p-adic Riemann-Hilbert correspondence of [Reference Liu and Zhu17]; as above, the pair $(\mathbb {D}_{\mathrm {dR}}(\mathbb {L}), \nabla )$ is isomorphic to the trivial pair $(V\otimes _{{\mathbb Q}_p}\mathcal {O}_{U}, 1\otimes d)$ . Hence, there is an invertible $n\times n$ matrix $g=g(\underline x)$ with entries in $\mathcal {O}_{U}(U)=\mathcal {O}_{U_0}(U_0)\otimes _{\breve {\mathbb Q}_p}F$ , such that

$$\begin{align*}dg\, g^{-1}-g\Omega g^{-1}=0. \end{align*}$$

This gives $\Omega =g^{-1}dg$ and since for each $\sigma \in \Gamma =\operatorname {\mathrm {Gal}}(F/\breve {\mathbb Q}_p)$ , $\sigma \Omega =\Omega $ , we obtain $\sigma (g)^{-1}d(\sigma (g))=g^{-1}dg$ . In turn, this implies $d(\sigma (g)g^{-1})=0$ . It follows that $\sigma (g)g^{-1}=:c_\sigma \in F^*$ and so $\sigma \mapsto c_\sigma $ gives a $1$ -cocycle $\Gamma \to F^*$ . By Hilbert $90$ , there is $\delta \in F^*$ such that $c_\sigma =\sigma (\delta )\delta ^{-1}$ . Then $h:=\delta ^{-1}g$ satisfies $\sigma (h)=h$ and $dh\, h^{-1}-h\Omega h^{-1}=0$ . Hence, h lies in $\mathrm {GL}_n(\mathcal {O}_{U_0}(U_0))$ and gives an isomorphism of $(\mathcal {V}_0, \nabla _0)$ with the trivial pair $(V\otimes _{{\mathbb Q}_p}\mathcal {O}_{U_0}, 1\otimes d)$ over $U_0$ . Under the identification $\mathcal {V}_0=V\otimes _{{\mathbb Q}_p}\mathcal {O}_{U_0}=\breve V\otimes _{\breve {\mathbb Q}_p}\mathcal {O}_{U_0}$ given by this isomorphism, the linearization $\phi ^\#_{\mathcal {V}_0}$ of $\phi _{\mathcal {V}_0}$ is given by $C\in \mathrm {End}_{\breve {\mathbb Q}_p}(\breve V)\otimes _{\breve {\mathbb Q}_p}\mathcal {O}_{U_0}(U_0)$ . Then $\phi _{\mathcal {V}_0}=C\cdot \phi _0$ . Since $\phi ^\#_{\mathcal {V}_0}$ is horizontal for the connection $\nabla _0=1\otimes d$ , we see that $dC=0$ , that is, C is constant, therefore $C=b\in \mathrm {GL}(\breve V)=\mathrm {GL}_n(\breve {\mathbb Q}_p)$ . Hence, the triple $(\mathcal {V}_0, \nabla _0, \phi _{\mathcal {V}_0})$ is indeed isomorphic to $(\breve V\otimes _{\breve {\mathbb Q}_p}\mathcal {O}_{U_0}, 1\otimes d, b\otimes \phi _0)$ , as claimed. Therefore, $\mathcal {E}_{\mathrm { crys}, \mathbb {L}}$ is constant and obtained by pull-back from the Frobenius isocrystal over $\mathrm{Spec }\ (k)$ given by b.

It then follows, using the functoriality of the construction of ${\mathbb B}_{\mathrm {crys}}(\mathcal {E})$ in [Reference Guo and Yang12, 4.1], that we have a Frobenius equivariant isomorphism of ${\mathbb B}_{\mathrm {crys}}$ -vector bundles

$$\begin{align*}{\mathbb B}_{\mathrm{crys}}(\mathcal {E}_{\mathrm{crys}, \mathbb {L}})\simeq \breve V\otimes_{\breve{\mathbb Q}_p}{\mathbb B}_{\mathrm{ crys}} \end{align*}$$

where the Frobenius action on $\breve V=V\otimes _{{\mathbb Q}_p}\breve {\mathbb Q}_p$ is given by $b\otimes \mathrm {Frob}$ . Hence, $\vartheta $ above amounts to a Frobenius equivariant isomorphism

(3.1) $$ \begin{align} \vartheta: \breve V\otimes_{\breve{\mathbb Q}_p}{\mathbb B}_{\mathrm{crys}}\xrightarrow{\sim} \mathbb {L}\otimes_{ {\mathbb{Z}}_p}{\mathbb B}_{\mathrm{crys}} \end{align} $$

of ${\mathbb B}_{\mathrm {crys}}$ -vector bundles over $U_{\mathrm {pro\acute {e}t}}$ . We will now use this isomorphism to obtain a b-framing of the shtuka $ \mathscr {V}(\mathbb {L})$ over $U^{\Diamond }$ . We set, for simplicity, $R=\mathcal {O}_F\langle x_1,\ldots , x_{s+t}\rangle $ , so that $U=\operatorname {\mbox {Spa }}(R[p^{-1}], R)$ and consider the perfectoid cover $\operatorname {\mbox {Spa }}(A, A^+)\to \operatorname {\mbox {Spa }}(R[p^{-1}], R)$ obtained as in §2.3.1. Evaluating the isomorphism (3.1) at the corresponding pro-étale cover of $U=\operatorname {\mbox {Spa }}(R[p^{-1}], R)$ gives

(3.2) $$ \begin{align} \breve V\otimes_{\breve{\mathbb Q}_p}{{\mathbb B}}_{\mathrm{crys}}(A, A^+)\xrightarrow{\ \sim\ } \Lambda\otimes_{ {\mathbb{Z}}_p} {{\mathbb B}}_{\mathrm{crys}}(A, A^+). \end{align} $$

Base changing (3.2) by ${{\mathbb B}}_{\mathrm {crys}}(A, A^+)\hookrightarrow {{\mathbb B}}_{\mathrm {max}}(A, A^+)$ gives

(3.3) $$ \begin{align} \breve V\otimes_{\breve{\mathbb Q}_p}{{\mathbb B}}_{\mathrm{max}}(A, A^+)\xrightarrow{\ \sim\ } \Lambda\otimes_{ {\mathbb{Z}}_p} {{\mathbb B}}_{\mathrm{max}}(A, A^+). \end{align} $$

These isomorphisms respect the actions of Frobenius $\phi $ and $\mathcal {G}_{R}$ : For (3.3) the Galois group $\mathcal {G}_{R}$ acts on the LHS only on ${{\mathbb B}}_{\mathrm {max}}(A,A^+)$ and diagonally on the RHS, while $\phi $ acts diagonally on the LHS and on the RHS only on ${{\mathbb B}}_{\mathrm {max}}(A, A^+)$ .

Set $S=\operatorname {\mbox {Spa }}(A^\flat , A^{\flat +})$ for the tilt of $\operatorname {\mbox {Spa }}(A, A^{+})$ ; this gives a cover $q: S=\operatorname {\mbox {Spa }}(A^\flat , A^{\flat +})\to U^{\Diamond }$ . Observe that

$$\begin{align*}{{\mathbb B}}^+_{\mathrm{max}}(A,A^+)=W(A^{\flat+})\Big\langle \frac{p-[p^\flat]}{p}\Big\rangle[p^{-1}]=W(A^{\flat+})\Big\langle \frac{[p^\flat]}{p}\Big\rangle[p^{-1}], \end{align*}$$

where $p^\flat =(p, p^{1/p}, \ldots )\,\mathrm {mod}\, p\in \mathcal {O}_{C^\flat }\subset A^{\flat +}$ is a pseudo-uniformizer of $A^\flat $ . Indeed, $\ker (\theta )\subset W(A^{\flat +})$ is generated by $\xi :=p-[p^\flat ]$ ([Reference Kedlaya and Liu16, Lem. 3.6.3]), and $|\xi |=|p-[p^\flat ]|\leq |p|$ is equivalent to $|[p^\flat ]|\leq |p|$ . Hence, (2.1) gives a natural Frobenius equivariant map

$$\begin{align*}{\beta}: {{\mathbb B}}^+_{\mathrm{max}}(A,A^+)\to \Gamma(\mathcal {Y}_{[1,\infty)}(S), \mathcal{O}). \end{align*}$$

This also gives

$$\begin{align*}\beta[t^{-1}]: {{\mathbb B}}_{\mathrm{max}}(A,A^+) \to \Gamma(\mathcal {Y}_{[1,\infty)}(S), \mathcal{O})[\beta(t)^{-1}]. \end{align*}$$

Recall that the completion of ${{\mathbb B}}^+_{\mathrm {max}}(A,A^+)$ along $\ker (\theta _{\mathrm {max}})$ identifies with ${{\mathbb B}}^+_{\mathrm {dR}}(A,A^+)$ . The base changes of the ${{\mathbb B}}^+_{\mathrm {max}}(A,A^+)$ -modules $\breve V\otimes _{\breve {\mathbb Q}_p}{{\mathbb B}}^+_{\mathrm { max}}(A, A^+)$ and $\Lambda \otimes _{ {\mathbb {Z}}_p} {{\mathbb B}}^+_{\mathrm {max}}(A,A^+)$ by ${{\mathbb B}}^+_{\mathrm { max}}(A,A^+)\to {{\mathbb B}}^+_{\mathrm {dR}}(A,A^+)$ give ${{\mathbb B}}^+_{\mathrm {dR}}(A,A^+)$ -lattices $\mathbb {M}_0(A,A^+)$ and $\mathbb {M}(A,A^+)$ , respectively, in

(3.4) $$ \begin{align} \breve V\otimes_{\breve{\mathbb Q}_p}{{\mathbb B}}_{\mathrm{dR}}(A, A^+)\simeq \Lambda\otimes_{ {\mathbb{Z}}_p} {{\mathbb B}}_{\mathrm{ dR}}(A, A^+), \end{align} $$

where the isomorphism is the base change of (3.3) by ${{\mathbb B}}^+_{\mathrm {max}}(A,A^+)\to {{\mathbb B}}^+_{\mathrm {dR}}(A,A^+)$ . Base changing (3.3) by $\beta [t^{-1}]$ gives a $\phi $ -equivariant meromorphic isomorphism

$$\begin{align*}\breve V\otimes_{\breve{\mathbb Q}_p} \mathcal{O}_{\mathcal {Y}_{[1,\infty)}(S)\setminus \cup_{n\geq 0} \phi^n(S^\sharp)} \xrightarrow{\ \sim\ } \Lambda\otimes_{ {\mathbb{Z}}_p} \mathcal{O}_{\mathcal {Y}_{[1,\infty)}(S)\setminus\cup_{n\geq 0} \phi^n(S^\sharp)}. \end{align*}$$

We can see, using the construction of the shtuka $ \mathscr {V}(\mathbb {L})$ , that this extends to an isomorphism

$$\begin{align*}\iota_{r}: \breve V\otimes_{\breve{\mathbb Q}_p} \mathcal{O}_{\mathcal {Y}_{[r,\infty)}(S)}\xrightarrow{\ \sim\ } \mathscr{V}(\mathbb {L})_{|\mathcal {Y}_{[r,\infty)} (S)} \end{align*}$$

for $r\gg 0$ , which is Frobenius equivariant, when the Frobenius on the LHS given by $b\otimes \mathrm {Frob}$ . Indeed, the claim follows from the construction of $ \mathscr {V}(\mathbb {L})$ in [Reference Pappas and Rapoport21, §2.5.1, §2.6], since the shtuka $ \mathscr {V}(\mathbb {L})$ over $U^{\Diamond }$ which is associated to the de Rham local system $\mathbb {L}$ over U, is given using the lattices $\mathbb {M}_0(A,A^+)$ and $\mathbb {M}(A,A^+)$ and the cover q, see also [Reference Pappas and Rapoport21, Def. 2.6.4, Prop. 2.6.3].

The isomorphism $\iota _r$ gives a b-framing of $ \mathscr {V}(\mathbb {L})$ over S. We now show that $\iota _r$ descends along the cover q to $U^{\Diamond }$ . Since $U^{\Diamond }=\mathrm{Spd }\ (A, A^+)/\underline {\mathcal {G}_{R}}$ by [Reference Scholze and Weinstein29, Lem. 10.1.7], it is enough to show that $\iota _{r}$ is equivariant for the action of the Galois group $\mathcal {G}_{R}$ . This follows from the construction and the fact that the isomorphism (3.2) is Galois equivariant. Hence, we obtain a b-framing of $ \mathscr {V}(\mathbb {L})$ over $U^{\Diamond }$ , which we will also denote by $\iota _r$ .

B3) Extending the framing:

It remains to show that we can adjust the framing $\iota _r$ of $ \mathscr {V}(\mathbb {L})$ we just constructed in the section above, so that it extends the original framing $\iota ^\times _r$ over $U^{\times {\Diamond }}$ given by $a^\times $ .

The discrepancy between the restriction $(\iota _r){|_{U^{\times {\Diamond }}}}$ and $\iota ^\times _r$ is given by a $U^{\times {\Diamond }}$ -section of the automorphism v-sheaf $\tilde {G}_b:=\underline {\mathrm {Aut}}(\mathcal {E}^b)$ , where $\mathcal {E}^b$ is the bundle corresponding to b over the (absolute) Fargues-Fontaine curve.

Let V be an analytic adic space over $\operatorname {\mbox {Spa }}(F,\mathcal {O}_F)$ with $|V|$ connected which admits an open cover

$$\begin{align*}V=\bigcup_{n\geq 1}\operatorname{\mbox{Spa }}(R_n[1/p], R_n). \end{align*}$$

We will assume that, for each n, $R_n$ is a Noetherian, normal and flat p-adically complete $\mathcal {O}_F$ -algebra and that the localizations of $R_n$ at height $1$ primes over p are dvrs which are formally smooth over $\mathcal {O}_F$ .

Note that we can write

$$\begin{align*}U^{\times}=\bigcup_{n\geq 1}\operatorname{\mbox{Spa }}(R_n[1/p], R_n) \end{align*}$$

with $ R_n=\mathcal {O}_F\langle x_1, p^n/x_1, \ldots , x_s, p^n/x_s, x_{s+1}, \ldots , x_{s+t}\rangle $ satisfying the above. The topological space $|U^\times |$ is connected. Hence, Proposition 3.15 applies to $V=U^{\times }$ and we obtain $ \tilde G_b( U^{\times {\Diamond }})= J_b({\mathbb Q}_p). $

Proof. By [Reference Fargues and Scholze10, Prop. III.5.1], the group sheaf $\tilde G_b$ is a semi-direct product

(3.5) $$ \begin{align} \tilde G_b= \mathcal {U}\rtimes \underline{J_b({\mathbb Q}_p)}, \end{align} $$

where $ \mathcal {U}$ is a unipotent group sheaf which is a successive extension of positive (absolute) Banach-Colmez spaces ${\mathcal BC}(\lambda )$ , cf. [Reference Pappas and Rapoport21, Prop. 4.2.5].

Set $A^+_n=\widehat {\overline R}_n$ , $A_n=\widehat {\overline R}_n[p^{-1}]$ , constructed as in §2.3.1. Then

(3.6) $$ \begin{align} \operatorname{\mbox{Spa }}(R_n[1/p], R_n)^{\Diamond}=\mathrm{Spd }\ (A_n, A^+_n)/\underline{\mathcal {G}_{R_{n}}}, \end{align} $$

with $\mathcal {G}_{R_n}=\operatorname {\mathrm {Gal}}({\overline R}_n[p^{-1}]/R_n[1/p])$ .

Lemma 3.16. Let $R_n$ be an $\mathcal {O}_F$ -algebra satisfying the above assumptions.

1. i) We have $\mathrm {B}^\nabla _{\mathrm {cris}}(R_n)^{\mathcal {G}_{R_n }}={\mathbb B}_{\mathrm {cris}}(A_n, A^+_n)^{\mathcal {G}_{R_n}}=\breve {\mathbb Q}_p$ .

2. ii) For $\lambda>0$ , we have $ {\mathcal BC}(\lambda )(\operatorname {\mbox {Spa }}(R_n[1/p], R_n)^{\Diamond })=(0). $

Proof. We remark that [Reference Pappas and Rapoport21, Rem. 4.2.8] sketches the proof of a similar result following a suggestion of Scholze, see also Remark 3.17 below.

Part i): Note that [Reference Shimizu30, Cor. 4.8], based on [Reference Brinon2, Cor. 5.3.7], gives a similar conclusion, but in there it is assumed that the ring satisfies [Reference Shimizu30, Set-up 3.1]. We give an argument which works in our situation: Set $R=R_n$ , $A=A_n$ , for simplicity. First notice that, as in loc. cit., it is enough to show $\mathrm {B}^\nabla _{\mathrm {dR}}(R)^{\mathcal {G}_{R}}={\mathbb B}_{\mathrm { dR}}(A, A^+)^{\mathcal {G}_{R}}=F$ . Let T be the set of height $1$ prime ideals of $\overline R$ that lie over p. Let $\widehat {\overline {R}_{\frak p}}$ denote the p-adic completion of the localization $\overline R_{\frak p}$ of $\overline R$ at $\frak p$ . Set $(A_{\frak p}, A^+_{\frak p})=(\widehat {\overline {R}_{\frak p}}[p^{-1}], \widehat {\overline {R}_{\frak p}})$ which also gives a perfectoid pair, cf. [Reference Shimizu30, Lem. 6.5]. Now observe that, since $\overline R$ is a union of Noetherian normal domains which are finite over R, the natural ring homomorphisms

$$\begin{align*}{\overline R}\to \prod_{\frak p\in T} {\overline{R}_{\frak p}}, \qquad \widehat{\overline R}\to \prod_{\frak p\in T}\widehat{\overline{R}_{\frak p}}, \end{align*}$$

are injective with p-torsion free cokernel, cf. the proof of [Reference Shimizu30, Lem. 6.13]. As in [Reference Shimizu30, Prop. 2.37], it follows that the natural ring homomorphism

$$\begin{align*}{\mathbb B}_{\mathrm{dR}}(A, A^+)\to \prod_{\frak p\in T}{\mathbb B}_{\mathrm{dR}}(A_{\frak p}, A^+_{\frak p}) \end{align*}$$

is injective. Hence,

$$\begin{align*}{\mathbb B}_{\mathrm{dR}}(A, A^+)^{\mathcal {G}_{R}}\subset \prod_{\frak p\in T}{\mathbb B}_{\mathrm{dR}}(A_{\frak p}, A^+_{\frak p})^{\mathcal {G}_{R}(\frak p)}, \end{align*}$$

where $\mathcal {G}_{R}(\frak p)=\operatorname {\mathrm {Gal}}(\overline {R}_{\frak p}[p^{-1}]/R[p^{-1}])$ is identified with the (decomposition) subgroup of $\mathcal {G}_R$ preserving the prime $\frak p$ .

For each height $1$ prime $\frak q$ of R above p, we let $\mathcal {K}_{\frak q}=\mathcal {O}_{\mathcal {K}_{\frak q}}[p^{-1}]$ be the fraction field of the completion $\mathcal {O}_{\mathcal {K}_{\frak q}}$ of the localization $R_{\frak q}$ . Fix an algebraic closure $\overline {\mathcal {K}}_{\frak q}$ of $\mathcal {K}_{\frak q}$ . Let $\mathcal {O}_{\overline {\mathcal {K}}_{\frak q}}$ denote the ring of integers of $\overline {\mathcal {K}}_{\frak q}$ and denote the p-adic completion of $\mathcal {O}_{\overline {\mathcal {K}}_{\frak q}}$ by $\widehat {\mathcal {O}}_{\overline {\mathcal {K}}_{\frak q}}$ . By [Reference Ohkubo19, Cor. 4.3],

(3.7) $$ \begin{align} {\mathbb B}_{\mathrm{dR}}(\widehat{\mathcal{O}}_{\overline{\mathcal{K}}_{\frak q}}[p^{-1}], \widehat{\mathcal{O}}_{\overline{\mathcal{K}}_{\frak q}})^{\operatorname{\mathrm{Gal}}(\overline{\mathcal{K}}_{\frak q}/\mathcal {K}_{\frak q})}=F. \end{align} $$

(cf. [Reference Shimizu30, Lem. 6.9]).

Suppose now $\frak p\in T$ lies above $\frak q$ and corresponds to an R-embedding $\overline R\hookrightarrow \mathcal {O}_{\overline {\mathcal {K}}_{\frak q}}$ . The resulting inclusion $\overline R_{\frak p}\hookrightarrow \mathcal {O}_{\overline {\mathcal {K}}_{\frak q}}$ is $\operatorname {\mathrm {Gal}}(\overline {\mathcal {K}}_{\frak q}/\mathcal {K}_{\frak q})$ -equivariant, where $\operatorname {\mathrm {Gal}}(\overline {\mathcal {K}}_{\frak q}/\mathcal {K}_{\frak q})$ acts on $\overline R_{\frak p}$ via a corresponding surjection $\operatorname {\mathrm {Gal}}(\overline {\mathcal {K}}_{\frak q}/\mathcal {K}_{\frak q})\to \mathcal {G}_{R}(\frak p)$ . By [Reference Shimizu30, Lem. 6.6], there is a corresponding $\operatorname {\mathrm {Gal}}(\overline {\mathcal {K}}_{\frak q}/\mathcal {K}_{\frak q})$ -equivariant inclusion

$$\begin{align*}{\mathbb B}_{\mathrm{dR}}(A_{\frak p}, A^+_{\frak p})\hookrightarrow {\mathbb B}_{\mathrm{dR}}(\widehat{\mathcal{O}}_{\overline{\mathcal {K}_{\frak q}}[p^{-1}], \widehat{\mathcal{O}}_{\overline{\mathcal{K}}_{\frak q}}}. \end{align*}$$

Hence, (3.7) above gives

$$\begin{align*}{\mathbb B}_{\mathrm{dR}}(A_{\frak p}, A^+_{\frak p})^{\mathcal {G}_R(\frak p)}=F. \end{align*}$$

(cf. [Reference Shimizu30, Lem. 6.10]). The result now follows.

Part ii): If $\lambda =r/s>0$ , with $(r,s)=1$ , write $E_{r/s}$ for the s-dimensional simple isocrystal of slope $r/s$ over $\breve {\mathbb Q}_p$ . Then

$$\begin{align*}{\mathcal BC}(\lambda)(\mathrm{Spd }\ (A_n, A^+_n))=(E_{r/s}\otimes_{\breve{\mathbb Q}_p}{\mathbb B}_{\mathrm{crys}}(A_n, A^+_n))^{\phi=1}=({\mathbb B}_{\mathrm{crys}}(A_n, A^+_n))^{\phi^s=p^r} \end{align*}$$

(cf. [Reference Scholze and Weinstein29, Thm. 15.2.5]). Hence, since by (3.6),

$$\begin{align*}{\mathcal BC}(\lambda)(\operatorname{\mbox{Spa }}(R_n[1/p], R_n)^{\Diamond})\subset ({\mathcal BC}(\lambda)(\mathrm{Spd }\ (A_n, A^{+}_n)))^{\mathcal {G}_{R_{n}}}, \end{align*}$$

we have

$$\begin{align*}{\mathcal BC}(\lambda)(\operatorname{\mbox{Spa }}(R_n[1/p], R_n)^{\Diamond})\subset({\mathbb B}_{\mathrm{crys}}(A_n, A^+_n)^{\mathcal {G}_{R_{n} }})^{\phi^s=p^r}. \end{align*}$$

By part (i), ${\mathbb B}_{\mathrm {crys}}(A_n, A^+_n)^{\mathcal {G}_{R_{n} }}=\breve {\mathbb Q}_p$ . Since $ \breve {\mathbb Q}_p^{\phi ^s=p^r}=(0)$ for $r\neq 0$ , the result follows.

Remark 3.17. In the above argument, we reduce the proof to showing a corresponding result for the complete dvrs $\mathcal {O}_{\mathcal {K}_{\frak q}}$ which have nonperfect residue field. An alternative proof which reduces to the dvrs with perfect residue field $\mathcal {O}_K$ , where K ranges over all finite extensions of F, can be given by considering the collection of all homomorphisms $R_n\to \mathcal {O}_K$ and using [Reference de Jong6, Prop. 7.3.6] to recover the analogue of [Reference Shimizu30, Lem. 6.13].

By covering V by $\operatorname {\mbox {Spa }}(R_n[1/p], R_n)$ , we see that Lemma 3.16 (ii) also implies that ${\mathcal BC}(\lambda )(V^{\Diamond })=(0)$ , for $\lambda>0$ . Since $ \mathcal {U}$ is given as a successive extension of ${\mathcal BC}(\lambda )$ , $\lambda>0$ , we also obtain $ \mathcal {U}(V^{\Diamond })=(0)$ . Now, by (3.5), we have

$$\begin{align*}\tilde G_b(V^{\Diamond})=\underline{J_b({\mathbb Q}_p)}(V^{\Diamond})=C^0(|V^{\Diamond}|, J_b({\mathbb Q}_p))=J_b({\mathbb Q}_p), \end{align*}$$

since $|V^{\Diamond }|=|V|$ is connected. This completes the proof of Proposition 3.15.

We can now complete the proof of the main result, Theorem 3.1: Proposition 3.15 applied to $V=U^\times $ implies

$$\begin{align*}(\iota_r){|_{U^{\times{\Diamond}}}}=j\cdot \iota^\times_r, \end{align*}$$

for some $j\in J_b({\mathbb Q}_p)$ . We can then redefine the framing to be $j^{-1}\cdot \iota _r$ so that $\iota ^\times _r$ extends. This completes the argument for extending the original framing.

We have now extended both the shtuka $ \mathscr {V}^\times = \mathscr {V}(\mathbb {L}^\times )$ and its framing $\iota _r$ , from $U^{\times {\Diamond }}$ to $U^{\Diamond }$ ; this gives an extension of $a^\times : U^{\times {\Diamond }}\to \mathrm {Sht}_K$ to $a: U^{{\Diamond }}\to \mathrm {Sht}_K$ and, by the above, completes the proof.

Proof. By an argument as in §3.3 (A), we can quickly reduce to the case $G=\mathrm {GL}_n$ , with $K=\mathrm {GL}_n( {\mathbb {Z}}_p)$ . By the discussion in the beginning of §3.3 (B1), rigid analytic GAGA, and the rigid analytic Riemann existence theorem of Lütkebohmert, we see that the pro-étale $ {\mathbb {Z}}_p$ -local system $\mathbb {L}=f^*\mathbb {L}^{\mathrm {univ}}$ obtained by pulling back the universal pro-étale $ {\mathbb {Z}}_p$ -local system over $\mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ is indeed defined in the algebraic category, as a pro-étale $ {\mathbb {Z}}_p$ -local system over the algebraic variety S.

Now fix an algebraic closure $\bar F$ of F and a point $\bar s\in S(\bar F)$ . By our assumption, there is a finite field extension $F\subset F'\subset \bar F$ and a finite étale cover $\pi : T\to S\times _FF'$ such that $T\times _{F'}\bar F$ is irreducible, $\bar s$ lifts to an $F'$ -rational point $t'$ of T, and $\pi ^*\mathbb {L}$ is isomorphic to the pull-back of the local system $\mathbb {L}_0=(t')^*\mathbb {L}$ over $\mathrm{Spec }\ (F')$ via the structure morphism $T\to \mathrm{Spec }\ (F')$ . Suppose we can show that the composition

$$\begin{align*}T^{\Diamond}\to (S\times_FF')^{\Diamond}\to \mathrm{Sht}_{(\mathrm{GL}_n, b, \mu, K)} \end{align*}$$

factors through a point $\mathrm{Spd }\ (F')\to \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ over $\mathrm{Spd }\ (\breve E)$ . Then, by applying Lemma 3.10 (a), we see that $(S\times _FF')^{\Diamond }\to \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ factors through $\mathrm{Spd }\ (F')\to \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ . Next, by Lemma 3.10 (b), $f: S^{\Diamond }\to \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ also factors through a point $\mathrm{Spd }\ (F)\to \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}$ and the proof is complete. Hence, by this argument we have reduced the proof to the special case in which S has an F-rational point $s\in S(F)$ , and the local system $\mathbb {L}=f^*\mathbb {L}^{\mathrm {unv}}$ is isomorphic to $g^*\mathbb {L}_0$ , with $\mathbb {L}_0=s^*\mathbb {L}$ , where $g:S\to \mathrm{Spec }\ (F)$ is the structure morphism. We will assume we are in this situation from now on.

By Proposition 2.1, $\mathbb {L}_0$ and $\mathbb {L}$ are horizontal de Rham local systems. In fact, the proof of Proposition 2.1 (ii) shows that the analytic bundle $(D_{\mathrm {dR}}(\mathbb {L}), \nabla )$ with connection over $S^{\mathrm { an}}$ associated to $\mathbb {L}$ by the p-adic Riemann-Hilbert functor admits a trivialization, that is, an isomorphism

$$\begin{align*}\alpha: (V\otimes_{{\mathbb Q}_p}\mathcal{O}_{S^{\mathrm{an}}}, 1\otimes d)\xrightarrow{\sim} (D_{\mathrm{dR}}(\mathbb {L}), \nabla) \end{align*}$$

which is constructed by using the framing of the corresponding shtuka. There is a similar isomorphism $\alpha _0: V\otimes _{{\mathbb Q}_p}F \xrightarrow {\sim } D_{\mathrm {dR}}(\mathbb {L}_0)$ over the point ${\mathrm {Sp}}(F)$ . We also have an isomorphism $c: D_{\mathrm { dR}}(\mathbb {L})\xrightarrow {\sim } D_{\mathrm {dR}}(g^*\mathbb {L}_0) \xrightarrow {\sim } g^*D_{\mathrm {dR}}(\mathbb {L}_0) $ respecting the connections. The composition $g^*(\alpha _0)^{-1}\circ c \circ \alpha $ is given by a global section A of $\mathrm { Aut}(V\otimes _{{\mathbb Q}_p}\mathcal {O}_{S^{\mathrm {an}}})\simeq \mathrm {GL}_n(\mathcal {O}_{S^{\mathrm {an}}})$ whose matrix entries are functions with zero differential, hence A belongs to $\mathrm {Aut}(V\otimes _{{\mathbb Q}_p}F)=\mathrm {GL}_n(F)$ .

Consider now the composition

$$\begin{align*}S^{\Diamond}\xrightarrow{ \ f \ } \mathrm{Sht}_{(\mathrm{GL}_n, b, \mu, K)}\xrightarrow{\pi_{\mathrm{GM}}} \mathrm{Gr}_{\mathrm{GL}_n, \leq \mu}. \end{align*}$$

By the definition of the period morphism $\pi _{\mathrm {GM}}$ , the value of $\pi _{\mathrm {GM}}\circ f$ at a $\operatorname {\mbox {Spa }}(R, R^+)$ -point of $S^{\Diamond }$ , is the point of $\mathrm {Gr}_{\mathrm {GL}_n,\leq \mu }$ obtained as follows: Recall the ${\mathbb B}_{\mathrm {dR}}^+(R^\sharp , R^{\sharp +})$ -lattices $\mathbb {M}$ and $\mathbb {M}_0$ associated to the shtuka, as in the proof of Proposition 2.1. We use the framing of the shtuka to obtain a trivialization of $\mathbb {M}_0$

$$\begin{align*}\beta: V\otimes_{{\mathbb Q}_p} {\mathbb B}_{\mathrm{dR}}^+(R^\sharp, R^{\sharp+})={\mathbb B}_{\mathrm{dR}}^+(R^\sharp, R^{\sharp+})^n\xrightarrow{\sim} \mathbb {M}_0 \end{align*}$$

and then consider the ${\mathbb B}_{\mathrm {dR}}^+(R^\sharp , R^{\sharp +})$ -lattice

$$\begin{align*}\beta^{-1}(\mathbb {M})\subset {\mathbb B}_{\mathrm{dR}}(R^\sharp, R^{\sharp+})^n; \end{align*}$$

this defines a point of $\mathrm {Gr}_{\mathrm {GL}_n, \leq \mu }$ .

Recall that, by [Reference Scholze24, §7], we have $\mathbb {M}_0=(\mathcal {E}\otimes _{\mathcal {O}_{S^{\mathrm { an}}}}\mathcal {O}{\mathbb B}_{\mathrm {dR}}^+)^{\nabla =0}$ , with $\mathcal {E}=\mathrm {D}_{\mathrm {dR}}(\mathbb {L})$ . We now observe that the trivialization $\beta $ of $\mathbb {M}_0$ is induced by the (horizontal) trivialization $\alpha $ of $\mathcal {E}=\mathrm {D}_{\mathrm {dR}}(\mathbb {L})$ above; similarly for the local system $\mathbb {L}_0$ over $\mathrm{Spd }\ (F)$ . Hence, it follows that the composition

$$\begin{align*}S^{\Diamond}\xrightarrow{ \ f \ } \mathrm{Sht}_{(\mathrm{GL}_n, b, \mu, K)}\xrightarrow{\pi_{\mathrm{GM}}} \mathrm{Gr}_{\mathrm{GL}_n, \leq \mu}\times_{\mathrm{Spd }\ ({\mathbb Q}_p)}\mathrm{Spd }\ (F) \end{align*}$$

is equal to $\mathrm{Spd }\ (F)^{\Diamond } \xrightarrow { \ f\circ s\ } \mathrm {Sht}_{(G, b, \mu , K)}\xrightarrow {\pi _{\mathrm {GM}}} \mathrm { Gr}_{\mathrm {GL}_n, \leq \mu }\times _{\mathrm{Spd }\ ({\mathbb Q}_p)}\mathrm{Spd }\ (F)$ composed with the action of $A\in \mathrm {GL}_n(F)$ . Hence, $\pi _{\mathrm {GM}}\circ f$ factors through a point $S^{\Diamond }\to \mathrm{Spd }\ (F)\to \mathrm {Gr}_{\mathrm {GL}_n, \leq \mu }$ . Since $\pi _{\mathrm {GM}}: \mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}\to \mathrm {Gr}_{\mathrm {GL}_n, \leq \mu }$ is étale and S is geometrically connected, $|f|: |S|\to |\mathrm {Sht}_{(\mathrm {GL}_n, b, \mu , K)}|$ also factors through a closed point and we can now conclude the proof as above.