2.1.1

As in [Reference Rapoport and ZinkRZ96], we will denote by $\text{Nilp}_{W}$ the category of $W$ -schemes $S$ such that $p$ is Zariski locally nilpotent in ${\mathcal{O}}_{S}$ . Denote by

$$\begin{eqnarray}\text{ANilp}_{W}\subset \text{Nilp}_{W}^{\mathit{op}}\end{eqnarray}$$

the full subcategory of Noetherian $W$ -algebras in which $p$ is nilpotent. We denote by $\text{ANilp}_{W}^{\text{f}}$ the category of Noetherian adic $W$ -algebras in which $p$ is nilpotent, and embed

$$\begin{eqnarray}\text{ANilp}_{W}\subset \text{ANilp}_{W}^{\text{f}}\end{eqnarray}$$

as a full subcategory by endowing any $W$ -algebra in $\text{ANilp}_{W}$ with its $p$ -adic topology.

We say that an adic $W$ -algebra $A$ is formally finitely generated if $A$ is Noetherian, and if $A/I$ is a finitely generated $W$ -algebra for some ideal of definition $I\subset A$ . Thus $\text{Spf}(A)$ is a formal scheme which is formally of finite type over $\text{Spf}(W)$ . If, in addition, $p$ is nilpotent in $A$ , then $A$ is a quotient of $W/(p^{n})[\![x_{1},\ldots ,x_{r}]\!][y_{1},\ldots ,y_{s}]$ for some $n$ , $r$ , and  $s$ .

We will denote by

$$\begin{eqnarray}\text{ANilp}_{W}^{\text{fsm}}\subset \text{ANilp}_{W}^{\text{f}}\end{eqnarray}$$

the full subcategory whose objects are $W$ -algebras that are formally finitely generated and formally smooth over $W/(p^{n})$ , for some $n\geqslant 1$ .

2.1.2

As in [Reference Rapoport and ZinkRZ96, § 2.1], every formal scheme $\mathfrak{X}$ over $\text{Spf}(W)$ defines a functor on $\text{Nilp}_{W}$ . We restrict this functor to $\text{ANilp}_{W}$ , and then extend to $\text{ANilp}_{W}^{\text{f}}$ as follows: for $A$ in $\text{ANilp}_{W}^{\text{f}}$ with ideal of definition $I$ , define $\mathfrak{X}(A)$ to be the set

$$\begin{eqnarray}\mathfrak{X}(A)=\text{Hom}_{\text{Spf}(W)}(\text{Spf}(A),\mathfrak{X})=\mathop{\varprojlim }\nolimits_{n}\mathfrak{X}(A/I^{n}).\end{eqnarray}$$

2.1.3

If $R$ is an object of $\text{ANilp}_{W}^{\text{fsm}}$ , the quotient $\bar{R}=R/pR$ satisfies the condition [Reference de JongdeJ95, (1.3.1.1)]. Thus, by [Reference de JongdeJ95, Lemma 1.3.3], $\bar{R}$ admits a divided power (PD) thickening

$$\begin{eqnarray}\widetilde{R}\rightarrow \widetilde{R}/p\widetilde{R}=\bar{R}\end{eqnarray}$$

by a formally smooth, $p$ -adically complete $W$ -algebra $\widetilde{R}$ , unique up to non-canonical isomorphism. The absolute Frobenius on $\bar{R}$ lifts to $\widetilde{R}$ . The formal smoothness of $R$ over $W/p^{n}W$ implies that $R\xrightarrow[{}]{{\sim}}\widetilde{R}/p^{n}\widetilde{R}$ , and hence $\widetilde{R}$ also provides a PD thickening $\widetilde{R}\rightarrow R$ . One can show that $\widetilde{R}$ is a quotient of a $W$ -algebra of the form

$$\begin{eqnarray}W[\![x_{1},\ldots ,x_{r}]\!]\{y_{1},\ldots ,y_{s}\}=\mathop{\varprojlim }\nolimits_{n}W/(p^{n})[\![x_{1},\ldots ,x_{r}]\!][y_{1},\ldots ,y_{s}],\end{eqnarray}$$

and hence is Noetherian.

2.1.4

Suppose that $R$ is any $k$ -algebra admitting a $p$ -basis in the sense of [Reference Berthelot and MessingBM90, § 1.1]. An explicit construction of a PD thickening $\widetilde{R}\rightarrow R$ is then explained in [Reference Berthelot and MessingBM90]. This applies in particular when $R=k^{\prime }$ is any field extension of $k$ , in which case $\widetilde{R}$ is isomorphic to the Cohen ring $W^{\prime }$ of  $k^{\prime }$ .

Recall that the Cohen ring $W^{\prime }$ is the unique, up to non-canonical isomorphism, discrete valuation ring with $k^{\prime }$ as a residue field and $p$ as uniformizer. It is flat over the Witt ring $W$ of $k$ . If $(x_{i})_{i}$ is a $p$ -basis of $k^{\prime }$ , then a choice of elements $y_{i}\in W^{\prime }$ with $x_{i}^{p}\equiv y_{i}~(\text{mod}~pW^{\prime })$ determines a lift $\unicode[STIX]{x1D70E}:W^{\prime }\rightarrow W^{\prime }$ of the absolute Frobenius $k^{\prime }\rightarrow k^{\prime }$ . Set $K^{\prime }=W^{\prime }[1/p]$ .

2.1.5

Continue with the above notation, and fix a lift $\unicode[STIX]{x1D70E}:W^{\prime }\rightarrow W^{\prime }$ of the Frobenius of the field $k^{\prime }$ . The triple $(W^{\prime },pW^{\prime },k^{\prime })$ gives a frame for $k^{\prime }$ in the sense of [Reference ZinkZin01]. As in [Reference ZinkZin01, Definition 2], a Dieudonné $W^{\prime }$ -window over $k^{\prime }$ consists of a triple $(M,M_{1},F),$ in which

1. – $M$ is a free finitely generated $W^{\prime }$ -module;

2. – $M_{1}\subset M$ is a $W^{\prime }$ -submodule such that $pM\subset M_{1}\subset M$ ;

3. – $F:M\rightarrow M$ is a $\unicode[STIX]{x1D70E}$ -semi-linear map such that $F(M_{1})\subset pM$ , and $p^{-1}F(M_{1})$ generates $M$ as an $W^{\prime }$ -module.

These conditions imply $F(M)\subset p^{-1}F(M_{1})$ , and so are equivalent to the conditions appearing in [Reference ZinkZin01, Definition 2]. If $(M,M_{1},F)$ is a Dieudonné $W^{\prime }$ -window then

$$\begin{eqnarray}M_{1}=F^{-1}(pM)=\{x\in M[1/p]:F(x)\in pM\}.\end{eqnarray}$$

If $M$ is a free finitely generated $W^{\prime }$ -module, and $F:M[1/p]\rightarrow M[1/p]$ is a $\unicode[STIX]{x1D70E}$ -semi-linear map such that

1. – $pM\subset F^{-1}(pM)\subset M$ , and

2. – $F(F^{-1}(M))$ generates $M$ as a $W^{\prime }$ -module,

then $F(M)\subset M$ and $(M,F^{-1}(pM),F)$ is a Dieudonné $W^{\prime }$ -window.

A Dieudonné $W^{\prime }$ -window is called simply a $W^{\prime }$ -window when the additional nilpotence condition of [Reference ZinkZin01, Definition 3] is satisfied.

2.1.6

Let $S$ be a scheme such that $p$ is Zariski locally nilpotent in ${\mathcal{O}}_{S}$ . Set

$$\begin{eqnarray}\bar{S}=S\otimes _{\mathbb{Z}_{p}}\mathbb{F}_{p},\end{eqnarray}$$

and denote by $\unicode[STIX]{x1D70E}:\bar{S}\rightarrow \bar{S}$ the absolute Frobenius morphism.

For a $p$ -divisible group $X$ over $S$ , we will denote by $\mathbb{D}(X)$ its contravariant Dieudonné crystal. It is a crystal of locally free ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -modules of rank equal to the height $h(X)$ of $X$ . We refer the reader to [Reference MessingMes72, Reference Berthelot, Breen and MessingBBM82, Reference de JongdeJ95] for background on the construction and properties of the Dieudonné crystal.

The crystal $\mathbb{D}(X)$ is equipped with the Hodge filtration

$$\begin{eqnarray}\text{Fil}^{1}(X)=\text{Lie}(X)^{\ast }\subset \mathbb{D}(X)_{S},\end{eqnarray}$$

where $\mathbb{D}(X)_{S}$ is the pull-back of $\mathbb{D}(X)$ to the Zariski site of $S$ ; it is a locally free ${\mathcal{O}}_{S}$ -module of rank $h(X)$ , and the ${\mathcal{O}}_{S}$ -submodule $\text{Fil}^{1}(X)$ is locally a direct summand. We also have the Frobenius morphism

(2.1.6.1) $$\begin{eqnarray}F:\unicode[STIX]{x1D70E}^{\ast }\mathbb{D}(X)\rightarrow \mathbb{D}(X),\end{eqnarray}$$

where the pull-back $\unicode[STIX]{x1D70E}^{\ast }\mathbb{D}(X)$ is defined as in the above references.

Define crystals

$$\begin{eqnarray}\mathbf{1}=\mathbb{D}(\mathbb{Q}_{p}/\mathbb{Z}_{p}),\quad \mathbf{1}(-1)=\mathbb{D}(\unicode[STIX]{x1D707}_{p^{\infty }}\!),\end{eqnarray}$$

and note that $\mathbf{1}$ is the structure sheaf ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ with the usual Frobenius structure and $\text{Fil}^{1}=(0)$ . We often confuse a global section $t$ of a crystal $\mathbb{D}$ with the corresponding morphism of crystals $t:\mathbf{1}\rightarrow \mathbb{D}$ .

We define $\mathbb{D}(X)^{\ast }$ to be the ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -linear dual with the dual filtration. Note that $\mathbb{D}(X^{\vee })^{\ast }=\mathbb{D}(X)(-1)$ , where $X^{\vee }$ is the dual $p$ -divisible group. There is a Frobenius structure on $\mathbb{D}(X)^{\ast }$ as in (2.1.6.1) but it is defined only ‘up to isogeny’, i.e. only after we view $\mathbb{D}(X)^{\ast }$ as an isocrystal as below.

We define the category of isocrystals over $S$ as follows.

1. – Objects are crystals $\mathbb{D}$ of locally free ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -modules. We write $\mathbb{D}[1/p]$ if we view $\mathbb{D}$ as an isocrystal.

2. – Morphisms $\mathbb{D}[1/p]\rightarrow \mathbb{D}^{\prime }[1/p]$ are given by global sections of the Zariski sheaf $\text{}\underline{\text{Hom}}(\mathbb{D},\mathbb{D}^{\prime })[1/p]$ over $S$ , where $\text{Hom}(\mathbb{D},\mathbb{D}^{\prime })$ is taken in the category of crystals of locally free ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -modules.

Every quasi-isogeny $\unicode[STIX]{x1D70C}:X{\dashrightarrow}X^{\prime }$ of $p$ -divisible groups over $S$ , in the sense of [Reference Rapoport and ZinkRZ96, Definition 2.8], induces an isomorphism of isocrystals

(2.1.6.2) $$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D70C}):\mathbb{D}(X^{\prime })[1/p]\xrightarrow[{}]{{\sim}}\mathbb{D}(X)[1/p].\end{eqnarray}$$

The total tensor algebra $\mathbb{D}(X)^{\otimes }$ is defined as the direct sum of all the crystals of locally free ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -modules which can be formed from $\mathbb{D}(X)$ using the operations of taking duals, tensor products, symmetric powers and exterior powers. It is a crystal of locally free ${\mathcal{O}}_{S}/\mathbb{Z}_{p}$ -modules over $S$ . The Hodge filtration on $\mathbb{D}(X)_{S}$ induces a natural filtration $\text{Fil}^{\bullet }(\mathbb{D}(X)_{S}^{\otimes })$ on $\mathbb{D}(X)_{S}^{\otimes }$ , and the Frobenius morphism (2.1.6.1) induces an isomorphism of isocrystals

$$\begin{eqnarray}F:\unicode[STIX]{x1D70E}^{\ast }\mathbb{D}(X)^{\otimes }[1/p]\xrightarrow[{}]{{\sim}}\mathbb{D}(X)^{\otimes }[1/p].\end{eqnarray}$$

For any quasi-isogeny $\unicode[STIX]{x1D70C}:X{\dashrightarrow}X^{\prime }$ of $p$ -divisible groups over $S$ , the isomorphism (2.1.6.2) extends to

$$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D70C}):\mathbb{D}(X^{\prime })^{\otimes }[1/p]\xrightarrow[{}]{{\sim}}\mathbb{D}(X)^{\otimes }[1/p].\end{eqnarray}$$

A similar discussion applies to formal schemes $S$ over $\text{Spf}(\mathbb{Z}_{p})$ , as in [Reference de JongdeJ95, ch. 2].

2.1.7

Suppose $X$ is a formal $p$ -divisible group over a field $k^{\prime }$ of characteristic $p$ . Again letting $W^{\prime }$ be the Cohen ring of $k^{\prime }$ , the evaluation

$$\begin{eqnarray}\mathbb{D}(X)(W^{\prime })=\mathop{\varprojlim }\nolimits_{n}\mathbb{D}(X)(W^{\prime }/p^{n}W^{\prime })\end{eqnarray}$$

of the crystal $\mathbb{D}(X)$ on $W^{\prime }$ has a natural structure of a $W^{\prime }$ -window over $k^{\prime }$ . (Combine the proof of [Reference ZinkZin01, Theorem 1.6] with [Reference ZinkZin02, Theorem 6].)

By [Reference ZinkZin01, Theorem 4], the functor $X\mapsto \mathbb{D}(X)(W^{\prime })$ gives an anti-equivalence of categories between formal $p$ -divisible groups over $k^{\prime }$ and $W^{\prime }$ -windows over $k^{\prime }$ . More precisely, the equivalence of [Reference ZinkZin01] uses the covariant Dieudonné crystal, and we compose the functor defined there with Cartier duality.

If $k^{\prime }$ is perfect, classical Dieudonné theory (or [Reference ZinkZin01, Theorem 3.2]) gives in the same way an anti-equivalence between (all) $p$ -divisible groups over $k^{\prime }$ and Dieudonné modules over $W^{\prime }=W(k^{\prime })$ .

2.2.1

Let $([b],\{\unicode[STIX]{x1D707}\})$ be a pair consisting of:

1. – a $G(\bar{K})$ -conjugacy class $\{\unicode[STIX]{x1D707}\}$ of cocharacters $\unicode[STIX]{x1D707}:\mathbb{G}_{m\bar{K}}\rightarrow G_{\bar{K}}$ ;

2. – a $\unicode[STIX]{x1D70E}$ -conjugacy class $[b]$ of elements $b\in G(K)$ .

Here $b$ and $b^{\prime }$ are $\unicode[STIX]{x1D70E}$ -conjugate if there is $g\in G(K)$ with $b^{\prime }=gb\unicode[STIX]{x1D70E}(g)^{-1}$ .

We let $E\subset \bar{K}$ be the field of definition of the conjugacy class $\{\unicode[STIX]{x1D707}\}$ . This is the local reflex field. Denote by ${\mathcal{O}}_{E}$ its valuation ring and by $k_{E}$ its (finite) residue field. In fact, under our assumption on $G$ , the field $E\subset \bar{K}$ is contained in $K$ and there is a cocharacter $\unicode[STIX]{x1D707}:\mathbb{G}_{mE}\rightarrow G_{E}$ in the conjugacy class $\{\unicode[STIX]{x1D707}\}$ that is defined over $E$ ; see [Reference KottwitzKot84, Lemma (1.1.3)]. In fact, we can find a representative $\unicode[STIX]{x1D707}$ that extends to an integral cocharacter

(2.2.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D707}:\mathbb{G}_{m{\mathcal{O}}_{E}}\rightarrow G_{{\mathcal{O}}_{E}},\end{eqnarray}$$

and the $G({\mathcal{O}}_{E})$ -conjugacy class of such an $\unicode[STIX]{x1D707}$ is well defined. In what follows, we usually assume that $\unicode[STIX]{x1D707}$ is such a representative.

To the conjugacy class $\{\unicode[STIX]{x1D707}\}$ we associate the homogeneous space

$$\begin{eqnarray}M_{G,\unicode[STIX]{x1D707}}=G_{{\mathcal{O}}_{E}}/P_{\unicode[STIX]{x1D707}}\end{eqnarray}$$

over ${\mathcal{O}}_{E}$ , in which $P_{\unicode[STIX]{x1D707}}\subset G_{{\mathcal{O}}_{E}}$ is the parabolic subgroup defined by $\unicode[STIX]{x1D707}$ . More precisely, $P_{\unicode[STIX]{x1D707}}$ is the parabolic subgroup such that $P_{\unicode[STIX]{x1D707}}\times _{{\mathcal{O}}_{E}}W$ contains exactly the root groups $U_{a}$ of the split group $G_{W}$ , for all roots $a$ with $a\cdot \unicode[STIX]{x1D707}\geqslant 0$ . The group $P_{\unicode[STIX]{x1D707}}\times _{{\mathcal{O}}_{E}}W$ stabilizes the filtration defined by $\unicode[STIX]{x1D707}$ in any representation of $G_{W}$ .

We write $\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D707})$ for the Frobenius conjugate of (2.2.1.1).

By [Reference Rapoport and RichartzRR96, Theorem 4.2], assumptions (i) and (ii) imply that $[b]$ lies in the set $B(G_{\mathbb{Q}_{p}},\{\unicode[STIX]{x1D707}\})$ of neutral acceptable elements for $\{\unicode[STIX]{x1D707}\}$ ; see [Reference Rapoport and ViehmannRV14, Definition 2.3]. In particular, $(G_{\mathbb{Q}_{p}},[b],\{\unicode[STIX]{x1D707}\})$ is a local Shimura datum in the sense of [Reference Rapoport and ViehmannRV14, Definition 5.1].

Definition 2.2.3. The local unramified Shimura datum $(G,[b],\{\unicode[STIX]{x1D707}\})$ is of Hodge type if there exists a closed group scheme embedding $\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GL}(C)$ , for a free $\mathbb{Z}_{p}$ -module $C$ of finite rank, with the following properties: the central torus $\mathbb{G}_{m}\subset \text{GL}(C)$ is contained in $G$ , and, after a choice of basis $C_{{\mathcal{O}}_{E}}\xrightarrow[{}]{{\sim}}{\mathcal{O}}_{E}^{n}$ , the composite cocharacter

$$\begin{eqnarray}\unicode[STIX]{x1D704}\circ \unicode[STIX]{x1D707}:\mathbb{G}_{m{\mathcal{O}}_{E}}\rightarrow \text{GL}_{n,{\mathcal{O}}_{E}}\end{eqnarray}$$

is the inverse of the minuscule cocharacterFootnote ¹

$$\begin{eqnarray}a\mapsto \text{diag}(a^{(r)},1^{(n-r)})\end{eqnarray}$$

for some $1\leqslant r<n$ .

By definition, there is a surjection $(G,b,\unicode[STIX]{x1D707},C)\mapsto (G,[b],\{\unicode[STIX]{x1D707}\})$ from the set of local unramified Shimura–Hodge data to the set of local unramified Shimura data of Hodge type.

Fix a local unramified Shimura–Hodge datum $(G,b,\unicode[STIX]{x1D707},C)$ , and set $D=\text{Hom}_{\mathbb{Z}_{p}}(C,\mathbb{Z}_{p})$ with the contragredient action of  $G$ .

Lemma 2.2.5. Up to isomorphism, there is a unique $p$ -divisible group

$$\begin{eqnarray}X_{0}=X_{0}(G,b,\unicode[STIX]{x1D707},C)\end{eqnarray}$$

over $k$ whose contravariant Dieudonné module is $\mathbb{D}(X_{0})(W)=D_{W}$ with Frobenius $F=b\circ \unicode[STIX]{x1D70E}$ . Moreover, the Hodge filtration

$$\begin{eqnarray}VD_{k}\subset D_{k}=\mathbb{D}(X_{0})(k)\end{eqnarray}$$

is induced by a conjugate of the reduction $\unicode[STIX]{x1D707}_{k}:\mathbb{G}_{mk}\rightarrow G_{k}$ .

Proof. By our assumption on $\unicode[STIX]{x1D707}$ in Definition 2.2.3, we have $\unicode[STIX]{x1D707}(p)D_{W}\subset D_{W}$ . Therefore, by (2.2.2.1), the lattice $D_{W}\subset D_{W}[1/p]$ is $F$ -stable. To determine $VD_{W}$ , write $b=h^{\prime }\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)h$ with $h,h^{\prime }\in G(W)$ , so that

$$\begin{eqnarray}\displaystyle VD_{W}=pF^{-1}D_{W} & = & \displaystyle p\unicode[STIX]{x1D70E}^{-1}(b^{-1}D_{W})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70E}^{-1}(h^{-1}p\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)^{-1}{h^{\prime }}^{-1}D_{W})\nonumber\\ \displaystyle & = & \displaystyle h_{1}p\unicode[STIX]{x1D707}(p)^{-1}h_{1}^{-1}D_{W}\nonumber\end{eqnarray}$$

for $h_{1}=\unicode[STIX]{x1D70E}^{-1}(h^{-1})\in G(W)$ . Observe that $p\unicode[STIX]{x1D707}(p)^{-1}D_{W}\subset D_{W}$ , and in fact the filtration

$$\begin{eqnarray}(p\unicode[STIX]{x1D707}(p)^{-1}D_{W})/pD_{W}\subset D_{W}/pD_{W}=D_{k}\end{eqnarray}$$

is induced by $\unicode[STIX]{x1D707}_{k}:\mathbb{G}_{mk}\rightarrow G_{k}$ .

The above calculation shows that $VD_{W}\subset D_{W}$ , and that the Hodge filtration $VD_{k}\subset D_{k}$ is induced by the conjugate $\bar{h}_{1}\unicode[STIX]{x1D707}_{k}\bar{h}_{1}^{-1}$ .◻

2.2.6

By [Reference KisinKis10, Proposition (1.3.2)], there is finite list $(s_{\unicode[STIX]{x1D6FC}})$ of tensors $s_{\unicode[STIX]{x1D6FC}}$ in the total tensor algebra $C^{\otimes }$ that ‘cut out’ the group $G$ , in the sense that

$$\begin{eqnarray}G(R)=\{g\in \text{GL}(C\otimes _{\mathbb{Z}_{p}}R):g\cdot (s_{\unicode[STIX]{x1D6FC}}\otimes 1)=(s_{\unicode[STIX]{x1D6FC}}\otimes 1),\forall \unicode[STIX]{x1D6FC}\}\end{eqnarray}$$

for all $\mathbb{Z}_{p}$ -algebras $R$ . Using the canonical isomorphism $C^{\otimes }=D^{\otimes }$ , the tensors $s_{\unicode[STIX]{x1D6FC}}\in C^{\otimes }$ determine tensors $s_{\unicode[STIX]{x1D6FC}}\otimes 1\in D^{\otimes }\otimes _{\mathbb{Z}_{p}}W$ . Thus, if $X_{0}$ is the $p$ -divisible group of Lemma 2.2.5, we obtain tensors

$$\begin{eqnarray}t_{\unicode[STIX]{x1D6FC},0}=s_{\unicode[STIX]{x1D6FC}}\otimes 1\in D^{\otimes }\otimes _{\mathbb{ Z}_{p}}W=\mathbb{D}(X_{0})(W)^{\otimes },\end{eqnarray}$$

which are Frobenius invariant when viewed in $\mathbb{D}(X_{0})(W)^{\otimes }[1/p]$ . The tensors $t_{\unicode[STIX]{x1D6FC},0}$ uniquely determine morphisms of crystals $t_{\unicode[STIX]{x1D6FC},0}:\mathbf{1}\rightarrow \mathbb{D}(X_{0})^{\otimes }$ over $\text{Spec}(k)$ , such that each

$$\begin{eqnarray}t_{\unicode[STIX]{x1D6FC},0}:\mathbf{1}[1/p]\rightarrow \mathbb{D}(X_{0})[1/p]^{\otimes }\end{eqnarray}$$

is Frobenius equivariant.Footnote ² Here, as before, we denote by $\mathbf{1}=\mathbb{D}(\mathbb{Q}_{p}/\mathbb{Z}_{p})$ the crystal determined by the Dieudonné module $W$ with $F=\unicode[STIX]{x1D70E}$ . Using Lemma 2.2.5, we easily see that $t_{\unicode[STIX]{x1D6FC},0}(k)\in \text{Fil}^{0}(\mathbb{D}(X_{0})(k)^{\otimes })$ .

2.2.7

By [Reference KottwitzKot85], every $\unicode[STIX]{x1D70E}$ -conjugacy class in $G(K)$ is decent in the sense of [Reference Rapoport and ZinkRZ96, Definition 1.8]. By [Reference Rapoport and ZinkRZ96, Proposition 1.12], any $b\in G(K)$ determines a smooth affine group scheme $J_{b}$ over $\mathbb{Q}_{p}$ with functor of points

$$\begin{eqnarray}J_{b}(R)=\{g\in G(R\otimes _{\mathbb{Q}_{p}}K):gb\unicode[STIX]{x1D70E}(g)^{-1}=b\}\end{eqnarray}$$

for any $\mathbb{Q}_{p}$ -algebra $R$ . Up to isomorphism, $J_{b}$ depends only on the $\unicode[STIX]{x1D70E}$ -conjugacy class  $[b]$ .

2.2.8

Let $\mathbb{T}$ be the pro-torus over $\mathbb{Z}_{p}$ with character group $\mathbb{Q}$ . For any $\mathbb{Z}_{p}$ -algebra $R$ , an $R$ -point $z\in \mathbb{T}(R)$ consists of a tuple

$$\begin{eqnarray}z=(z_{m}\in R^{\times })_{m\in \mathbb{Z}_{{>}0}}\end{eqnarray}$$

such that $z_{m}=z_{md}^{d}$ for all positive $m$ and $d$ . The character indexed by the rational number $s/t$ sends $z\mapsto z_{t}^{s}$ .

Kottwitz [Reference KottwitzKot85] attaches to every $b\in G(K)$ a slope cocharacter

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{b}:\mathbb{T}_{K}\rightarrow G_{K}\end{eqnarray}$$

such that for any representation $\unicode[STIX]{x1D713}:G_{\mathbb{Q}_{p}}\rightarrow \text{GL}(M)$ on a $\mathbb{Q}_{p}$ -vector space $M$ , the decomposition

$$\begin{eqnarray}M_{K}=\bigoplus _{s/t\in \mathbb{Q}}M_{K}^{s/t}\end{eqnarray}$$

of $M_{K}$ determined by the cocharacter $\unicode[STIX]{x1D713}\circ \unicode[STIX]{x1D708}_{b}:\mathbb{T}_{K}\rightarrow \text{GL}(M_{K})$ agrees with the slope decomposition of the isocrystal $(M_{K},\unicode[STIX]{x1D713}(b)\circ \unicode[STIX]{x1D70E})$ . The slope cocharacter depends only on the $\unicode[STIX]{x1D70E}$ -conjugacy class  $[b]$ .

An element $b\in G(K)$ is basic if its slope cocharacter $\unicode[STIX]{x1D708}_{b}$ factors through the center of $G_{K}$ . By [Reference KottwitzKot85], $b$ is basic if and only if the group $J_{b}$ is an inner form of  $G$ .

2.3.1

Suppose that $X_{0}$ is any $p$ -divisible group over $k$ . The Rapoport–Zink space $\text{RZ}(X_{0})$ of deformations of $X_{0}$ up to quasi-isogeny is, as in [Reference Rapoport and ZinkRZ96], the formal scheme over $\text{Spf}(W)$ that represents the functor assigning to each scheme $S$ in $\text{Nilp}_{W}$ the set of isomorphism classes of pairs $(X,\unicode[STIX]{x1D70C})$ in which

1. – $X$ is a $p$ -divisible group over $S$ ;

2. – $\unicode[STIX]{x1D70C}:X_{0}\times _{k}\bar{S}{\dashrightarrow}X\times _{S}\bar{S}$ is a quasi-isogeny, where $\bar{S}=S\otimes _{W}k$ .

Suppose now that $X_{0}$ comes with a principal polarization $\unicode[STIX]{x1D706}_{0}:X_{0}\xrightarrow[{}]{{\sim}}X_{0}^{\vee }$ . The symplectic Rapoport–Zink space $\text{RZ}(X_{0},\unicode[STIX]{x1D706}_{0})$ is the formal scheme over $\text{Spf}(W)$ that represents the functor that assigns to each $S$ in $\text{Nilp}_{W}$ the set of isomorphism classes of triples $(X,\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})$ in which

1. – $X$ is a $p$ -divisible group over $S$ ;

2. – $\unicode[STIX]{x1D706}:X\xrightarrow[{}]{{\sim}}X^{\vee }$ is a principal polarization;

3. – $\unicode[STIX]{x1D70C}:X_{0}\times _{k}\bar{S}{\dashrightarrow}X\times _{S}\bar{S}$ is a quasi-isogeny that respects polarizations up to a scalar, in the sense that, Zariski locally on $\bar{S}$ , we have

   $$\begin{eqnarray}\unicode[STIX]{x1D70C}^{\vee }\circ \unicode[STIX]{x1D706}\circ \unicode[STIX]{x1D70C}=c^{-1}(\unicode[STIX]{x1D70C})\cdot \unicode[STIX]{x1D706}_{0},\end{eqnarray}$$
   for some $c(\unicode[STIX]{x1D70C})\in \mathbb{Q}_{p}^{\times }$ .

By [Reference Rapoport and ZinkRZ96] the formal schemes $\text{RZ}(X_{0})$ and $\text{RZ}(X_{0},\unicode[STIX]{x1D706}_{0})$ are formally smooth and locally formally of finite type over $W$ , and forgetting the polarization defines a closed immersion $\text{RZ}(X_{0},\unicode[STIX]{x1D706}_{0})\rightarrow \text{RZ}(X_{0})$ .

2.3.2

Suppose that $(G,b,\unicode[STIX]{x1D707},C)$ is a local unramified Shimura–Hodge datum. Choose tensors $(s_{\unicode[STIX]{x1D6FC}})$ that cut out $G$ as in § 2.2.6. Denote by

$$\begin{eqnarray}X_{0}=X_{0}(G,b,\unicode[STIX]{x1D707},C)\end{eqnarray}$$

the corresponding $p$ -divisible group over $k$ of Lemma 2.2.5, with its Frobenius invariant crystalline tensors $(t_{\unicode[STIX]{x1D6FC},0})$ .

Definition 2.3.3. Consider the functor

$$\begin{eqnarray}\text{RZ}_{G}^{\text{nilp}}=\text{RZ}_{G,b,\unicode[STIX]{x1D707},C,(s_{\unicode[STIX]{x1D6FC}})}^{\text{nilp}}:\text{ANilp}_{W}\rightarrow \text{Sets}\end{eqnarray}$$

that assigns to each $R\in \text{ANilp}_{W}$ the set of isomorphism classes of triples $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))$ in which

1. – $(X,\unicode[STIX]{x1D70C})$ consists of a $p$ -divisible group over $\text{Spec}(R)$ and a quasi-isogeny

   $$\begin{eqnarray}\unicode[STIX]{x1D70C}:X_{0}\otimes _{k}\bar{R}{\dashrightarrow}X\otimes _{R}\bar{R},\end{eqnarray}$$
   with $\bar{R}=R/pR$ , as in the definition of the Rapoport–Zink formal scheme $\text{RZ}(X_{0})$ ;

2. – the collection $(t_{\unicode[STIX]{x1D6FC}})$ consists of morphisms of crystals $t_{\unicode[STIX]{x1D6FC}}:\mathbf{1}\rightarrow \mathbb{D}(X)^{\otimes }$ over $\text{Spec}(R)$ with $t_{\unicode[STIX]{x1D6FC}}:\mathbf{1}[1/p]\rightarrow \mathbb{D}(X)^{\otimes }[1/p]$ Frobenius equivariant,

satisfying the following properties.

1. (i) For some nilpotent ideal $J\subset R$ with $p\in J$ , the pull-back of $t_{\unicode[STIX]{x1D6FC}}$ over $\text{Spec}(R/J)$ is identified with $t_{\unicode[STIX]{x1D6FC},0}$ under the isomorphism of isocrystals

   $$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D70C}):\mathbb{D}(X_{R/J})^{\otimes }[1/p]\xrightarrow[{}]{{\sim}}\mathbb{D}(X_{0}\times _{k}R/J)^{\otimes }[1/p]\end{eqnarray}$$
   induced by the quasi-isogeny $\unicode[STIX]{x1D70C}$ .

2. (ii) The sheaf of $G_{W}$ -sets over $\text{CRIS}(\text{Spec}(R)/W)$ given by isomorphisms

   $$\begin{eqnarray}\text{}\underline{\text{Isom}}_{t_{\unicode[STIX]{x1D6FC}},s_{\unicode[STIX]{x1D6FC}}\otimes 1}(\mathbb{D}(X),D\otimes _{\mathbb{Z}_{p}}R)\end{eqnarray}$$
   that respect the tensors as indicated, is a crystal of $G_{W}$ -torsors, i.e. a crystal fppf locally isomorphic to the crystal defined by  $G_{W}$ .

3. (iii) There exists an étale cover $\{U_{i}\}$ of $\text{Spec}(R)$ , and for each $i$ an isomorphism

   $$\begin{eqnarray}\mathbb{D}(X_{U_{i}})_{U_{i}}\xrightarrow[{}]{{\sim}}D\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}_{U_{i}}\end{eqnarray}$$
   of vector bundles respecting the tensors $t_{\unicode[STIX]{x1D6FC}}$ and $s_{\unicode[STIX]{x1D6FC}}\otimes 1$ as in (ii), such that the Hodge filtration
   $$\begin{eqnarray}\text{Fil}^{1}(X_{U_{i}})\subset \mathbb{D}(X_{U_{i}})_{U_{i}}\xrightarrow[{}]{{\sim}}D\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}_{U_{i}}\end{eqnarray}$$
   is induced by a cocharacter that is $G(U_{i})$ -conjugate to  $\unicode[STIX]{x1D707}$ .

Two triples $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))$ and $(X^{\prime },\unicode[STIX]{x1D70C}^{\prime },(t_{\unicode[STIX]{x1D6FC}}^{\prime }))$ are identified if there is an isomorphism $X\xrightarrow[{}]{{\sim}}X^{\prime }$ of $p$ -divisible groups that respects the rest of the data in the obvious manner.

Above, $\text{CRIS}(\text{Spec}(R)/W)$ denotes the big fppf crystalline site of $\text{Spec}(R)$ over $(W,(p),\unicode[STIX]{x1D6FE})$ with $\unicode[STIX]{x1D6FE}$ the natural PD structure, as in [Reference Berthelot, Breen and MessingBBM82, 1.1]. Condition (ii) implies that, for any nilpotent PD thickening $R^{\prime }\rightarrow R$ of $R$ , the $\text{Spec}(R^{\prime })$ -scheme of isomorphisms of finite locally free $R^{\prime }$ -modules

$$\begin{eqnarray}T_{R^{\prime }}=\text{}\underline{\text{Isom}}_{R^{\prime },t_{\unicode[STIX]{x1D6FC}}(R^{\prime }),s_{\unicode[STIX]{x1D6FC}}\otimes 1}(\mathbb{D}(X)(R^{\prime }),D\otimes _{\mathbb{ Z}_{p}}R^{\prime })\end{eqnarray}$$

is a $G_{R^{\prime }}$ -torsor.

Remark 2.3.4. There is a distinguished point

$$\begin{eqnarray}x_{0}=(X_{0},\unicode[STIX]{x1D70C}_{0},(t_{\unicode[STIX]{x1D6FC},0}))\in \text{RZ}^{\text{nilp}}(k),\end{eqnarray}$$

defined by taking $\unicode[STIX]{x1D70C}_{0}$ to be the identity quasi-isogeny $X_{0}{\dashrightarrow}X_{0}$ .

Remark 2.3.5. (a) Suppose $(p)\subset J^{\prime }\subset J$ with $J^{\prime }$ also nilpotent. Then a power of the Frobenius of  $R/J^{\prime }$ factors through $R/J$ . Since $t_{\unicode[STIX]{x1D6FC}}$ are Frobenius equivariant we obtain that condition (i) is independent of the ideal $J$ . In particular, we can simply take $J=(p)$ .

(b) Conditions (ii) and (iii) together imply the following. The $\text{Spec}(R)$ -scheme of $R$ -linear isomorphisms

$$\begin{eqnarray}\mathbb{D}(X)(R)\xrightarrow[{}]{{\sim}}D\otimes _{\mathbb{Z}_{p}}R\end{eqnarray}$$

identifying $t_{\unicode[STIX]{x1D6FC}}(R)$ with $s_{\unicode[STIX]{x1D6FC}}\otimes 1$ , and identifying the Hodge filtration $\text{Fil}^{1}(X)\subset \mathbb{D}(X)(R)$ with standard filtration $F_{\unicode[STIX]{x1D707}}\otimes _{W}R\subset D\otimes _{\mathbb{Z}_{p}}R$ defined by $\unicode[STIX]{x1D707}$ , is a $P_{\unicode[STIX]{x1D707}}\times _{W}R$ -torsor.

(c) For $R$ in $\text{ANilp}_{W}^{\text{f}}$ , the categories of $p$ -divisible groups over $\text{Spf}(R)$ and over $\text{Spec}(R)$ are naturally equivalent, by [Reference de JongdeJ95, Lemma 2.4.4]. For $R$ in $\text{ANilp}_{W}^{\text{fsm}}$ , the argument in the proof of [Reference de JongdeJ95, Proposition 2.4.8] shows that each morphism of crystals $\hat{t}_{\unicode[STIX]{x1D6FC}}:\mathbf{1}\rightarrow \mathbb{D}(X)^{\otimes }$ over $\text{Spf}(R)$ is induced by a unique morphism of crystals $t_{\unicode[STIX]{x1D6FC}}:\mathbf{1}\rightarrow \mathbb{D}(X)^{\otimes }$ over $\text{Spec}(R)$ .

(d) For $\bar{R}$ of finite type over $k$ , we will see that it is enough to verify (i) over one closed point of each connected component of $\text{Spec}(R)$ ; see Lemma 3.2.8 and its proof.

2.3.6

Define a functor $\text{RZ}_{G}^{\text{fsm}}$ on $\text{ANilp}_{W}^{\text{fsm}}$ by setting

$$\begin{eqnarray}\text{RZ}_{G}^{\text{fsm}}(A)=\mathop{\varprojlim }\nolimits_{n}\text{RZ}_{G}^{\text{nilp}}(A/I^{n}),\end{eqnarray}$$

where $I$ is an ideal of definition of $A$ .

Assume that $I$ is chosen with $p\in I$ . By Remark (c) above and the rigidity of quasi-isogenies [Reference Drinfel’dDri76], we see that elements of $\text{RZ}_{G}^{\text{fsm}}(A)$ correspond to isomorphism classes of triples $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))$ , in which $X$ is a $p$ -divisible group over $\text{Spec}(A)$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:X_{0}\times _{k}A/I{\dashrightarrow}X\times _{A}A/I\end{eqnarray}$$

is a quasi-isogeny, and $t_{\unicode[STIX]{x1D6FC}}$ a morphism of crystals over $\text{Spec}(A)$ , such that (i), (ii), and (iii) above are satisfied. The definition is independent of the choice of  $I$ .

Since any object $A$ of $\text{ANilp}_{W}^{\text{fsm}}$ is also an object of $\text{ANilp}_{W}$ , it makes sense to consider $\text{RZ}_{G}^{\text{nilp}}(A)$ . We will rarely do this unless $A$ is discrete, in which case

$$\begin{eqnarray}\text{RZ}_{G}^{\text{nilp}}(A)=\text{RZ}_{G}^{\text{fsm}}(A).\end{eqnarray}$$

The difference between $\text{RZ}_{G}^{\text{fsm}}(A)$ and $\text{RZ}_{G}^{\text{nilp}}(A)$ is that, in the former, we ask that the quasi-isogeny $\unicode[STIX]{x1D70C}$ only exists over $A/I$ , with $I$ an ideal of definition of the adic algebra $A$ . For $A$ in $\text{ANilp}_{W}^{\text{fsm}}$ it will often be the case that $\text{RZ}_{G}^{\text{nilp}}(A)=\emptyset$ , while $\text{RZ}_{G}^{\text{fsm}}(A)\neq \emptyset$ .

2.3.7

The closed immersion $\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GL}(C)$ induces an injective homomorphism from the group $J_{b}(\mathbb{Q}_{p})$ into the group $\text{Aut}_{\mathbb{Q}_{p}}(X_{0})$ of quasi-automorphisms of the $p$ -divisible group $X_{0}$ , i.e. of automorphisms of $X_{0}$ up to isogeny. In addition, we can see that the induced action of $J_{b}(\mathbb{Q}_{p})$ on $\mathbb{D}(X_{0})(W)^{\otimes }[1/p]$ preserves the tensors $t_{\unicode[STIX]{x1D6FC},0}$ . Therefore, the group $J_{b}(\mathbb{Q}_{p})$ acts on the functors $\text{RZ}_{G}^{\text{nilp}}$ and $\text{RZ}_{G}^{\text{fsm}}$ on the left by

(2.3.7.1) $$\begin{eqnarray}g\cdot (X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))=(X,\unicode[STIX]{x1D70C}\circ g^{-1},(t_{\unicode[STIX]{x1D6FC}})).\end{eqnarray}$$

2.4.1

For $G$ , $b$ and $\unicode[STIX]{x1D707}:\mathbb{G}_{mW}\rightarrow G_{W}$ as in the beginning of § 2.2.1, the ‘classical’ affine Deligne–Lusztig set is

$$\begin{eqnarray}X_{G,b,\unicode[STIX]{x1D707}}(k)=\{g\in G(K):g^{-1}b\unicode[STIX]{x1D70E}(g)\in G(W)\unicode[STIX]{x1D707}(p)G(W)\}/G(W).\end{eqnarray}$$

We will define an analogous set for any extension field $k^{\prime }/k$ .

Let $W^{\prime }$ be the Cohen ring of $k^{\prime }$ , let $K^{\prime }=W^{\prime }[1/p]$ be its fraction field, and let $\unicode[STIX]{x1D70E}:W^{\prime }\rightarrow W^{\prime }$ be a lift of the absolute Frobenius as in § 2.1.4. Consider the set

(2.4.1.1) $$\begin{eqnarray}\{g\in G(K^{\prime }):g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}(p)^{-1}\in G(W^{\prime })\},\end{eqnarray}$$

and define

$$\begin{eqnarray}Q_{\unicode[STIX]{x1D707}}(W^{\prime })=G(W^{\prime })\cap \unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p)^{-1}G(W^{\prime })\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p),\end{eqnarray}$$

the intersection taking place in $G(K^{\prime })$ .

The right translation action of $Q_{\unicode[STIX]{x1D707}}(W^{\prime })$ on $G(K^{\prime })$ preserves (2.4.1.1). Indeed, if $q\in Q_{\unicode[STIX]{x1D707}}(W^{\prime })$ and $g$ belongs to (2.4.1.1), then $U_{g}=g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}(p)^{-1}$ belongs to $G(W^{\prime })$ , and hence so does

$$\begin{eqnarray}(gq)^{-1}b\unicode[STIX]{x1D70E}(gq)\unicode[STIX]{x1D707}(p)^{-1}=q^{-1}g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D70E}(q)\unicode[STIX]{x1D707}(p)^{-1}=q^{-1}U_{g}\unicode[STIX]{x1D707}(p)\unicode[STIX]{x1D70E}(q)\unicode[STIX]{x1D707}(p)^{-1}.\end{eqnarray}$$

Thus $gq$ belongs to (2.4.1.1).

Definition 2.4.2. The refined affine Deligne–Lusztig set is the quotient

$$\begin{eqnarray}X_{G,b,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}}(k^{\prime })=\{g\in G(K^{\prime }):g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}(p)^{-1}\in G(W^{\prime })\}/Q_{\unicode[STIX]{x1D707}}(W^{\prime }).\end{eqnarray}$$

Similarly, we have the naive affine Deligne–Lusztig set

$$\begin{eqnarray}X_{G,b,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}}^{\text{naive}}(k^{\prime })=\{g\in G(K^{\prime }):g^{-1}b\unicode[STIX]{x1D70E}(g)\in G(W^{\prime })\unicode[STIX]{x1D707}(p)G(W^{\prime })\}/G(W^{\prime }).\end{eqnarray}$$

For simplicity, we will often omit $\unicode[STIX]{x1D70E}$ from the list of subscripts. However, we do not know if the set $X_{G,b,\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}}(k^{\prime })$ is independent of the choice of the lift of Frobenius  $\unicode[STIX]{x1D70E}$ .

Proof. We first show (i). The condition for $g$ in the refined notion is stronger than the condition in the naive notion; since $Q_{\unicode[STIX]{x1D707}}(W^{\prime })\subset G(W^{\prime })$ the map is well defined. It remains to show that it is injective. Let $g$ , $g^{\prime }\in G(K^{\prime })$ , and assume

$$\begin{eqnarray}\displaystyle U_{g} & := & \displaystyle g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}(p)^{-1}\in G(W^{\prime }),\nonumber\\ \displaystyle U_{g^{\prime }} & := & \displaystyle {g^{\prime }}^{-1}b\unicode[STIX]{x1D70E}(g^{\prime })\unicode[STIX]{x1D707}(p)^{-1}\in G(W^{\prime }).\nonumber\end{eqnarray}$$

Suppose there is an $h\in G(W^{\prime })$ such that $g^{\prime }=gh$ . Then we obtain

$$\begin{eqnarray}U_{g^{\prime }}=h^{-1}g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D70E}(h)\unicode[STIX]{x1D707}(p)^{-1}=h^{-1}U_{g}\unicode[STIX]{x1D707}(p)\unicode[STIX]{x1D70E}(h)\unicode[STIX]{x1D707}(p)^{-1}.\end{eqnarray}$$

As $h^{-1}U_{g}$ and $U_{g^{\prime }}$ are in $G(W^{\prime })$ , we see that the element

$$\begin{eqnarray}\unicode[STIX]{x1D707}(p)\unicode[STIX]{x1D70E}(h)\unicode[STIX]{x1D707}(p)^{-1}=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p)h\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p)^{-1})\in G(K)\end{eqnarray}$$

actually lies in $G(W^{\prime })$ . Since $\unicode[STIX]{x1D70E}:W^{\prime }/p^{i}W^{\prime }\rightarrow W^{\prime }/p^{i}W^{\prime }$ is injective for all $i$ , and $G$ is affine and flat over $\mathbb{Z}_{p}$ , this implies that

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p)h\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}^{-1}}(p)^{-1}\in G(W^{\prime }).\end{eqnarray}$$

It follows that $h\in Q_{\unicode[STIX]{x1D707}}(W^{\prime })$ . This shows the injectivity of the map $\unicode[STIX]{x1D719}(k^{\prime })$ .

Now suppose that $k^{\prime }$ is perfect so that $\unicode[STIX]{x1D70E}^{-1}$ makes sense on $W^{\prime }=W(k^{\prime })$ . If $g\in G(K^{\prime })$ is such that $g^{-1}b\unicode[STIX]{x1D70E}(g)=h_{1}\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)h_{2}$ with $h_{i}\in G(W^{\prime })$ , then $g^{\prime }=g\unicode[STIX]{x1D70E}^{-1}(h_{2})h_{1}$ satisfies the refined condition. Hence $\unicode[STIX]{x1D719}(k^{\prime })$ is surjective.

Part (ii) is routine. To show part (iii), observe that for $h=\unicode[STIX]{x1D70E}^{-1}(b^{-1}g)$ , we have

$$\begin{eqnarray}h^{-1}b\unicode[STIX]{x1D70E}(h)=\unicode[STIX]{x1D70E}^{-1}(g^{-1}b)bb^{-1}g=\unicode[STIX]{x1D70E}^{-1}(g^{-1})\unicode[STIX]{x1D70E}^{-1}(b)g=\unicode[STIX]{x1D70E}^{-1}(g^{-1}b\unicode[STIX]{x1D70E}(g))\end{eqnarray}$$

and the result follows. ◻

2.4.4

Suppose $\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GL}_{n}$ is a closed immersion of group schemes over $\mathbb{Z}_{p}$ , and set $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D704}\circ \unicode[STIX]{x1D707}:\mathbb{G}_{mW}\rightarrow \text{GL}_{n,W}$ . For $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70E}^{-1}\in \text{Aut}(W)$ , we have

$$\begin{eqnarray}G(W^{\prime })=G(K^{\prime })\cap \text{GL}_{n}(W^{\prime })\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70F}}(p)^{-1}G(W^{\prime })\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70F}}(p)=G(K^{\prime })\cap \unicode[STIX]{x1D708}^{\unicode[STIX]{x1D70F}}(p)^{-1}\text{GL}_{n}(W^{\prime })\unicode[STIX]{x1D708}^{\unicode[STIX]{x1D70F}}(p),\end{eqnarray}$$

the intersections taking place in $\text{GL}_{n}(K^{\prime })$ . The embedding $\unicode[STIX]{x1D704}$ then induces injections

$$\begin{eqnarray}G(K^{\prime })/G(W^{\prime }){\hookrightarrow}\text{GL}_{n}(K^{\prime })/\text{GL}_{n}(W^{\prime })\end{eqnarray}$$

and

$$\begin{eqnarray}G(K^{\prime })/Q_{\unicode[STIX]{x1D707}}(W^{\prime }){\hookrightarrow}\text{GL}_{n}(K^{\prime })/Q_{\unicode[STIX]{x1D708}}^{\text{GL}_{n}}(W^{\prime }).\end{eqnarray}$$

Moreover, $g\in G(K^{\prime })$ satisfies $g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}(p)^{-1}\in G(W^{\prime })$ if and only if $\unicode[STIX]{x1D704}(g)\in \text{GL}_{n}(K^{\prime })$ satisfies the corresponding condition with $(G,b,\unicode[STIX]{x1D707})$ replaced by $(\text{GL}_{n},\unicode[STIX]{x1D704}(b),\unicode[STIX]{x1D708})$ . It follows that $\unicode[STIX]{x1D704}$ defines an injection

$$\begin{eqnarray}X_{G,b,\unicode[STIX]{x1D707}}(k^{\prime }){\hookrightarrow}X_{\text{GL}_{n},\unicode[STIX]{x1D704}(b),\unicode[STIX]{x1D708}}(k^{\prime }).\end{eqnarray}$$

2.4.5

We now return to the set-up of § 2.3. Assume that $(G,[b],\{\unicode[STIX]{x1D707}\})$ is an unramified local Shimura datum of Hodge type. Fix a corresponding local Shimura–Hodge datum $(G,b,\unicode[STIX]{x1D707},C)$ and a set of tensors $(s_{\unicode[STIX]{x1D6FC}})$ that cuts out $G\subset \text{GL}(C)$ . We then have the functor $\text{RZ}_{G}^{\text{nilp}}$ as before.

Fix a point

$$\begin{eqnarray}(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))\in \text{RZ}_{G}^{\text{nilp}}(k^{\prime }).\end{eqnarray}$$

Consider the value $M=\mathbb{D}(X)(W^{\prime })$ of the crystal $\mathbb{D}(X)$ on the Cohen ring $W^{\prime }$ of $k^{\prime }$ , viewed as a PD thickening of $k^{\prime }$ . We have the tensors $t_{\unicode[STIX]{x1D6FC}}(W^{\prime })\in M^{\otimes }$ , which are Frobenius invariant in $M^{\otimes }[1/p]$ , and the quasi-isogeny $\unicode[STIX]{x1D70C}$ , which induces an isomorphism

$$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D70C}):M[1/p]\xrightarrow[{}]{{\sim}}M_{0}[1/p]\end{eqnarray}$$

such that $\mathbb{D}(\unicode[STIX]{x1D70C})(t_{\unicode[STIX]{x1D6FC}}(W^{\prime }))=t_{\unicode[STIX]{x1D6FC},0}(W)\otimes 1$ , by (i) of Definition 2.3.3.

Lemma 2.4.6. Under these assumptions, the scheme

$$\begin{eqnarray}T=\text{}\underline{\text{Isom}}_{W^{\prime },t_{\unicode[STIX]{x1D6FC}}(W^{\prime }),s_{\unicode[STIX]{x1D6FC}}\otimes 1}(M,D\otimes _{\mathbb{Z}_{p}}W^{\prime })\end{eqnarray}$$

is a trivial $G_{W}^{\prime }$ -torsor over $\text{Spec}(W^{\prime })$ .

2.4.7

We now describe $\text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ in terms of a refined affine Deligne–Lusztig set. The following may be standard, but we could not find a reference.

Proof. Suppose that $R$ is a finitely generated smooth $k$ -algebra which is a domain and is such that $k^{\prime }$ is the fraction field of $R$ . By replacing $R$ by a localization we can assume that the differentials $\unicode[STIX]{x1D6FA}_{R/k}$ are a free $R$ -module of rank equal to the Krull dimension of $R$ ; let $dx_{i}$ , $i=1,\ldots ,d$ , be an $R$ -basis of $\unicode[STIX]{x1D6FA}_{R/k}$ . In this situation, the absolute Frobenius $\unicode[STIX]{x1D719}_{R}:R\rightarrow R$ is injective and makes $R$ into a finitely generated $R$ -module. Therefore, by [Reference GrothendieckEGAIV, ch. 0, Proposition (21.1.7)], the tuple $(x_{i})$ with $i=1,\ldots ,d$ is a system of $p$ -generators of $R$ over $k$ , i.e.  $R=k[R^{p},(x_{i})]$ . In fact, we can easily see that, since $dx_{i}$ are $R$ -linearly independent, the $x_{i}$ are $p$ -independent (cf. [Reference MatsumuraMat80, p. 276]). Therefore, the $x_{i}$ form a $p$ -basis of $R$ over $k$ .

If we start with a $p$ -basis $(x_{i})$ of $k^{\prime }$ , then [Reference MatsumuraMat80, Theorem 86] implies that $(dx_{i})$ are a basis of the $k^{\prime }$ -vector space $\unicode[STIX]{x1D6FA}_{k^{\prime }/k}$ . If $R\subset k^{\prime }$ is any smooth finite type $k^{\prime }$ -algebra with $k^{\prime }=\text{Frac}(R)$ such that $x_{i}\in R$ and $(dx_{i})$ generate $\unicode[STIX]{x1D6FA}_{R/k}$ , then, by the above, $(x_{i})$ also provide a $p$ -basis of  $R$ .

Since $k$ is perfect, we can write $k^{\prime }=\varinjlim R$ , where $R$ is as above. Now, as in [Reference Berthelot and MessingBM90, § 1.1], using the $p$ -basis $(x_{i})$ we obtain a concrete construction of the Cohen ring

$$\begin{eqnarray}W^{\prime }=\mathop{\varprojlim }\nolimits_{n}A_{n}(k^{\prime }),\end{eqnarray}$$

and of a $W$ -flat lift

$$\begin{eqnarray}\widetilde{R}=\mathop{\varprojlim }\nolimits_{n}A_{n}(R)\end{eqnarray}$$

of $R$ . Here $A_{n}(R)$ and $A_{n}(k^{\prime })$ , are certain subrings of the truncated Witt vector rings $W_{n}(R)$ and $W_{n}(k^{\prime })$ . By [Reference Berthelot and MessingBM90], sending $x_{i}\in R$ to $x_{i}\in k^{\prime }$ gives ring homomorphisms $i_{n}:A_{n}(R)\rightarrow A_{n}(k^{\prime })$ . Since $R\rightarrow k^{\prime }$ is injective, $A_{n}(R)\rightarrow A_{n}(k^{\prime })$ is injective, and so also $\widetilde{i}:\widetilde{R}\rightarrow W^{\prime }$ is injective. Recall that, by [Reference Berthelot and MessingBM90, Proposition 1.2.6], a lift of Frobenius on $A_{n}(R)$ , respectively $W_{n}(k^{\prime })$ , is uniquely determined by giving (arbitrary) lifts $y_{i,n}\in A_{n}(R)$ , respectively $A_{n}(k^{\prime })$ , of all the elements $x_{i}^{p}$ . Therefore, we can choose this way lifts of Frobenius on $\widetilde{R}$ and $W^{\prime }$ that are compatible under $\widetilde{i}:\widetilde{R}{\hookrightarrow}W^{\prime }$ .◻

2.4.9

Let $(G,b,\unicode[STIX]{x1D707},C)$ be a local unramified Shimura–Hodge datum as in Definition 2.2.4, and let $(s_{\unicode[STIX]{x1D6FC}})$ be tensors in $C^{\otimes }$ that cut out $G\subset \text{GL}(C)$ , as in § 2.2.6.

Let $k^{\prime }/k$ be a finitely generated field extension, and suppose that the lift $\unicode[STIX]{x1D70E}:W^{\prime }\rightarrow W^{\prime }$ of the Frobenius is chosen as in Proposition 2.4.8. Suppose also that the Dieudonné module structure on $D_{W}$ determined by $\unicode[STIX]{x1D704}(b)\in \text{GL}(C_{W})$ has no zero slopes (equivalently, the base point $p$ -divisible group $X_{0}$ over $k$ defined in Lemma 2.2.5 is formal).

Theorem 2.4.10. Under the above assumptions, there are natural bijections

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\mathop{\varinjlim }\nolimits_{R}\text{RZ}_{G}^{\text{nilp}}(R)\xrightarrow[{}]{{\sim}}\text{RZ}_{G}^{\text{nilp}}(k^{\prime })\xrightarrow[{}]{{\sim}}X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime }),\end{eqnarray}$$

where the limit over $R$ is as in Proposition 2.4.8 above.

Proof. Let $X_{0}$ be the $p$ -divisible group of Lemma 2.2.5, and recall that $\text{RZ}(X_{0})$ is the (undecorated) Rapoport–Zink formal scheme from § 2.3.1. Notice that $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))\mapsto (X,\unicode[STIX]{x1D70C})$ defines an injection $\text{RZ}_{G}^{\text{nilp}}(k^{\prime }){\hookrightarrow}\text{RZ}(X_{0})(k^{\prime }),$ as $t_{\unicode[STIX]{x1D6FC}}$ is determined by $t_{\unicode[STIX]{x1D6FC}}(W^{\prime })$ .

Similarly, $\text{RZ}_{G}^{\text{nilp}}(R)$ injects to $\text{RZ}(X_{0})(R)$ . Indeed, $t_{\unicode[STIX]{x1D6FC}}$ is determined by $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})$ , and, as $\widetilde{R}$ is torsion-free, $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})$ is determined by

$$\begin{eqnarray}t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})[1/p]=\mathbb{D}(\unicode[STIX]{x1D70C})^{-1}(t_{\unicode[STIX]{x1D6FC},0}[1/p]).\end{eqnarray}$$

Since $\text{RZ}(X_{0})$ is formally locally of finite type over $W$ , we have

$$\begin{eqnarray}\text{RZ}(X_{0})(k^{\prime })=\mathop{\varinjlim }\nolimits_{R}\text{RZ}(X_{0})(R),\end{eqnarray}$$

and so $\mathop{\varinjlim }\nolimits_{R}\text{RZ}_{G}^{\text{nilp}}(R){\hookrightarrow}\text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ .

Pick a point $x=(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))\in \text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ , and consider the value $M:=\mathbb{D}(X)(W^{\prime })$ of the crystal $\mathbb{D}(X)$ on $W^{\prime }$ , endowed with the tensors $t_{\unicode[STIX]{x1D6FC}}(W^{\prime })\in M^{\otimes }$ . By Lemma 2.4.6, the scheme

$$\begin{eqnarray}T_{x}=\text{}\underline{\text{Isom}}_{W^{\prime },t_{\unicode[STIX]{x1D6FC}}(W^{\prime }),s_{\unicode[STIX]{x1D6FC}}\otimes 1}(M,D\otimes _{\mathbb{Z}_{p}}W^{\prime })\end{eqnarray}$$

is a trivial $G$ -torsor over $\text{Spec}(W^{\prime })$ .

If $k^{\prime }=k$ and $x=x_{0}$ is the base point of Remark 2.3.4, then there is an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{0}:M_{0}:=\mathbb{D}(X_{0})(W)\xrightarrow[{}]{{\sim}}D\otimes _{\mathbb{Z}_{p}}W\end{eqnarray}$$

with $\unicode[STIX]{x1D6FD}_{0}^{\otimes }(t_{\unicode[STIX]{x1D6FC},0}(W))=s_{\unicode[STIX]{x1D6FC}}\otimes 1$ and we can use this to identify $M_{0}=D\otimes _{\mathbb{Z}_{p}}W$ . In general, the generic fiber of $T_{x}$ has a section constructed using $\mathbb{D}(\unicode[STIX]{x1D70C})$ and $\unicode[STIX]{x1D6FD}_{0}$ . Since the $G$ -torsor $T_{x}$ is trivial there is $\unicode[STIX]{x1D6FD}:M\xrightarrow[{}]{{\sim}}D_{W}^{\prime }$ such that $\unicode[STIX]{x1D6FD}^{\otimes }(t_{\unicode[STIX]{x1D6FC}}(W^{\prime }))=s_{\unicode[STIX]{x1D6FC}}\otimes 1$ . Using $\mathbb{D}(\unicode[STIX]{x1D70C})$ , we can identify

$$\begin{eqnarray}M_{\mathbb{Q}}=M_{0}\otimes _{W}K^{\prime }=D\otimes _{\mathbb{ Z}_{p}}K^{\prime }\end{eqnarray}$$

and therefore think of $M\subset M_{\mathbb{Q}}$ as a $W^{\prime }$ -lattice in $M_{0}\otimes _{W}K^{\prime }=D\otimes _{\mathbb{Z}_{p}}K^{\prime }$ . Under this identification, the choice of $\unicode[STIX]{x1D6FD}$ is equivalent to picking $g\in G(K^{\prime })$ such that $M=g\cdot (D\otimes _{\mathbb{Z}_{p}}W^{\prime })$ ; then $\unicode[STIX]{x1D6FD}$ is given by left multiplication by $g^{-1}$ . Notice that the coset $gG(W^{\prime })$ is independent of the choice of $\unicode[STIX]{x1D6FD}_{0}$ and $\unicode[STIX]{x1D6FD}$ and so we have a well-defined map

$$\begin{eqnarray}\text{RZ}_{G}^{\text{nilp}}(k^{\prime })\rightarrow G(K^{\prime })/G(W^{\prime })\end{eqnarray}$$

given by $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))\mapsto gG(W^{\prime })$ .

By Zink’s theory, as in § 2.1.5 (see especially [Reference ZinkZin01, Theorem 4]), and using the inclusion

$$\begin{eqnarray}\text{RZ}_{G}^{\text{nilp}}(k^{\prime })\subset \text{RZ}(X_{0})(k^{\prime }),\end{eqnarray}$$

we see that the $W^{\prime }$ -lattice $M\subset M_{0}\otimes _{W}W^{\prime }[1/p]$ uniquely determines the point $(X,\unicode[STIX]{x1D70C},(t_{\unicode[STIX]{x1D6FC}}))\in \text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ . On the other hand,

$$\begin{eqnarray}M=g\cdot (M_{0}\otimes _{W}W^{\prime })\subset M_{0}\otimes _{W}W^{\prime }[1/p]\end{eqnarray}$$

is uniquely determined by $gG(W^{\prime })$ , and so this construction gives an injection

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:\text{RZ}_{G}^{\text{nilp}}(k^{\prime }){\hookrightarrow}G(K^{\prime })/G(W^{\prime }).\end{eqnarray}$$

We have to show that the image of this injection is exactly the refined affine Deligne–Lusztig set $X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ .

This amounts to showing that $M$ is the $W^{\prime }$ -window corresponding to a point in $\text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ if and only if we can pick $g\in G(K^{\prime })$ such that $M=g\cdot (M_{0}\otimes _{W}W^{\prime })$ and $g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)^{-1}\in G(W^{\prime })$ .

Suppose that $M$ indeed corresponds to the point $x\in \text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ as above. Then

$$\begin{eqnarray}\mathbb{D}(X)(k^{\prime })=M\otimes _{W^{\prime }}k^{\prime }=M/pM\end{eqnarray}$$

with Hodge filtration given by $\text{Fil}^{1}(X)=M_{1}/pM\subset M/pM$ . Since the Hodge filtration is a $G$ -filtration of type $\unicode[STIX]{x1D707}$ by Definition 2.3.3(iii), we can pick a trivialization $\bar{\unicode[STIX]{x1D6FD}}:M/pM\xrightarrow[{}]{{\sim}}D_{k}^{\prime }$ that preserves the tensors (i.e. a $k^{\prime }$ -section of the trivial torsor $T_{x}$ above), and such that

$$\begin{eqnarray}M_{1}/pM\subset M/pM\cong D_{k}^{\prime }\end{eqnarray}$$

is actually the filtration given by $\unicode[STIX]{x1D707}$ . (Indeed, since $G_{W}$ is split, the quotient morphism $G_{W}\rightarrow G_{W}/P_{\unicode[STIX]{x1D707}}$ splits locally for the Zariski topology over $W$ . Therefore, $G(k^{\prime })$ acts transitively on the set of $G$ -filtrations of $D_{k}^{\prime }$ of type $\unicode[STIX]{x1D707}$ .) By Hensel’s lemma, we can lift $\bar{\unicode[STIX]{x1D6FD}}$ to $\unicode[STIX]{x1D6FD}:M\xrightarrow[{}]{{\sim}}D_{W}^{\prime }$ . Then $\unicode[STIX]{x1D6FD}(M_{1})=p\unicode[STIX]{x1D707}(p)^{-1}D_{W}^{\prime }$ . As before, $\unicode[STIX]{x1D6FD}$ corresponds to $g\in G(K^{\prime })$ such that

$$\begin{eqnarray}M=g\cdot (M_{0}\otimes _{W}W^{\prime })=g\cdot D_{W}^{\prime }\end{eqnarray}$$

and then $\unicode[STIX]{x1D6FD}$ is multiplication by $g^{-1}$ . Hence,

$$\begin{eqnarray}M=g\cdot D_{W}^{\prime },\quad M_{1}=g\cdot p\unicode[STIX]{x1D707}(p)^{-1}D_{W}^{\prime }.\end{eqnarray}$$

The quadruple $(M,M_{1},F=b\circ \unicode[STIX]{x1D70E})$ defines a $W^{\prime }$ -window if and only if the $W^{\prime }$ -submodule $\langle p^{-1}F(M_{1})\rangle$ of $M[1/p]$ generated by $p^{-1}F(M_{1})$ is equal to $M$ . Indeed, if $(M,M_{1},F=b\circ \unicode[STIX]{x1D70E})$ is a $W^{\prime }$ -window this condition follows from the definition in § 2.1.5. Conversely, assume that $\langle p^{-1}F(M_{1})\rangle =M$ . Then

$$\begin{eqnarray}pM\subset M_{1}\subset F^{-1}(pM)\subset M\end{eqnarray}$$

and $\langle F(M_{1})\rangle =M=\langle F(F^{-1}(pM))\rangle$ . Hence $M_{1}=F^{-1}(pM)$ , and then $(M,M_{1},F=b\circ \unicode[STIX]{x1D70E})$ gives a $W^{\prime }$ -module by the observation in § 2.1.5 (the nilpotence condition also follows by a slope argument).

The condition that $\langle p^{-1}F(M_{1})\rangle =M$ reads

$$\begin{eqnarray}\langle p^{-1}b\unicode[STIX]{x1D70E}(gp\unicode[STIX]{x1D707}(p)^{-1}\cdot D\otimes _{\mathbb{ Z}_{p}}W^{\prime })\rangle =g\cdot (D\otimes _{\mathbb{ Z}_{p}}W^{\prime }),\end{eqnarray}$$

which translates to

(2.4.10.1) $$\begin{eqnarray}\langle g^{-1}b\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)^{-1}\cdot (D\otimes _{\mathbb{ Z}_{p}}\unicode[STIX]{x1D70E}(W^{\prime }))\rangle =D\otimes _{\mathbb{ Z}_{p}}W^{\prime }.\end{eqnarray}$$

Set $u=gb\unicode[STIX]{x1D70E}(g)\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)^{-1}\in G(K^{\prime })$ . Since

$$\begin{eqnarray}\langle h\cdot (D\otimes _{\mathbb{Z}_{p}}\unicode[STIX]{x1D70E}(W^{\prime }))\rangle =h\cdot (D\otimes _{\mathbb{ Z}_{p}}W^{\prime }),\end{eqnarray}$$

for any $h\in \text{GL}_{n}(K^{\prime })$ , the equation (2.4.10.1) above amounts to $u\in \text{GL}_{n}(W^{\prime })$ . We obtain that $u$ is in $G(W^{\prime })=G(K^{\prime })\cap \text{GL}_{n}(W^{\prime })$ , and thus $g\in X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ . Therefore, the image of $\unicode[STIX]{x1D70B}$ is contained in $X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ .

Let us now discuss the converse. Start with $g\in X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ , and set

$$\begin{eqnarray}M=g\cdot D_{W}^{\prime },\quad M_{1}=gp\unicode[STIX]{x1D707}(p)^{-1}\cdot D_{W}^{\prime }.\end{eqnarray}$$

By the argument above, $(M,M_{1},b\circ \unicode[STIX]{x1D70E})$ is a $W^{\prime }$ -window. By Zink’s theory, there is a corresponding $p$ -divisible group $X$ over $k^{\prime }$ with a quasi-isogeny $\unicode[STIX]{x1D70C}:X_{0}\times _{k}k^{\prime }{\dashrightarrow}X$ , and $(X,\unicode[STIX]{x1D70C})$ gives a $k^{\prime }$ -point of the Rapoport–Zink space $\text{RZ}(X_{0})$ .

Since the underlying reduced scheme $\text{RZ}(X_{0})^{\text{red}}$ is locally of finite type, we can find a smooth domain $R$ over $k$ as above with $k^{\prime }=\text{Frac}(R)$ and a $p$ -divisible group with quasi-isogeny $(X_{R},\unicode[STIX]{x1D70C}_{R})$ over $R$ that extends $(X,\unicode[STIX]{x1D70C})$ . By replacing $R$ by a localization we can further assume that $R$ has a lift $\widetilde{R}$ such that $\unicode[STIX]{x1D70E}(\widetilde{R})\subset \widetilde{R}$ and that $\mathbb{D}(X_{R})(\widetilde{R})$ is $\widetilde{R}$ -free. We have

$$\begin{eqnarray}\mathbb{D}(X_{R})(\widetilde{R})\otimes _{\widetilde{R}}W^{\prime }\cong \mathbb{D}(X)(W^{\prime })\cong M=g\cdot (M_{0}\otimes _{W}W^{\prime }).\end{eqnarray}$$

We will now produce a corresponding $R$ -valued point $(X_{R},\unicode[STIX]{x1D70C}_{R},(t_{\unicode[STIX]{x1D6FC}}))$ of $\text{RZ}_{G}^{\text{nilp}}$ . For this, we will construct a morphism of crystals $t_{\unicode[STIX]{x1D6FC}}:\mathbf{1}\rightarrow \mathbb{D}(X_{R})^{\otimes }$ such that $t_{\unicode[STIX]{x1D6FC}}:\mathbf{1}\rightarrow \mathbb{D}(X_{R})^{\otimes }[1/p]$ are Frobenius invariant and then check that $t_{\unicode[STIX]{x1D6FC}}$ satisfy (i), (ii), and (iii) of Definition 2.3.3.

By [Reference Berthelot and MessingBM90, Proposition 1.3.3] or [Reference de JongdeJ95, Corollary 2.2.3], to give $t_{\unicode[STIX]{x1D6FC}}$ as above, it suffices to give $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})(1)\in \mathbb{D}(X_{R})(\widetilde{R})^{\otimes }$ which are horizontal for the connection and are Frobenius invariant in $\mathbb{D}(X_{R})(\widetilde{R})^{\otimes }[1/p]$ . Consider the images $t_{\unicode[STIX]{x1D6FC}}$ of the ‘constant’ tensors $s_{\unicode[STIX]{x1D6FC}}\otimes 1$ under the isomorphism of isocrystals

$$\begin{eqnarray}\mathbb{D}(\unicode[STIX]{x1D70C})^{-1}:\mathbb{D}(X_{0,R})^{\otimes }[1/p]\xrightarrow[{}]{{\sim}}\mathbb{D}(X_{R})^{\otimes }[1/p]\end{eqnarray}$$

induced by the quasi-isogeny $\unicode[STIX]{x1D70C}$ . Since $\unicode[STIX]{x1D70C}$ is defined over $R$ , we obtain $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})\in \mathbb{D}(X_{R})(\widetilde{R})^{\otimes }[1/p]$ . However, since $g$ is in $G(K^{\prime })$ , we actually have $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})=s_{\unicode[STIX]{x1D6FC}}\otimes 1$ in

$$\begin{eqnarray}M^{\otimes }[1/p]=\mathbb{D}(X_{R})(\widetilde{R})^{\otimes }\otimes _{\widetilde{R}}W^{\prime }[1/p],\end{eqnarray}$$

and these lie in

$$\begin{eqnarray}M^{\otimes }=\mathbb{D}(X_{R})(\widetilde{R})^{\otimes }\otimes _{\widetilde{R}}W^{\prime }.\end{eqnarray}$$

Since $\mathbb{D}(X_{R})(\widetilde{R})$ is $\widetilde{R}$ -free and $\widetilde{R}[1/p]\cap W^{\prime }=\widetilde{R}$ , we can see that our tensors $t_{\unicode[STIX]{x1D6FC}}(\widetilde{R})=s_{\unicode[STIX]{x1D6FC}}\otimes 1$ lie in $\mathbb{D}(X_{R})(\widetilde{R})^{\otimes }$ . They are horizontal and Frobenius invariant since this is true over $W^{\prime }$ , and $\widetilde{R}\subset W^{\prime }$ . Moreover, conditions (i), (ii), and (ii) can now be seen to be satisfied after possibly further localizing $R$ . We have now produced an $R$ -valued point in  $\text{RZ}_{G}^{\text{nilp}}$ .

These two constructions are inverses of each other. This shows that $\mathop{\varinjlim }\nolimits_{R}\text{RZ}_{G}^{\text{nilp}}(R)=\text{RZ}_{G}^{\text{nilp}}(k^{\prime })$ and the image of $\unicode[STIX]{x1D70B}$ is $X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ .◻

Remark 2.4.11. Observe that $\text{RZ}_{G}^{\text{fsm}}(R)=\text{RZ}_{G}^{\text{nilp}}(R)$ , and so Theorem 2.4.10 also gives

$$\begin{eqnarray}\mathop{\varinjlim }\nolimits_{R}\text{RZ}_{G}^{\text{fsm}}(R)\xrightarrow[{}]{{\sim}}X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime }).\end{eqnarray}$$

This will be important for the application to Rapoport–Zink formal schemes. As we will see, these are defined via the functor $\text{RZ}_{G}^{\text{fsm}}$ which can be evaluated at $R$ as above but not at  $k^{\prime }$ .

We can directly obtain a bijection $\text{RZ}_{G}^{\text{nilp}}(k^{\prime })\xrightarrow[{}]{{\sim}}X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D70E}}(k^{\prime })$ , without assuming that $k^{\prime }/k$ is finitely generated, by a simpler version of the above argument.

3.1.1

Let $(G,{\mathcal{H}})$ be a Hodge type Shimura datum in the sense of [Reference DeligneDel79], so that $G$ is a connected reductive group over $\mathbb{Q}$ and ${\mathcal{H}}=\{h\}$ is the $G(\mathbb{R})$ -conjugacy class of a Deligne cocharacter $h:\text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}\rightarrow G_{\mathbb{R}}$ . Define $\unicode[STIX]{x1D707}_{h}:\mathbb{G}_{m\mathbb{C}}\rightarrow G_{\mathbb{C}}$ , as usual, by $\unicode[STIX]{x1D707}_{h}(z)=h_{\mathbb{C}}(z,1)$ . The reflex field $E\subset \bar{\mathbb{Q}}\subset \mathbb{C}$ is the field of definition of the conjugacy class $\{\unicode[STIX]{x1D707}_{h}\}$ .

The condition of Hodge type means that there is an algebraic group embedding

(3.1.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GSp}_{2g}\end{eqnarray}$$

over $\mathbb{Q}$ inducing a morphism of Shimura data $(G,{\mathcal{H}})\rightarrow (\text{GSp}_{2g},{\mathcal{H}}_{2g})$ . Here ${\mathcal{H}}_{2g}$ is the union of the usual Siegel upper and lower half-spaces of genus $g$ . The composition $\unicode[STIX]{x1D704}\circ \unicode[STIX]{x1D707}_{h}$ is conjugate to the standard minuscule cocharacter $\unicode[STIX]{x1D707}_{\text{std}}:\mathbb{G}_{m}\rightarrow \text{GSp}_{2g}$ given by $\unicode[STIX]{x1D707}_{\text{std}}(a)=\text{diag}(a^{(g)},1^{(g)})$ . As $G(\mathbb{R})$ contains the image of the weight homomorphism $w_{h}$ , we see that $G$ has to contain the torus of scalars (diagonal matrices) of $\text{GSp}_{2g}$ .

We assume that $G$ extends to a connected reductive group over $\mathbb{Z}_{(p)}$ , which we again denote by $G$ . As in [Reference KisinKis10, Lemma (2.3.1)], this implies that there is a rank $2g$ symplectic space $(C,\unicode[STIX]{x1D713})$ over $\mathbb{Z}_{(p)}$ and a closed immersion $\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GL}(C)$ of reductive groups over $\mathbb{Z}_{(p)}$ whose generic fiber factors through the subgroup $\text{GSp}(C_{\mathbb{Q}},\unicode[STIX]{x1D713})\subset \text{GL}(C_{\mathbb{Q}})$ and induces (3.1.1.1) after fixing an identification $\text{GSp}(C_{\mathbb{Q}},\unicode[STIX]{x1D713})=\text{GSp}_{2g}$ . By Zarhin’s trick, after replacing $C$ by $\text{Hom}_{\mathbb{Z}_{(p)}}(C,C)^{\oplus 4}$ and enlarging $g$ , we may assume that $C$ is self-dual with respect to $\unicode[STIX]{x1D713}$ . We then have a closed immersion of reductive group schemes

(3.1.1.2) $$\begin{eqnarray}\unicode[STIX]{x1D704}:G{\hookrightarrow}\text{GSp}(C,\unicode[STIX]{x1D713})\end{eqnarray}$$

over $\mathbb{Z}_{(p)}$ with generic fiber (3.1.1.1).

As in (1.1.1.3), let $D$ be the $G$ -representation contragredient to $C$ . By [Reference KisinKis10, (1.3.2)] there is finite list $(s_{\unicode[STIX]{x1D6FC}})$ of tensors $s_{\unicode[STIX]{x1D6FC}}\in C^{\otimes }=D^{\otimes }$ that cut out $G\subset \text{GL}(C)$ , in the sense that

$$\begin{eqnarray}G(R)=\{g\in \text{GL}(C_{R}):g\cdot (s_{\unicode[STIX]{x1D6FC}}\otimes 1)=(s_{\unicode[STIX]{x1D6FC}}\otimes 1),\forall \unicode[STIX]{x1D6FC}\},\end{eqnarray}$$

for all $\mathbb{Z}_{(p)}$ -algebras $R$ . In particular, $G(W)=G(K)\cap \text{GL}(C_{W})$ . In what follows, we take the set of tensors $(s_{\unicode[STIX]{x1D6FC}})$ to always include the tensor corresponding to the perfect symplectic form $\unicode[STIX]{x1D713}$ ; see the proof of Theorem 3.2.1 below.

A choice of field embedding $\bar{\mathbb{Q}}{\hookrightarrow}\bar{K}$ determines a place $v\mid p$ of $E$ . The completion $E_{v}$ is the field of definition of $\{\unicode[STIX]{x1D707}_{h}\}$ , now regarded as a $G(\bar{K})$ -conjugacy class of cocharacters $\mathbb{G}_{m\bar{K}}\rightarrow G_{\bar{K}}$ .

Let

$$\begin{eqnarray}\text{Sh}_{U_{p}}(G,{\mathcal{H}})=\mathop{\varprojlim }\nolimits_{U^{p}}\text{Sh}_{U^{p}U_{p}}(G,{\mathcal{H}})\end{eqnarray}$$

be the canonical model over $E\subset K$ of the corresponding Shimura variety for the hyperspecial subgroup

$$\begin{eqnarray}U_{p}=G(\mathbb{Z}_{p})\subset G(\mathbb{Q}_{p}).\end{eqnarray}$$

Here the limit is over compact open subgroups $U^{p}$ of $G(\mathbb{A}_{f}^{p})$ .

3.1.2

For a $\mathbb{Z}_{(p)}$ -scheme $S$ and an abelian scheme $A\rightarrow S$ we set

$$\begin{eqnarray}\text{Ta}^{p}(A)=\mathop{\varprojlim }\nolimits_{p\nmid n}A[n],\end{eqnarray}$$

viewed as an étale local system on $S$ , and write

$$\begin{eqnarray}\text{Ta}^{p}(A)_{\mathbb{ Q}}=\text{Ta}^{p}(A)\otimes _{\mathbb{ Z}}\mathbb{Q}.\end{eqnarray}$$

Consider the category obtained from the category of abelian schemes over $S$ by tensoring the Hom groups by $\mathbb{Z}_{(p)}$ . An object in this category will be called an abelian scheme over $S$ up to prime-to-p-isogeny. An isomorphism in this category will be called a $p^{\prime }$ -quasi-isogeny. Note that $\text{Ta}^{p}(A)_{\mathbb{Q}}$ is functorial for $p^{\prime }$ -quasi-isogenies.

If $A$ is an abelian scheme up to prime-to- $p$ -isogeny, we write $A^{\vee }$ for the dual abelian scheme up to prime-to- $p$ -isogeny. A weak polarization of $A$ is an equivalence class of $p^{\prime }$ -quasi-isogenies $\unicode[STIX]{x1D706}:A\xrightarrow[{}]{{\sim}}A^{\vee }$ such that some $\mathbb{Z}_{(p)}^{\times }$ -multiple of $\unicode[STIX]{x1D706}$ is a polarization. Here two such $\unicode[STIX]{x1D706}$ are equivalent if they differ by multiplication by an element of $\mathbb{Z}_{(p)}^{\times }$ .

Let $U^{\prime p}\subset \text{GSp}_{2g}(\mathbb{A}_{f}^{p})$ be any compact open subgroup, let

$$\begin{eqnarray}U_{p}^{\prime }=\text{GSp}(C,\unicode[STIX]{x1D713})(\mathbb{Z}_{p})\subset \text{GSp}_{2g}(\mathbb{Q}_{p})\end{eqnarray}$$

be the hyperspecial subgroup determined by the self-dual symplectic space $(C,\unicode[STIX]{x1D713})$ over $\mathbb{Z}_{(p)}$ , and set

$$\begin{eqnarray}U^{\prime }=U^{\prime p}U_{p}^{\prime }\subset \text{GSp}_{2g}(\mathbb{A}_{f}).\end{eqnarray}$$

Assume $(A,\unicode[STIX]{x1D706})$ is an abelian scheme up to prime-to- $p$ -isogeny with a weak polarization. A  $U^{\prime p}$ -level structure on $(A,\unicode[STIX]{x1D706})$ is a global section

$$\begin{eqnarray}\unicode[STIX]{x1D716}_{U^{\prime }}^{p}\in \unicode[STIX]{x1D6E4}(S,\text{}\underline{\text{Isom}}(C\otimes \mathbb{A}_{f}^{p},\text{Ta}^{p}(A)_{\mathbb{ Q}})/U^{\prime p}).\end{eqnarray}$$

Here $\text{}\underline{\text{Isom}}(C\otimes \mathbb{A}_{f}^{p},\text{Ta}^{p}(A)_{\mathbb{Q}})/U^{\prime p}$ is the étale sheaf on $S$ of $U^{\prime p}$ -orbits of isomorphisms

$$\begin{eqnarray}C\otimes \mathbb{A}_{f}^{p}\xrightarrow[{}]{{\sim}}\text{Ta}^{p}(A)_{\mathbb{ Q}}\end{eqnarray}$$