2.1 Hodge and Abelian type

Let $(G, b, \{\mu \})$ be a local Shimura datum over $\mathbb {Q}_p$ , cf. [Reference Rapoport and ViehmannRV14]. Recall that this means that G is a reductive group, that $b\in G(\breve {\mathbb {Q}}_p)$ and that $\{\mu \}$ is a conjugacy class of a minuscule cocharacter $\mu :\mathbb {G}_{m/\bar {\mathbb {Q}}_p}\to G_{\bar {\mathbb {Q}}_p}$ . It is assumed that b is neutral acceptable, i.e., the $\sigma $ -conjugacy class $[b]$ lies in $B(G, \mu ^{-1})$ . We will often omit the bracket and simply write $(G, b, \mu )$ .

We will be concerned with local Shimura data of a particular type.

Definition 2.1.1. The local Shimura datum $(G, b, \mu )$ is called of Hodge type if there is an embedding

$$\begin{align*}i: (G, \mu)\hookrightarrow (\mathrm{GL}_h, \mu_d), \end{align*}$$

for some $0\leq d\leq h$ , where $\mu _d(a)=\operatorname {\mathrm {diag}}(a^{(d)}, 1^{(h-d)})$ is the standard minuscule cocharacter of $\mathrm {GL}_h$ .

2.2 Quasi-parahoric subgroups

Let $\breve G$ be a reductive group over $\breve {\mathbb {Q}}_p$ . By definition, a quasi-parahoric subgroup of $\breve G(\breve {\mathbb {Q}}_p)$ is a subgroup $\breve K$ which is squeezed as

$$ \begin{align*}\breve G(\breve{\mathbb{Q}}_p)^0\cap \mathrm{Stab}_{\frak F}\subset \breve K\subset \breve G(\breve{\mathbb{Q}}_p)^1\cap \mathrm{Stab}_{\frak F}. \end{align*} $$

Here $\mathrm {Stab}_{\frak F}$ is the stabiliser of a facet $\frak F$ in the building $\mathscr {B} (\breve G_{\mathrm {ad}}, \breve {\mathbb {Q}}_p)$ , and

$$ \begin{align*}\breve G(\breve{\mathbb{Q}}_p)^0=\ker(\kappa: \breve G(\breve{\mathbb{Q}}_p)\to \pi_1(\breve G)_I), \quad \breve G(\breve{\mathbb{Q}}_p)^1=\ker(\kappa: \breve G(\breve{\mathbb{Q}}_p)\to \pi_1(\breve G)_I\otimes\mathbb{Q}). \end{align*} $$

Here, $\kappa $ is the Kottwitz homomorphism. Equivalently, it is a subgroup of finite index in $\breve G(\breve {\mathbb {Q}}_p)^1\cap \mathrm {Stab}_{{\frak F}}$ . Still another way of characterising quasi-parahoric subgroups is to say that they are of finite index in the stabiliser of a point in the extended building $\mathscr {B}^e(\breve G, \breve {\mathbb {Q}}_p)$ . In the case that the quasi-parahoric subgroup coincides with $\breve G(\breve {\mathbb {Q}}_p)^1\cap \mathrm {Stab}_{\mathbf {x}}$ , with $\mathbf {x}\in \mathscr {B}(\breve G_{\mathrm {ad}}, \breve {\mathbb {Q}}_p)$ (equivalently, if it coincides with the stabiliser subgroup in $\breve G(\breve {\mathbb {Q}}_p)$ of a point in the extended building $\mathscr {B}^e(\breve G, \breve {\mathbb {Q}}_p)$ ), it is called a stabiliser quasi-parahoric. The parahoric subgroup $\breve K^o=\breve G(\breve {\mathbb {Q}}_p)^0\cap \mathrm { Stab}_{\frak F}$ is called the parahoric subgroup associated to the quasi-parahoric subgroup $\breve K$ . Note that if $\pi _1(\breve G)_I$ is torsion-free, then any quasi-parahoric is a parahoric.

By Bruhat-Tits theory [Reference Bruhat and TitsBTII], there is a unique smooth group scheme $\breve {\mathcal {G}}$ over $\breve {\mathbb {Z}}_p$ with generic fibre $\breve G$ such that $\breve {\mathcal {G}} (\breve {\mathbb {Z}}_p)=\breve K$ .

Now let G be a reductive group over $\mathbb {Q}_p$ . Then, in analogy with the Bruhat-Tits definition of a $\mathbb {Q}_p$ -parahoric subgroup [Reference Bruhat and TitsBTII, right after Def. 5.2.6], there is the notion of a $\mathbb {Q}_p$ -quasi-parahoric subgroup of G. It can be equivalently defined as a quasi-parahoric subgroup $\breve K$ of $G(\breve {\mathbb {Q}}_p)$ which is invariant under Frobenius, or as a subgroup of finite index in $G(\breve {\mathbb {Q}}_p)^1\cap \mathrm {Stab}_{{\frak F}}$ , where $ \mathrm {Stab}_{\frak F}$ is the stabiliser of a facet $\frak F$ in the building $\mathscr {B} (G_{\mathrm {ad}}, \mathbb {Q}_p)$ . By descent from $\breve {\mathbb {Z}}_p$ to $\mathbb {Z}_p$ , we have a corresponding smooth group scheme ${\mathcal {G}} $ over $\operatorname {\mbox {Spec }}(\mathbb {Z}_p)$ . Then ${\mathcal {G}} (\mathbb {Z}_p)=\breve K\cap G(\mathbb {Q}_p)$ .

In the sequel, we simply call ${\mathcal {G}} $ a quasi-parahoric group scheme for G. Note that a parahoric group scheme for G is uniquely defined by its associated subgroup ${\mathcal {G}} (\mathbb {Z}_p)$ of $G(\mathbb {Q}_p)$ , cf. [Reference Bruhat and TitsBTII, Prop. 5.2.8]. However, for general quasi-parahoric group schemes for G, the association ${\mathcal {G}} \mapsto {\mathcal {G}} (\mathbb {Z}_p)$ ceases to be injective (for instance ${\mathcal {G}} ^o(\mathbb {Z}_p)= {\mathcal {G}} (\mathbb {Z}_p)$ when $\pi _0({\mathcal {G}} )^\phi $ is trivial, which may happen even when $\pi _0({\mathcal {G}} )$ is non-trivial). Still, we write $K={\mathcal {G}} (\mathbb {Z}_p)=\breve K\cap G(\mathbb {Q}_p)$ and sometimes use the notation K to refer to our choice of ${\mathcal {G}}$ . This does not lead to confusions.

2.3 The local model

RecallFootnote ¹ the Scholze-Weinstein v-sheaf local model $\mathbb {M}^v_{{\mathcal {G}}, \mu }$ over $\mathrm{Spd}(O_E)$ (in [Reference Scholze and WeinsteinSW20] it is denoted by $\mathrm {Gr}_{{\mathcal {G}}, \mathrm{Spd}(O_E),\leq \mu }$ ).

Assume $\mu $ is minuscule. In this case, $\mathrm {Gr}_{\mathcal {G}, \mathrm{Spd}(O_E),\leq \mu }=\mathrm {Gr}_{\mathcal {G}, \mathrm{Spd}(O_E), \mu }$ and the v-sheaf local model $\mathbb {M}^v_{\mathcal {G}, \mu }=\mathrm {Gr}_{\mathcal {G}, \mathrm{Spd}(O_E), \mu }$ is given as the closure inside the Beilinson-Drinfeld style affine Grassmannian $\mathrm {Gr}_{\mathcal {G}, \mathrm{Spd}(O_E)}$ of the v-sheaf $X{}_\mu {}^{\Diamond }$ associated to the symmetric space $X_\mu =\mathrm {Gr}_{G,\mu }$ of parabolics of type $\mu $ over $\operatorname {\mbox {Spec }}(E)$ . Then ${\mathbb{M}^{v}_{\mathcal {G}, \mu}}$ is representable by a normal flat projective $O_E$ -scheme ${\mathbb{M}^{\mathrm {loc}}_{\mathcal {G}, \mu}}$ with reduced special fibre. This was conjectured by Scholze-Weinstein [Reference Scholze and WeinsteinSW20, Conj. 21.4.1] and was shown in [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22] and [Reference Gleason and LourençoGLo24] (which settled the normality in some remaining cases in characteristics $2$ and $3$ ). Here ${\mathbb{M}^{\mathrm {loc}}_{\mathcal {G}, \mu}}$ is uniquely determined. Assume in addition that $(G, \mu )$ is of abelian type and satisfies Condition (A) or (B), cf. Definition 2.5.2 below. Then the normality of ${\mathbb{M}^{\mathrm {loc}}_{\mathcal {G}, \mu}}$ is given by [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Thm. 7.23]. Furthermore, in the case (A) ${\mathbb{M}^{\mathrm {loc}}_{\mathcal {G}, \mu}}$ can be obtained by the procedure of [Reference Pappas and ZhuPZ13], extended to restrictions of scalars from wild extensions in [Reference LevinLe01] and as modified in [Reference He, Pappas and RapoportHPR20] when $p\mid |\pi _1(G_{\mathrm {der}})|$ . This statement uses Remark 2.5.3. In the case (B) of Definition 2.5.2, ${\mathbb{M}^{\mathrm {loc}}_{\mathcal {G}, \mu }}$ can also be obtained by taking the closure of the generic fibre in the naive local model of [Reference Rapoport and ZinkRZ96]. There is no difference when discussing quasi-parahorics because the natural map is an isomorphism,

(2.3.1)

cf. [Reference Scholze and WeinsteinSW20, Prop. 21.4.3]. (This proposition assumes that $\mu $ is minuscule, but the argument applies to general $\mu $ , see also [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, §4]). If $\mu $ is minuscule, this also yields an isomorphism for the corresponding scheme local models,

2.4 The integral local Shimura variety $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$

Let ${\mathcal {G}} $ be a smooth affine group scheme over $\mathbb {Z}{}_p$ with generic fibre a reductive group G, let $b\in G(\breve {\mathbb {Q}}{}_p)$ , and let $\mu $ be a conjugacy class of cocharacters of G. It is assumed that the Frobenius-conjugacy class of b lies in $B(G, \mu ^{-1})$ . We call the triple $({\mathcal {G}}, b, \mu )$ an integral local shtuka datum and $(G, b, \mu )$ a rational local shtuka datum. As usual, we denote by E the field of definition of $\mu $ and by $\breve E$ the completion of the maximal unramified extension.

Then Scholze-Weinstein associate to $({\mathcal {G}}, b, \mu )$ an ‘integral’ moduli space of shtukas ${\mathcal {M}^{\mathrm { int}}_{\mathcal {G}, b, \mu }}$ over $O_{\breve E}$ , [Reference Scholze and WeinsteinSW20, Def. 25.1.1]. It is given as a ‘v-sheaf moduli space’ of certain ${\mathcal {G}}$ -shtukas with one leg bounded by $\mu $ with a fixed associated Frobenius element, cf. [Reference Scholze and WeinsteinSW20, §§23.1, 23.2, 23.3], as follows.

Definition 2.4.1. The integral moduli space of local shtuka ${\mathcal{M}^{\mathrm {int}}_{\mathcal {G}, b, \mu }}$ is the functor that sends $S\in \mathrm {Perfd}_k$ to the set of isomorphism classes of tuples

$$ \begin{align*} (S^\sharp, \mathscr{P}, \phi_{\mathscr{P}}, i_r) , \end{align*} $$

where

1. 1) $S^\sharp $ is an untilt of S over $\mathrm{Spa}( O_{\breve E})$ ,

2. 2) $(\mathscr {P}, \phi _{\mathscr {P}})$ is a ${\mathcal {G}}$ -shtuka over S with one leg along $S^\sharp $ bounded by $\mu $ ,

3. 3) $i_r$ is a ‘framing’, i.e., an isomorphism of G-torsors

   (2.4.1) $$ \begin{align} i_r: G_{\mathcal{Y}{}_{[r,\infty)}(S)}\xrightarrow{\sim} \mathscr{P}{}_{\, |\mathcal{Y}{}_{[r,\infty)}(S)} \end{align} $$
   for large enough r (for an implicit choice of pseudouniformiser $\varpi $ ), under which $\phi _{\mathscr {P}}$ is identified with $\phi _b=b\times \mathrm {Frob}_S$ .

Here, $\mathcal {Y}{}_{[r,\infty )}(S)$ is as defined in [Reference Scholze and WeinsteinSW20], [Reference Fargues and ScholzeFS21, II]. We have denoted by $G_{\mathcal {Y}{}_{[r,\infty )}(S)}$ the trivial G-torsor over ${\mathcal {Y}{}_{[r,\infty )}(S)}$ (denoted $G\times {\mathcal {Y}{}_{[r,\infty )}(S)}$ in [Reference Scholze and WeinsteinSW20, App. to §19]), and by

(2.4.2) $$ \begin{align} \phi_b=\phi_{G, b}\colon \phi^*(G_{\mathcal{Y}{}_{[r,\infty)}(S)})=G_{\mathcal{Y}{}_{[\frac{1}{p}r,\infty)}(S)}\xrightarrow{b\phi} G_{\mathcal{Y}{}_{[r,\infty)}(S)} \end{align} $$

the isomorphism given by the $\phi $ -linear isomorphism induced by right multiplication by b (denoted by $b\times \mathrm {Frob}$ in [Reference Scholze and WeinsteinSW20, Def. 23.1.1]). In 3) we mean more precisely an equivalence class, where $i_r$ and $i^{\prime }_{r'}$ are called equivalent if there exists $r"\geq r, r'$ such that ${i_r}_{\, |\mathcal {Y} {}_{[r",\infty )}(S)}={i^{\prime }_{r'}}_{\, |\mathcal {Y} {}_{[r",\infty )}(S)}$ . Also, in 2) the precise definition of “bounded by $\mu $ ” is given via the local model $\mathbb {M}{}^v_{{\mathcal {G}}, \mu }$ (see [Reference Scholze and WeinsteinSW20, Def. 25.1.1], [Reference Pappas and RapoportPR24, p. 46]).

Note that the formation of $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b,\mu }$ is functorial, i.e., for a morphism $({\mathcal {G}}, b, \mu )\to ({\mathcal {G}}', b', \mu ')$ we have $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b,\mu }\to \mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}', b',\mu '}\times _{\mathrm{Spd}(O_{\breve E})} \mathrm{Spd}(O_{\breve E'})$ . It is also compatible with products, i.e., $ \mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}{}_1\times {\mathcal {G}}{}_2, b_1\times b_2,\mu _1\times \mu _2}$ is isomorphic to the product $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}{}_1, b_1,\mu _1}\times \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}_2, b_2,\mu _2} $ after base changing to the compositum of the reflex fields (in this, we omit the base changes from the notation, for simplicity). In addition, the $\phi $ -centraliser group $J_b(\mathbb {Q} {}_p)$ acts on $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b,\mu }$ by changing the framing.

By loc. cit., $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ is a v-sheaf over $\mathrm{Spd}( O_{\breve E})$ . In fact, as in [Reference GleasonGl21, Prop. 2.23], $\mathcal {M} {}^{\mathrm{int}}_{{\mathcal {G}}, b, \mu }$ is a small v-sheaf. In addition, $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b,\mu }$ supports a Weil descent datum from $\mathcal{O} {}_{\breve E}$ down to $O_E$ , as explained in [Reference Pappas and RapoportPR24, §3.1, §3.2]. In this paper, even though this is not always pointed out explicitly, the smooth group scheme ${\mathcal {G}}$ will always be a quasi-parahoric group scheme for G.

In fact, most of the time we are interested in the case that $(G, b, \mu )$ is a local Shimura datum, i.e., we also assume that $\mu $ is minuscule. In addition, we take ${\mathcal {G}} $ to be a quasi-parahoric group scheme for G. In this case we call ${\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }}$ the integral local Shimura variety associated to the local Shimura datum $(G, b, \mu )$ and the quasi-parahoric group scheme ${\mathcal {G}} $ .

2.5 Statement of the main results

Our concern is with the following conjecture.

Conjecture 2.5.1 (Scholze).

Let $(G, b, \mu )$ be a local Shimura datum and ${\mathcal {G}} $ a quasi-parahoric group scheme for G. There exists a formal scheme $\mathscr {M} {}_{{\mathcal {G}}, b, \mu }$ which is normal and flat locally formally of finite type over $O_{\breve E}$ with

$$\begin{align*}\mathcal{M}{}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}=\mathscr{M} {}_{{\mathcal{G}}, b, \mu}^{\Diamond}\end{align*}$$

as v-sheaves over $\mathrm{Spd}(O_{\breve E})$ . Then, $\mathscr {M} {}_{{\mathcal {G}}, b, \mu }$ is unique ([Reference Scholze and WeinsteinSW20, Prop. 18.4.1]).

Our main result is a proof of this conjecture when $(G, b, \mu )$ is of abelian type under certain mild hypotheses.

Theorem 2.5.4 is closely related to the following result. In fact, it was shown in [Reference Pappas and RapoportPR24] that the following result implies Theorem 2.5.4 in the Hodge type case under some mild additional hypotheses, cf. [Reference Pappas and RapoportPR24, proof of Thm. 3.7.1].

Theorem 2.5.5. Let $(G, b, \mu )$ be a local Shimura datum of abelian type satisfying Condition (A) or (B), and let ${\mathcal {G}} $ be a quasi-parahoric group scheme for G. For any $\mathrm{Spd}(k)$ -valued point x of $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ , there is a k-valued point y of $\mathbb {M}{}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }$ such that

(2.5.1) $$ \begin{align} \mathcal{M}{}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu/x} \simeq (\mathbb{M}{}^{\mathrm{loc}}_{{\mathcal{G}},\mu/y})^{\Diamond}, \end{align} $$

as v-sheaves over $\mathrm{Spd}(O_{\breve E})$ .

In this, y is a k-valued point of $\mathbb {M}{}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }$ obtained by fixing a trivialisation of the ${\mathcal {G}} $ -torsor underlying the ${\mathcal {G}} $ -shtuka at x (more precisely, y is in the orbit $\ell (x)$ given by (3.4.2)). The notation $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu /x}$ stands for the formal completion of the v-sheaf $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ at x, as defined in [Reference GleasonGl25, Def. 4.18], cf. [Reference Pappas and RapoportPR24, §3.3.1], see also §3.4. On the RHS, $(\mathbb {M}{}^{\mathrm {loc}}_{{\mathcal {G}}, \mu /y})^{\Diamond }$ denotes the v-sheaf attached to the adic space $\mathrm{Spa}(A, A)$ , where A is the completion of the local ring of the scheme $\mathbb {M}{}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }$ at y, taken with the topology defined by the maximal ideal.

The combination of these two results also gives

Note here that, by the full-faithfulness result of [Reference Scholze and WeinsteinSW20, Prop. 18.4.1], the Weil descent datum for the v-sheaf $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b,\mu }$ gives a corresponding Weil descent datum for the formal scheme $\mathscr {M} {}_{{\mathcal {G}}, b, \mu }$ , from $\mathcal {O}{}_{\breve E}$ down to $O_E$ .

The following result shows that $\mathscr {M} {}_{{\mathcal {G}}, b, \mu }$ is an integral model of the local Shimura variety in a certain sense. Recall from the introduction that to ${\mathcal {G}}$ is associated the finite abelian group $\Pi _{{\mathcal {G}}}$ , and that to every $\bar {\beta }\in \Pi _{{\mathcal {G}}}$ , there is associated a quasi-parahoric group scheme ${\mathcal {G}}{}_\beta $ over $\mathbb {Z}{}_p$ , as well as its associated parahoric group scheme ${\mathcal {G}}{}^o_\beta $ , and the finite group of its connected components $\pi _0({\mathcal {G}}{}_\beta )$ with its action by the Frobenius $\phi $ . Also we let $K_\beta ={\mathcal {G}}{}_\beta (\mathbb {Z} {}_p)$ .

Theorem 2.5.7. Let $(G, b, \mu )$ be a local Shimura datum of abelian type satisfying Condition (A) or (B), and let ${\mathcal {G}}$ be a quasi-parahoric group scheme for G. There is an isomorphism of rigid-analytic varieties over $\breve E$ ,

$$ \begin{align*}\mathscr{M}{}_{{\mathcal{G}}, b, \mu}^{\mathrm{rig}}\simeq \bigsqcup_{\bar{\beta}\in\Pi_{{\mathcal{G}}}}\mathrm{Sht}_{K_\beta}(G, b, \mu). \end{align*} $$

This isomorphism is induced by an isomorphism of formal schemes:

$$ \begin{align*}\mathscr{M}{}_{{\mathcal{G}}, b, \mu}\simeq \bigsqcup_{\bar{\beta}\in\Pi_{{\mathcal{G}}}}{\mathscr{M}}_{{\mathcal{G}}{}^o_\beta, b, \mu}/\pi_0({\mathcal{G}}{}_\beta)^\phi. \end{align*} $$

We finally mention the following result. Let $(G, b, \mu )$ be a local shtuka datum, and let ${\mathcal {G}} $ be a quasi-parahoric group scheme for G. Consider the scheme-theoretic v-sheaf $(\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{\mathrm {red}} $ , cf. [Reference GleasonGl21]. Then $(\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{\mathrm {red}}$ is representable by a perfect k-scheme $X_{{\mathcal {G}}}(b, \mu ^{-1})$ which can be identified with the admissible set (3.3.2) contained in the Witt vector affine Grassmannian $X_{{\mathcal {G}}} =LG/L^+{\mathcal {G}} $ , cf. Proposition 3.3.1. The problem of determining the set of connected components of $X_{{\mathcal {G}}} (b, \mu ^{-1})$ is of considerable interest.

Let ${\mathcal {G}} $ be a parahoric. Then there is a surjective map

(2.5.2) $$ \begin{align} X_{{\mathcal{G}}} (b, \mu^{-1})\to c_{b, \mu}+\pi_1(G)_I^\phi , \end{align} $$

where $c_{b, \mu }\in \pi _1(G)_I$ is a certain element unique up to $\pi _1(G)_I^\phi $ , cf. [Reference He and ZhouHZ20, Lem. 6.1]. By a recent result of Gleason-Lim-Xu [Reference Gleason, Lim and XuGLX22, Thm. 1.2], the fibres of (2.5.2) are the connected components of $X_{{\mathcal {G}}} (b, \mu ^{-1})$ when $(b, \mu )$ is Hodge-Newton irreducible (proving a conjecture of X. He). The map (2.5.2) is equivariant for the natural action of the $\phi $ -centraliser group $J_b(\mathbb {Q} {}_p)$ on source and target. We prove the following result.

Composing (2.5.2) with the specialisation map (3.4.1) gives a $J_b(\mathbb {Q} {}_p)$ -equivariant map

(2.5.3) $$ \begin{align} \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}\to \underline{c_{b, \mu}+\pi_1(G)_I^\phi}, \end{align} $$

where the target is the corresponding constant v-sheaf. Hence, if the centre of G is connected, any two fibres of the map (2.5.3) are also isomorphic.

2.6 The plan of the proof

The proof of the main theorems proceeds in the following steps.

1. • Step 1. We develop a devissage procedure that allows us to pass

   a) between ${\mathcal {G}} $ and ${\mathcal {G}}{}^o$ .

   b) between groups linked via an ad-isomorphism.

   In most of the statements in this step, we deal with integral local shtuka data $({\mathcal {G}}, b, \mu )$ , i.e., we do not assume that $\mu $ is minuscule.

2. • Step 2. We show the results in ‘good’ Hodge type cases: In such cases, the proof follows from the case of $\mathrm {GL}_n$ (already treated in [Reference Scholze and WeinsteinSW20] by relating to RZ formal schemes) and from the constructions in [Reference Kisin and PappasKP18], [Reference Kisin and ZhouKZ21], [Reference Kisin, Pappas and ZhouKPZ24] (which use Zink displays and Breuil-Kisin modules).

3. • Step 3. Using Step 1 we reduce the general case to:

   a) the Hodge type cases handled in Step 2, (in case (A)),

   b) EL/PEL cases for which we give directly $\mathscr {M}{}_{{\mathcal {G}}, b,\mu }$ as a Rapoport-Zink formal scheme, (in case (B)).

2.7 The lay-out of the paper

In §3, we generalise the Anschütz purity theorem from parahoric group schemes to quasi-parahoric group schemes, and use this to transfer to this more general context the formalism of specialisation, formal completion and v-sheaf local model diagram that Gleason had established for hyperspecial parahoric ${\mathcal {G}}$ . We also make the link between the reduced locus of $ \mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ and affine Deligne-Lusztig varieties for quasi-parahorics. In §4, we study the devissage step of varying the quasi-parahorics with a given associated parahoric. We also prove at this point Theorem 2.5.7. In §5, we study the devissage step of varying the groups with a given adjoint group. In §6 we treat the case when $\mu _{\mathrm {ad}}$ is trivial. In §7 we consider the crucial Hodge type case mentioned in the introduction. In §7.2, we develop methods to relate Hodge type cases and abelian type cases to the crucial Hodge type case. Everything comes together in §8, where we give the proofs of Theorems 2.5.4 and 2.5.5; this is done separately for G satisfying Condition (A) and (B).

3.1 Torsors under quasi-parahoric group schemes

Let $\Omega =\Omega _G=\pi _1(G)_I$ (coinvariants of the algebraic fundamental group under the inertia group, cf., e.g., [Reference Pappas and RapoportPR08, §3]). We recall the Kottwitz homomorphism

$$ \begin{align*}\kappa\colon G(\breve{\mathbb{Q}}{}_p)\to \Omega_G. \end{align*} $$

A parahoric subgroup lies in the kernel of $\kappa $ and in fact, the kernel of $\kappa $ is generated by all parahoric subgroups, cf. [Reference Haines and RapoportHR08, Lem. 17]. Also, for any quasi-parahoric subgroup $\breve K$ of $G(\breve {\mathbb {Q}}{}_p)$ with associated parahoric subgroup $\breve K^o$ we have $\breve K^o=\ker (\kappa _{| \breve K}\colon \breve K\to \Omega )$ . Let ${\mathcal {G}}$ be the corresponding quasi-parahoric group scheme and consider the injective map

(3.1.1) $$ \begin{align} \pi_0({\mathcal{G}})=\breve K/\breve K^o\hookrightarrow \Omega_G. \end{align} $$

In [Reference Scholze and WeinsteinSW20, 25.3.1], Scholze-Weinstein point out the importance of the finite abelian group,

(3.1.2) $$ \begin{align} \Pi_{{\mathcal{G}}}:=\mathrm{ker}(\pi_0({\mathcal{G}})_\phi\to \Omega_\phi) \end{align} $$

(coinvariants under the action of $\phi $ ).

Lemma 3.1.1. There is a natural isomorphism

$$\begin{align*}\Pi_{{\mathcal{G}}} \cong \ker(\mathrm{H}^1_{{{{\acute{\mathrm{e}}}\mathrm{t}}}}(\mathbb{Z}_p, {\mathcal{G}})\to \mathrm{ H}^1_{{{{\acute{\mathrm{e}}}\mathrm{t}}}}(\mathbb{Q}_p, G)) \end{align*}$$

Proof. This is sketched in [Reference Scholze and WeinsteinSW20, 25.3]. The exact sequence $0\to \breve K^o\to \breve K\to \pi _0({\mathcal {G}} )\to 0$ induces an exact sequence

$$\begin{align*}\mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Z}{}_p, {\mathcal{G}}{}^o)\to \mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Z}{}_p, {\mathcal{G}})\to \mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Z}{}_p, \pi_0({\mathcal{G}})). \end{align*}$$

Here the left term vanishes by Lang’s theorem. The second map is surjective since $\mathrm {H}^1_{{\text {\'{e}t}}}(\mathbb {Z} {}_p, {\mathcal {G}} )=\breve K/_{\!\!\phi }\breve K$ . We therefore obtain

$$ \begin{align*}\mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Z} {}_p, {\mathcal{G}} )\simeq \mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{F} {}_p, \pi_0({\mathcal{G}} ))\simeq \mathrm{H}^1(\langle \phi\rangle, \pi_0({\mathcal{G}} ))\cong \pi_0({\mathcal{G}} )_\phi. \end{align*} $$

On the other hand, we have an injection

(3.1.3) $$ \begin{align} \mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Q} {}_p, G)\hookrightarrow \mathrm{H}^1(\langle \phi\rangle, G(\breve{\mathbb{Q}} {}_p))\cong B(G) \end{align} $$

with the image landing in the basic elements $B(G)_{\mathrm {basic}}\subset B(G)$ , cf. [Reference KottwitzKo97]. Now

$$\begin{align*}B(G)_{\mathrm{basic}}\simeq \pi_1(G)_\Gamma=(\pi_1(G)_I)_\phi=\Omega_\phi, \end{align*}$$

and the map (3.1.3) induces an identification

$$\begin{align*}\mathrm{H}^1_{{\text{\'{e}t}}}(\mathbb{Q}{}_p, G)\simeq \Omega_{\phi, \mathrm{tors}} , \end{align*}$$

where $\Omega _{\phi , \mathrm {tors}}$ is the torsion subgroup of $\Omega _\phi $ . It remains to observe that the following diagram is commutative,

(3.1.4)

This commutativity follows from the fact that the vertical homomorphisms on the outer left and the outer right are both induced by the Kottwitz homomorphism $G(\breve {\mathbb {Q}} {}_p)\to \Omega $ : this is obvious from the construction for the outer left map and follows from [Reference KottwitzKo97, §7.5] for the outer right map.

3.2 Purity of torsors

The following purity/extension result is crucial. It is a quick generalisation of the corresponding result for parahoric subgroups due to Anschütz ([Reference AnschützAn22]).

Proof. We have an exact sequence

$$\begin{align*}1\to {\mathcal{G}} {}^o\to {\mathcal{G}} \to i_*(\pi_0({\mathcal{G}} ))\to 1 \end{align*}$$

of étale sheaves, where $i: \operatorname {\mbox {Spec }}(k)\hookrightarrow \operatorname {\mbox {Spec }}(\breve {\mathbb {Z}} {}_p)$ . Note $U\times _{\operatorname {\mbox {Spec }}(\breve {\mathbb {Z}} {}_p)}\operatorname {\mbox {Spec }}(k)=\operatorname {\mbox {Spec }}(C^+[1/\varpi ])=\operatorname {\mbox {Spec }}(C)$ . This gives the exact sequence

$$\begin{align*}\mathrm{H}^1(U, {\mathcal{G}} {}^o)\to \mathrm{H}^1(U, {\mathcal{G}} )\to \mathrm{H}^1(U, i_*(\pi_0({\mathcal{G}} )))=\mathrm{H}^1(\operatorname{\mbox{Spec }}(C), \pi_0({\mathcal{G}} ))=(0). \end{align*}$$

By Anschütz’s theorem [Reference AnschützAn22], $\mathrm {H}^1(U, {\mathcal {G}} {}^o)=(0)$ and the result follows.

This extends as follows to strictly totally disconnected affinoid perfectoids which are given as ‘products of points’.

Proof. In the parahoric case ${\mathcal {G}} ={\mathcal {G}} {}^o$ , the extension follows from [Reference AnschützAn22, Prop. 11.5], see also [Reference GleasonGl21, Thm. 2.8]. Observe that all étale covers of

$$\begin{align*}\operatorname{\mbox{Spec }}(R^+)=\operatorname{\mbox{Spec }}(\prod_{i\in I}C_i^+ ) \end{align*}$$

split. Indeed, as in the proof of [Reference Bhatt and ScholzeBS17, Lem. 6.2], we can consider

$$\begin{align*}\operatorname{\mbox{Spec }}(\prod_{i\in I}C_i^+)\to \pi_0(\operatorname{\mbox{Spec }}(\prod_i C_i^+))=\pi_0(\operatorname{\mbox{Spec }}(\prod_i k_i))=\beta I \end{align*}$$

where $k_i=C_i^+/\mathfrak {m} {}_i$ is the (algebraically closed) residue field. Here, $\beta I$ is the Stone-Čech compactification of the discrete set I. Each connected component of $\operatorname {\mbox {Spec }}(R^+)$ (i.e., fibre of this map) is the spectrum of a valuation ring V with algebraically closed fraction field K; this valuation ring is an ultraproduct of $C^+_i$ . It follows that all étale covers of $\operatorname {\mbox {Spec }}(V)$ and then also of $\operatorname {\mbox {Spec }}(R^+)$ split. (We see that $R^+$ is ‘strictly w-local’ in the terminology of [Reference Bhatt and ScholzeBS15, 2.2], and, in fact, $\operatorname {\mbox {Spec }}(R^+)$ is ‘w-contractible’, [Reference Bhatt and ScholzeBS15, Lem. 2.4.8].)

A similar picture holds for $R=R^+[1/\varpi ]$ : in this case, we have

$$\begin{align*}\pi_0(\operatorname{\mbox{Spec }}(R))\simeq \pi_0(\operatorname{\mbox{Spec }}(R^+))\simeq \beta I\end{align*}$$

(by considering idempotents) and

$$\begin{align*}\operatorname{\mbox{Spec }}(R)\to \pi_0(\operatorname{\mbox{Spec }}(R))\simeq \beta I, \end{align*}$$

has every fibre isomorphic to $\operatorname {\mbox {Spec }}(V[1/\varpi ])$ , with $V[1/\varpi ]$ a valuation ring with (algebraically closed) fraction field K. Étale covers over each such $\operatorname {\mbox {Spec }}(V[1/\varpi ])$ split, and then étale covers over $\operatorname {\mbox {Spec }}(R)$ also split.

Now it also follows that every ${\mathcal {G}} $ -torsor over $\operatorname {\mbox {Spec }}(W(R^+))$ is trivial. Indeed, since ${\mathcal {G}} $ is smooth and $W(R^+)$ is p-adically complete, it is enough to show that all ${\mathcal {G}} $ -torsors over $R^+=\prod _{i\in I} C^+_i$ are trivial. As above, we see that all étale covers of $\operatorname {\mbox {Spec }}(R^+)$ split and so this follows using the smoothness of ${\mathcal {G}} $ ; the same applies of course to ${\mathcal {G}} {}^o$ -torsors. The argument in the proof of Proposition 3.2.1 above now extends to $U=\operatorname {\mbox {Spec }}(W(R^+))\setminus V(p,[\varpi ])$ , to complete the proof.

3.3 The affine Witt Grassmannian and affine Deligne-Lusztig varieties

Let

(3.3.1) $$ \begin{align} X_{{\mathcal{G}}} :=\mathrm{Gr}^W_{{\mathcal{G}}} =LG/L^+{\mathcal{G}} \end{align} $$

be the Witt vector affine Grassmannian for ${\mathcal {G}} $ , cf. [Reference Bhatt and ScholzeBS17, Reference ZhuZhu17]. Here $LG(R)=G(W(R)[\frac {1}{p}])$ and $L^+{\mathcal {G}} (R)={\mathcal {G}} (W(R))$ for any perfect k-algebra R. Note that here ${\mathcal {G}} $ is only assumed to be a quasi-parahoric group scheme, whereas in loc. cit. it is assumed that ${\mathcal {G}} $ is a parahoric group scheme. Also, note that k-points of $X_{{\mathcal {G}}} $ are given by isomorphism classes of pairs $(\mathcal {P}, \alpha )$ of a ${\mathcal {G}} $ -torsor $\mathcal {P} $ over $W(k)$ with a trivialisation $\alpha $ of its restriction to $W(k)[1/p]$ .

Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is a quasi-parahoric group scheme for G. Inside the Witt vector affine Grassmannian we consider the affine Deligne-Lusztig variety $X_{{\mathcal {G}} }(b, \mu ^{-1})$ (the $\mu ^{-1}$ -admissible locus). This is a perfect scheme which is locally (perfectly) of finite type (cf. [Reference ZhuZhu17], [Reference Hamacher and ViehmannHV20]) over k with

(3.3.2) $$ \begin{align} X_{{\mathcal{G}} }(b, \mu^{-1})(k)=\{g\breve K\in G(\breve{\mathbb{Q}} {}_p)/\breve K\mid g^{-1}b\phi(g)\in \mathrm{ {Adm}}^K(\mu^{-1})\}\subset X_{{\mathcal{G}}} (k). \end{align} $$

Here $\mathrm {{Adm}}^K(\mu ^{-1})=\breve K\mathrm {{Adm}}(\mu ^{-1})\breve K$ , where $\mathrm {{Adm}}(\mu ^{-1})\subset \widetilde W$ is the $\mu ^{-1}$ -admissible subset of the Iwahori Weyl group of G.

Proposition 3.3.1. Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is a quasi-parahoric group scheme for G. Then the reduced locus $(\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{\mathrm {red}} $ is represented by the perfect k-scheme $X_{{\mathcal {G}} }(b, \mu ^{-1}) $ , and hence

$$\begin{align*}\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}(\mathrm{Spd}(k))=X_{{\mathcal{G}}}(b, \mu^{-1})(k). \end{align*}$$

3.4 The specialisation map and formal completions

Gleason explains certain conditions on a small v-sheaf $\mathcal {F} $ to construct a continuous specialisation map on the underlying spaces,

$$ \begin{align*} { \mathrm sp}_{\mathcal{F}} \colon |\mathcal{F} |\to |\mathcal{F} {}_{\mathrm{red}} | , \end{align*} $$

cf. [Reference GleasonGl25, §4.2], see also [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, §2.3].

Furthermore, he proves that these conditions are satisfied when $\mathcal {F} =\mathbb {M} {}^v_{{\mathcal {G}}, \mu }$ with ${\mathcal {G}} $ reductive (hyperspecial parahoric). This result is extended to parahoric ${\mathcal {G}} $ by [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Prop. 4.14]. In view of (2.3.1), it also holds for quasi-parahoric ${\mathcal {G}} $ . The scheme-theoretic v-sheaf $(\mathbb {M} {}^v_{{\mathcal {G}}, \mu })_{\mathrm {red}} $ is represented by a perfect k-scheme (by [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22] this is the $\mu $ -admissible locus) which is a closed subscheme of the Witt affine Grassmannian $X_{{\mathcal {G}}} =\mathrm {Gr}^W_{{\mathcal {G}}} $ .

By [Reference GleasonGl21, Prop. 2.30], these conditions are also satisfied in the case when $\mathcal {F} =\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ where ${\mathcal {G}} $ is a hyperspecial parahoric subgroup. Again, one main ingredient is the Anschütz purity theorem for ${\mathcal {G}} $ and its extension to product of points. By Propositions 3.2.1 and 3.2.2, these purity statements remain true for quasi-parahoric group schemes, and the proof extends.

In fact, by Proposition 3.3.1, $(\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{\mathrm {red}} $ is represented by the affine Deligne-Lusztig variety (ADLV) $X_{{\mathcal {G}}} (b, \mu ^{-1})$ in the Witt affine Grassmannian $X_{{\mathcal {G}}} $ . Hence we get a continuous map

(3.4.1) $$ \begin{align} \mathrm{sp}_{\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}}\colon |\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}|\to |(\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu})_{\mathrm{red}}|=|X_{{\mathcal{G}}}(b,\mu^{-1})|. \end{align} $$

Using this, we define the formal completion of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ along a point $x\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))=X_{{\mathcal {G}}} (b, \mu ^{-1})(k)$ as the sub-v-sheaf of $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ , with

$$ \begin{align*}\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu/x}(S)=\{y\colon S\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}\mid { \mathrm sp}_{\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}}\circ y(|S|)\subset \{x\} \}. \end{align*} $$

Similarly, we can define the formal completion $\mathbb {M} {}^v_{{\mathcal {G}}, \mu /y}$ of $\mathbb {M} {}^v_{{\mathcal {G}}, \mu }$ along a point

$$\begin{align*}y\in \mathbb{M} {}^v_{{\mathcal{G}} , \mu}(\mathrm{Spd}(k))\subset X_{{\mathcal{G}}}(k)=G(W(k)[1/p])/{\mathcal{G}}(W(k)). \end{align*}$$

Using the condition ‘bounded by $\mu $ ’, we obtain a map

(3.4.2) $$ \begin{align} \ell: \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}(\mathrm{Spd}(k))\to {\mathcal{G}}(W(k))\backslash \mathbb{M} {}^v_{{\mathcal{G}} , \mu}(\mathrm{Spd}(k)). \end{align} $$

The set of orbits which appears as the target of $\ell $ is the $\mu $ -admissible set for K. The image of a point in $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))$ under $\ell $ is obtained by choosing a trivialisation of the ${\mathcal {G}}$ -shtuka and taking the coset given by the inverse of the Frobenius map.

If $\mu $ is minuscule, then $\mathbb {M}{}^v_{{\mathcal {G}}, \mu }$ is representable by the $O_E$ -scheme $\mathbb {M} {}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }$ ([Reference Anschütz, Gleason, Lourenço and RicharzAGLR22]), and the formal completion $\mathbb {M} {}^v_{{\mathcal {G}}, \mu /y}$ is given as $\mathrm{Spd}(A, A)$ , where A is the completion of the local ring of $\mathbb {M} {}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }\otimes _{O_E}{\breve O_E}$ at the corresponding point, taken with the topology given by the maximal ideal. In this case, we also have

$$\begin{align*}{\mathcal{G}} (W(k))\backslash \mathbb{M} {}^v_{{\mathcal{G}} , \mu}(\mathrm{Spd}(k))={\mathcal{G}} (k)\backslash \mathbb{M} {}^{\mathrm{loc}}_{{\mathcal{G}} , \mu}(k), \end{align*}$$

so $\ell (x)$ above can be also considered as a ${\mathcal {G}} (k)$ -orbit in $\mathbb {M} {}^{\mathrm {loc}}_{{\mathcal {G}}, \mu }(k)$ .

3.5 Change of base point

Note that, when $b\in \mathrm {Adm}^K(\mu ^{-1})$ , then $\mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ has a canonical $\mathrm{Spd}(k)$ -valued ‘base point’ $x_0$ . Under the identification $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))=X_{{\mathcal {G}}} (b, \mu ^{-1})(k)\subset X_{{\mathcal {G}}}(k)$ it corresponds to the ‘base point’ of the ADLV given by the trivial coset.

In general, let $x\in \mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))=X_{{\mathcal {G}}} (b, \mu ^{-1})(k)\subset X_{{\mathcal {G}}} (k)$ correspond to the isomorphism class of a pair $(\mathcal {P}, \alpha )$ , where $\mathcal {P} $ is a ${\mathcal {G}} $ -torsor over $W(k)$ and

$$\begin{align*}\alpha: \mathcal{P} [1/p] \xrightarrow{\sim} {\mathcal{G}} \times \operatorname{\mbox{Spec }}(W(k)[1/p]) \end{align*}$$

is a trivialisation of the restriction $\mathcal {P} [1/p]$ of $\mathcal {P} $ to $\operatorname {\mbox {Spec }}(W(k)[1/p])$ such that

$$\begin{align*}\phi_{\mathcal{P}} =\alpha^{-1}\cdot \phi_b\cdot \phi^*(\alpha): \phi^*(\mathcal{P} )[1/p]\xrightarrow{\sim} \mathcal{P}[1/p] \end{align*}$$

has pole at $p=0$ bounded by $\mu $ . Choose a trivialisation of the ${\mathcal {G}} $ -torsor $\mathcal {P} $ over $W(k)$ . Then $\alpha $ is given by $g\in {\mathcal {G}} (W(k)[1/p])$ and $\phi _{\mathcal {P}} $ by $b_x\times \phi \colon {\mathcal {G}} [1/p]\to {\mathcal {G}} [1/p]$ such that $b_x=g^{-1}b\phi (g)$ . Hence we have $b_x=g^{-1}b\phi (g)\in \mathrm { Adm}^K(\mu ^{-1})$ . We obtain an isomorphism

$$\begin{align*}\tau_g\colon \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b_x, \mu} ,\quad (\mathscr{P}, \phi_{\mathscr{P}}, i_r)\mapsto (\mathscr{P} , \phi_{\mathscr{P}} , i_r\circ g^{-1}) \end{align*}$$

which sends x to the base point $x_0$ of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b_x, \mu }$ , cf. [Reference Pappas and RapoportPR24, proof of Prop. 3.4.1]. This gives an isomorphism (depending on our choices)

(3.5.1) $$ \begin{align} \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu/x}\simeq \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b_x, \mu/x_0}. \end{align} $$

In particular, $ \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu /x}$ is representable (by a normal complete Noetherian local ring) for given ${\mathcal {G}} $ and any b in a given $\phi $ -conjugacy class in $B(G)$ and arbitrary $x\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))$ if and only if this holds for the base point $x_0$ of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ , for given ${\mathcal {G}} $ and any $b\in \mathrm {Adm}^K(\mu ^{-1})$ in the given $\phi $ -conjugacy class. Note that, by He’s theorem [Reference HeHe16], any $\phi $ -conjugacy class $[b]\in B(G, \mu ^{-1})$ contains elements $b\in \mathrm {Adm}^K(\mu ^{-1})$ .

3.6 A v-sheaf ‘local model diagram’

Let $\mathcal {H} $ be an affine group scheme over $\breve {\mathbb {Z}} {}_p$ . For an affinoid perfectoid $(R, R^+)$ over k, consider

$$\begin{align*}\mathbb{W} {}^+\mathcal{H} (\mathrm{Spa}(R, R^+))=\mathcal{H} (W(R^+)), \end{align*}$$

and

$$\begin{align*}\widehat{\mathbb{W}} {}^+\mathcal{H} (R, R^+)=\{h\in \mathcal{H} (W(R^+))\ |\ h\equiv 1\ \mathrm{mod}\, [\varpi_h]\}, \end{align*}$$

( $\varpi _h$ stands for some pseudouniformiser of $R^+$ that depends on h).

These definitions extend to perfectoid spaces over k and define v-sheaves of groups $\mathbb {W} {}^+\mathcal {H} $ and $ \widehat {\mathbb {W}} {}^+\mathcal {H} $ on $\mathrm {Perfd}_k$ . We set

$$\begin{align*}\widehat {L_W^+}\mathcal{H} = \widehat{\mathbb{W}} {}^+\mathcal{H} \times \mathrm{Spd}(\mathbb{Z} {}_p) \end{align*}$$

which is a v-sheaf of groups over $\mathrm{Spd}(\mathbb {Z} {}_p)$ . (This definition appears in [Reference GleasonGl25, 2.3.15].) We have

(3.6.1) $$ \begin{align} \widehat {L_W^+}\mathcal{H} (S)=\{((S^\sharp, y), h)\ |\ h\in \mathcal{H} (W(R^+)),\ h\equiv 1\ {\mathrm {mod}}\, [\varpi_h] \}, \end{align} $$

where $S=\mathrm{Spa}(R, R^+)$ and $(S^\sharp , y)$ is an S-valued point of $\mathrm{Spd}(\mathbb {Z} {}_p)$ , and where $\varpi _h$ is a pseudouniformiser of $R^+$ (that depends on h). Here, $S^\sharp =\mathrm{Spa}(R^\sharp ,R^{\sharp +})$ together with $y:(S^\sharp )^\flat \xrightarrow {\sim } S$ is an untilt of S, see [Reference Scholze and WeinsteinSW20, Prop. 11.3.1].

We will later need the following lemma.

Proof. The ring $W(R^+)$ is complete and separated for the $[\varpi ]$ –topology, where $\varpi $ is a pseudouniformiser of $R^+$ . Set inductively $\lambda _0=1$ , $\eta _0=h^{-1}$ , and

$$\begin{align*}\eta_n=\lambda^{-1}_n\phi(\lambda_n)\cdot h^{-1},\quad \lambda_{n+1}=\lambda_n\cdot\eta_n, \end{align*}$$

Then we have

$$\begin{align*}\eta_{n+1}=h\cdot \phi(\eta_n)\cdot h^{-1}. \end{align*}$$

Since $\eta _0\equiv 1\ \mathrm {mod}\, [\varpi _h]$ , this gives $\eta _n\equiv 1\ \mathrm {mod}\, [\varpi ^{p^n}_h]$ . Hence, $\eta _n$ converges to $1$ and $\lambda _n$ converges to an element $\lambda $ with

$$\begin{align*}h=\lambda^{-1}\phi(\lambda), \end{align*}$$

(cf. [Reference GleasonGl21, Lem. 2.15].) If $\lambda =\phi (\lambda )$ with $\lambda \equiv 1\ \mathrm {mod}\, [\varpi ]$ , then we easily see inductively $\lambda \equiv 1\ \mathrm {mod}\, [\varpi ^{p^n}]$ for all n, so $\lambda =1$ , so uniqueness follows.

Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is quasi-parahoric. We define as follows a functor $\widehat {L{\mathcal {G}} }_{b, \mu }$ on $\mathrm {Perfd}_k$ over $\mathrm{Spd}(O_{\breve E})$ , cf. [Reference GleasonGl21, §2.4]. It assigns to an affinoid perfectoid $S=\mathrm{Spa}(R, R^+)$ over k the set

$$\begin{align*}\widehat{L{\mathcal{G}} }_{b, \mu}(S)=\{\mathrm{isomorphism\ classes\ of}\ ((S^\sharp, y), \mathcal{P} , \psi, \sigma)\} \end{align*}$$

where

• • $(S^\sharp , y)$ is an untilt of S over $ O_{\breve E}$ ,

• • $\mathcal {P} $ is a ${\mathcal {G}} $ -torsor over $\operatorname {\mbox {Spec }}(W(R^+))$ ,

• • $ \psi : \mathcal {P} [1/\xi _{R^\sharp }]\xrightarrow {\sim } {\mathcal {G}} \times \operatorname {\mbox {Spec }}(W(R^+)[1/\xi _{R^\sharp }])$ , and $\sigma : \mathcal {P} \xrightarrow {\sim } \phi ^*({\mathcal {G}} \times \operatorname {\mbox {Spec }}(W(R^+))) $

  are both ${\mathcal {G}} $ -isomorphisms such that:

  1) $(\mathcal {P}, \psi )$ is bounded by $\mu $ along $\xi _{R^\sharp }=0$ ,

  2) there is a pseudouniformiser $\varpi \in R^+$ such that $\phi _b \circ \sigma \equiv \psi \ \mathrm {mod}\, [\varpi ]$ .

Here, we denote by $\xi _{R^\sharp }$ a generator of the map $W(R^+)\to R^{\sharp +}$ given by the untilt $(S^\sharp , y)$ of S. Also ‘bounded by $\mu $ ’ means, by definition, that the point of the ${\mathbb {B} }_{\mathrm {dR}}$ -affine Grassmannian $\mathrm {Gr}_{{\mathcal {G}}, \mathrm{Spd}(\breve O_E)}$ given by $((S^\sharp , y), \mathcal {P}, \psi )$ factors through the v-sheaf local model $\mathbb {M}{}^v_{{\mathcal {G}}, \mu }\subset \mathrm {Gr}_{{\mathcal {G}}, \mathrm{Spd}(\breve O_E)}$ . More precisely, choose (locally) a section $\tau : {\mathcal {G}} \xrightarrow {\sim }\mathcal {P} $ and consider $g=\psi \circ \tau (1)$ ; we ask that $g^{-1}{\mathcal {G}} ({\mathbb {B} }^+_{\mathrm {dR}}(S^\sharp ))$ lies in $\mathbb {M} {}^v_{{\mathcal {G}}, \mu }(R^\sharp )$ .

As in [Reference GleasonGl21], we can see that $\widehat {L{\mathcal {G}}}_{b, \mu }$ is a v-sheaf over $\mathrm{Spd}(O_{\breve E})$ (this v-sheaf is denoted by $\widehat {\mathrm {WSht}}^{\mathcal {D}, \leq \mu }$ in [Reference GleasonGl21, Def. 2.34 (3), (4)], and by $L\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu /x_0}$ in [Reference Pappas and RapoportPR24]).

It is useful to observe that by using the mapping

$$\begin{align*}(\mathcal{P}, \psi, \sigma)\mapsto h=(\psi\circ \sigma^{-1})(1)\in {\mathcal{G}} (W(R^+)[1/\xi_{R^\sharp}]) , \end{align*}$$

we obtain the following simpler description:

(3.6.2) $$ \begin{align} \widehat{L{\mathcal{G}}}_{b, \mu}(S)=\{((S^\sharp, y), h)\ |\ h\in {\mathcal{G}} (W(R^+)[1/\xi_{R^\sharp}]),\ h \equiv b\ {\mathrm {mod}}\, [\varpi_h], \ [h^{-1}]\in \mathbb{M} {}^v_{{\mathcal{G}}, \mu}(S)\}, \end{align} $$

where $(S^\sharp , y)$ is an untilt of S over $ O_{\breve E}$ , $\varpi _h$ is a pseudouniformiser of $R^+$ , and $[h^{-1}]$ is the S-point of the ${\mathbb {B}}_{\mathrm {dR}}$ -affine Grassmannian $\mathrm {Gr}_{{\mathcal {G}}, \mathrm{Spd}(\breve O_E)}$ defined by the coset $h^{-1} {\mathcal {G}} (\mathbb {B} {}^+_{\mathrm {dR}}(R^\sharp ))$ .

Theorem 3.6.2. There is a diagram of v-sheaves over $\mathrm{Spd }(\breve O_E)$

(3.6.3)

where both $\pi _{\bullet }$ , $\pi _\star $ are $\widehat {{L}^+_W{\mathcal {G}}}$ -torsors (for the v-topology) for two corresponding actions (see (d) below).

a) Here $x_0\in \mathcal {M}{}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(\mathrm{Spd}(k))$ denotes the base point as in §3.5 above. Similarly $y_0\in \mathbb {M} {}^v_{{\mathcal {G}}, \mu }(\mathrm{Spd}(k))$ denotes the corresponding point of $\mathbb {M} {}^v_{{\mathcal {G}}, \mu }(\mathrm{Spd}(k))\subset X_{{\mathcal {G}} }(k)$ given by the pair $({\mathcal {G}}, b^{-1})$ , i.e., by the coset $b^{-1}\,{\mathcal {G}} (W(k))$ . Also, $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu /x_0}$ , resp. $\mathbb {M} {}^v_{{\mathcal {G}}, \mu /y_0}$ , denotes the v-sheaf given by the formal completion of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ , resp. $\mathbb {M} {}^v_{{\mathcal {G}}, \mu }$ , at these points. (See §3.4 above.)

b) The map $\pi _\star $ is given by

$$\begin{align*}\pi_\star(((S^\sharp, y), \mathcal{P}, \psi, \sigma))=((S^\sharp, y), \mathcal{P}, \psi). \end{align*}$$

c) The map $\pi _\bullet $ is given by

$$\begin{align*}\pi_\bullet((S^\sharp, y), \mathcal{P} \psi, \sigma)=((S^\sharp, y), (\mathscr{P} ,\phi_{\mathscr{P} }), i_r). \end{align*}$$

Here, $(\mathscr {P}, \phi _{\mathscr {P} })$ is the ${\mathcal {G}} $ -shtuka over S with leg at y given as follows: The ${\mathcal {G}} $ -torsor $\mathscr {P} $ is $(\phi ^{-1})^*(\mathcal {P} )$ restricted to $\mathcal {Y} {}_{[0,\infty )}(S)$ , i.e.,

$$\begin{align*}\mathscr{P} =(\phi^{-1})^*(\mathcal{P} )_{|\mathcal{Y} {}_{[0,\infty)}(S)}. \end{align*}$$

The Frobenius $\phi _{\mathscr {P} }$ is given by a similar restriction of the composition

$$\begin{align*}\Phi: \mathcal{P} [1/\xi_{R^\sharp}]\xrightarrow{\psi} {\mathcal{G}} \times \operatorname{\mbox{Spec }}(W(R^+)[1/\xi_{R^\sharp}])\xrightarrow{((\phi^{-1})^*\sigma)^{-1}} (\phi^{-1})^*(\mathcal{P} )[1/\xi_{R^\sharp}]. \end{align*}$$

(Note that $\phi ^*(\mathscr {P} )=\mathcal {P} {}_{|\mathcal {Y} {}_{[0,\infty )}(S)}$ .) The framing $i_r$ is constructed in [Reference GleasonGl21].

d) The v-sheaf of groups $\widehat {{L}^+_W{\mathcal {G}} }$ acts on $\widehat {L{\mathcal {G}} }_{b, \mu } $ on the right by

$$\begin{align*}( \mathcal{P} , \psi, \sigma)\star g=( \mathcal{P} , \psi, \phi^*(r_{\phi^{-1}(g)})\circ \sigma), \quad ( \mathcal{P}, \psi, \sigma)\bullet g= ( \mathcal{P}, r_{g}\circ \psi, \phi^*(r_{g})\circ\sigma), \end{align*}$$

where, $r_a$ is the map given by right multiplication by a, and, for simplicity, we omit the untilt $(S^\sharp , y)$ from the notation.

Note that the composition $ \Phi : \mathcal {P} [1/\xi _{R^\sharp }] \to (\phi ^{-1})^*(\mathcal {P})[1/\xi _{R^\sharp }] $ in c), is fixed under the $\bullet $ -action. This follows from the identity

$$\begin{align*}((\phi^{-1})^*(\phi^*(r_{g})\circ \sigma))^{-1}\circ (r_g\circ \psi)=(r_g\circ (\phi^{-1})^*\sigma)^{-1}\circ r_g\circ \psi=((\phi^{-1})^*\sigma)^{-1}\circ\psi. \end{align*}$$

It is convenient to express the torsors $\pi _\star $ , $\pi _\bullet $ , using the description (3.6.2) of $\widehat {L{\mathcal {G}} }_{b, \mu } $ . In the description (3.6.2), $\pi _\star $ is given by projection to the coset $[h^{-1}]=h^{-1} {\mathcal {G}} (\mathbb {B} {}^+_{\mathrm {dR}}(R^\sharp ))$ , and $\pi _\bullet $ is given by sending h to the shtuka $\mathscr {P} (h)={\mathcal {G}} \times \mathcal {Y} {}_{[0,\infty )}(S)={\mathcal {G}} {}_{ \mathcal {Y} {}_{[0,\infty )}(S)}$ (the trivial torsor) with Frobenius defined by $\phi _{\mathscr {P} (h)}=\phi _h=r_h\phi : (\phi ^*{\mathcal {G}} {}_{\mathcal {Y} {}_{[0,\infty )}(S)})[1/\xi _{R^\sharp }]\xrightarrow {\sim } {\mathcal {G}} {}_{\mathcal {Y} {}_{[0,\infty )}(S)}[1/\xi _{R^\sharp }]$ . To check this description of $\pi _\bullet $ , note that, since $\phi _h=r_h \phi = \psi \circ \sigma ^{-1}$ , and $\Phi =((\phi ^{-1})^*(\sigma ))^{-1}\circ \psi $ , the following diagram commutes

Hence, the ${\mathcal {G}} $ -shtukas $(\mathscr {P} (h), \phi _{\mathscr {P} (h)})$ and $(\mathscr {P}, \phi _{\mathscr {P}} )$ given above, are isomorphic.

Under the above isomorphism, the framing $i_r$ of $(\mathscr {P}, \phi _{\mathscr {P}} )$ corresponds to a framing $i_r(h)$ of $(\mathscr {P} (h), \phi _{\mathscr {P} (h)})$ ; this is given by the unique lift of the identity trivialisation modulo $[\varpi _h]$ , cf. Lemma 3.6.1. More precisely, for $((S^\sharp , y), h)\in \widehat {L{\mathcal {G}} }_{b, \mu }(S)$ , the framing of $(\mathscr {P} (h), \phi _{\mathscr {P} (h)})$ is given by the unique element $ i_r(h)\in G(B^{[r,\infty )}_{(R, R^+)})$ with $i(h) \equiv 1\ {\mathrm {mod}}\, [\varpi _h]$ and the property

$$\begin{align*}h=i_r(h)^{-1}\cdot b\cdot \phi(i_r(h)), \end{align*}$$

(see [Reference GleasonGl21, Lem. 2.15].) Finally, the actions correspond to $h\star g=g^{-1}h$ and $h\bullet g=\phi (g)^{-1}h g$ . Note that $[(h\star g)^{-1}]=[(g^{-1}h)^{-1}]=[h^{-1}g]=[h^{-1}]$ .

4.1 The aim of this section

The aim of this section is to prove the following devissage result.

Here by a stabiliser group scheme we understand the BT-group scheme associated to the stabiliser of a point in the extended building $\mathscr {B} {}^e(G, \breve {\mathbb {Q}} {}_p)$ (a particular type of quasi-parahoric group schemes, cf. section 2.2). The proof proceeds by comparing formal completions of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ and of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu }$ (cf. Section 4.2) and of their underlying reduced schemes (cf. Section 4.3).

4.2 Formal completions

Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is quasi-parahoric. For simplicity of notation, we set $O=O_{\breve E}$ with residue field $k=\bar k_E$ . We consider the natural v-sheaf morphism

(4.2.1) $$ \begin{align} \pi: {\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}^o, b, \mu}}\to {\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}} \end{align} $$

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

Proposition 4.2.1. a) The map $\pi $ is qcqs (quasi-compact quasi-separated [Reference ScholzeSch17]).

b) The map $\pi $ induces an isomorphism

$$\begin{align*}\mathcal{M}{}^{\mathrm{int}}_{{\mathcal{G}} {}^o, b,\mu/x}\xrightarrow{\sim} \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b,\mu/\pi(x)} \end{align*}$$

for each $x\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b,\mu }(\mathrm{Spd}(k))$ .

Proof. (a) Observe that ${\mathcal {G}} {}^o\to {\mathcal {G}} $ is an isomorphism on the generic fibres and can be identified with the dilation ${\mathcal {G}} {}^{\mathrm {dil}}$ of ${\mathcal {G}} $ along its closed subscheme given by the neutral component $({\mathcal {G}} \otimes _{\mathbb {Z} {}_p}{\mathbb F}_p)^o$ of the special fibre ${\mathcal {G}} \otimes _{\mathbb {Z} {}_p}{\mathbb F}_p$ . (Indeed, both ${\mathcal {G}} {}^o$ and the dilation ${\mathcal {G}} {}^{\mathrm {dil}}$ are smooth, have the same generic fibre and the same $\breve {\mathbb {Z}} {}_p$ -points.) In particular, we have $\mathcal {O} {}_{{\mathcal {G}} }\hookrightarrow \mathcal {O} {}_{{\mathcal {G}} {}^o}=\mathcal {O} {}_{{\mathcal {G}} }[f_1,\ldots , f_m]$ , and there is $N\geq 1$ , such that $p^Nf_i\in \mathcal {O} {}_{{\mathcal {G}} }$ , for all i. Then the argument in [Reference Pappas and RapoportPR24, proof of Prop. 3.6.2] applies to prove the claim.

(b) As in section 3.5 we have an isomorphism

(4.2.2) $$ \begin{align} \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu/x}\simeq \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b_x, \mu/x_0} \end{align} $$

and similarly for ${\mathcal {G}} {}^o$ . Hence it is enough to show the isomorphism for the base point $x=x_0\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b,\mu }(\mathrm{Spd}(k))$ (at the cost of changing $b=b_{x_0}$ to $b_x$ ). By Theorem 3.6.2

$$\begin{align*}\widehat{L{\mathcal{G}} }^o_{b,\mu}\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}^o, b,\mu/x_0} \end{align*}$$

is a $\widehat {\mathbb {W}} {}^+{\mathcal {G}} {}^o\times \mathrm{Spd}(O_{\breve E})$ -torsor. Also, $\widehat {L{\mathcal {G}} }^o_{b,\mu }$ is a $\widehat {\mathbb {W}} {}^+{\mathcal {G}} {}^o\times \mathrm{Spd}(O_{\breve E})$ -torsor over $\mathbb {M} {}^{v}_{{\mathcal {G}} {}^o, \mu /x_0}$ , and there are corresponding statements for ${\mathcal {G}} $ . By (2.3.1), the natural map ${\mathcal {G}} {}^o\to {\mathcal {G}} $ induces an isomorphism

$$\begin{align*}\mathbb{M} {}^{v}_{{\mathcal{G}} {}^o, \mu}\xrightarrow{\sim} \mathbb{M} {}^{v}_{{\mathcal{G}} , \mu}. \end{align*}$$

On the other hand, it is easy to see that ${\mathcal {G}} {}^o\to {\mathcal {G}} $ induces $\widehat {\mathbb {W}} {}^+{\mathcal {G}} {}^o\xrightarrow {\sim } \widehat {\mathbb {W}} {}^+{\mathcal {G}} $ and, hence, $\widehat {L{\mathcal {G}}}^o_{b,\mu }\xrightarrow {\sim } \widehat {L{\mathcal {G}}}_{b,\mu }$ . The result follows.

Remark 4.2.2. Suppose $\kappa $ is an algebraically closed field over k and set $O(\kappa )= O_{\breve E}\otimes _{W(k)}W(\kappa )$ . We can consider the base change

$$\begin{align*}(\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu})_{O(\kappa)}:=\mathcal{M}{}^{\mathrm{int}}_{{\mathcal{G}}, b, \mu}\times_{\mathrm{Spd}(O)}\mathrm{Spd}( O(\kappa)), \end{align*}$$

and similarly for ${\mathcal {G}} {}^o$ . Given $x\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu }(\mathrm{Spd}(\kappa ))$ , we obtain a corresponding point $x\in (\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu })_{O(\kappa )}(\mathrm{Spd}(\kappa ))$ . The formal completions $( \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu })_{ O(\kappa )/x}$ and $( \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{ O(\kappa )/\pi (x)}$ now make sense and the argument in the proof of (b) above also applies to give

(4.2.3) $$ \begin{align} ( \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}^o, b, \mu})_{ O(\kappa)/x}\xrightarrow{\sim}( \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu})_{ O(\kappa)/\pi(x)}. \end{align} $$

4.3 ADLV for quasi-parahorics

In this subsection, we express the ADLV $X_{{\mathcal {G}} }(b, \mu )$ for a quasi-parahoric ${\mathcal {G}} $ in terms of ADLV attached to parahoric subgroups. Note that the results will eventually be applied for $\mu $ replaced by $\mu ^{-1}$ , to relate to integral local Shimura varieties via Prop. 3.3.1.

Recall the Kottwitz map $\kappa _G: G(\breve {\mathbb {Q}} {}_p)\to \Omega _G$ . This map can be enhanced to a morphism $LG\to \Omega _G$ which factors through $X_{{\mathcal {G}} {}^o}=LG/L^+{\mathcal {G}} {}^o$ , and induces a map with connected fibres,

(4.3.1) $$ \begin{align} \kappa_G\colon X_{{\mathcal{G}} {}^o}\to \Omega_G , \end{align} $$

cf. [Reference ZhuZhu17, Prop. 1.21], comp. also [Reference Pappas and RapoportPR08, §5].

Recall $\pi _0({\mathcal {G}} )=\breve K/\breve K^o\subset \Omega $ , cf. (3.1.1). Then $\pi _0({\mathcal {G}} )$ acts on $X_{{\mathcal {G}} {}^o}$ by $g\breve K^o\mapsto g\dot \gamma \breve K^o$ , where $\dot \gamma \in \breve K$ is a representative of a given element in $\pi _0({\mathcal {G}} )$ . This action is permuting connected components and the quotient is $X_{{\mathcal {G}}} $ ,

$$\begin{align*}X_{{\mathcal{G}} {}^o}\to X_{{\mathcal{G}}} =X_{{\mathcal{G}} {}^o}/\pi_0({\mathcal{G}} ). \end{align*}$$

Each connected component of $X_{{\mathcal {G}} {}^o}$ maps isomorphically to a connected component of $X_{{\mathcal {G}}} $ .

Recall that there exists $c_{b, \mu }\in \Omega $ with $\phi (c_{b, \mu })-c_{b, \mu }=\kappa _G(b)-\kappa _G(\mu )$ , comp. [Reference He and ZhouHZ20, §6]. Here $\kappa _G(\mu )\in \Omega $ is the element associated to the conjugacy class $\mu $ , i.e., the residue class of $\{\mu \}$ modulo the affine Weyl group. Furthermore, the class modulo $\Omega ^\phi $ of $c_{b, \mu }$ is uniquely determined.

Now $X_{{\mathcal {G}} {}^o}(b, \mu )$ lies in the union of certain connected components of $X_{{\mathcal {G}} {}^o}$ . From $-\kappa _G(g)+\phi (\kappa _G(g))=-\kappa _G(b)+\kappa _G(\mu )$ , we obtain $\kappa _G(X_{{\mathcal {G}} {}^o}(b, \mu ))\subset c_{b, \mu }+\Omega ^\phi .$ In fact, by [Reference He and ZhouHZ20, Lem. 6.1], we have an equality,

(4.3.2) $$ \begin{align} \kappa_G(X_{{\mathcal{G}} {}^o}(b, \mu))= c_{b, \mu}+\Omega^\phi. \end{align} $$

We now extend these considerations to quasi-parahorics. Let $C=C_{{\mathcal {G}}} =\Omega /\pi _0({\mathcal {G}} )$ . Then the Kottwitz homomorphism induces a map

(4.3.3) $$ \begin{align} \kappa_G\colon X_{{\mathcal{G}}} \to C_{{\mathcal{G}}}. \end{align} $$

The image of $X_{{\mathcal {G}}} (b, \mu )$ in $C_\phi $ is equal to the residue class of $\kappa _G(b)-\kappa _G(\mu )$ , so

(4.3.4) $$ \begin{align} \kappa_G(X_{{\mathcal{G}}} (b, \mu))\subset \bar c_{b, \mu}+C^\phi , \end{align} $$

where $\bar c_{b, \mu }\in C_{{\mathcal {G}}} $ is the image of $ c_{b, \mu }$ . This inclusion will turn out to be an equality, cf. Corollary 4.3.6 below.

Recall $\Pi _{{\mathcal {G}}} =\ker (\pi _0({\mathcal {G}} )_\phi \to \Omega _\phi )$ , cf. (3.1.2). From the exact sequence $0\to \pi _0({\mathcal {G}} )\to \Omega \to C_{{\mathcal {G}}} \to 0$ we have an exact sequence

(4.3.5) $$ \begin{align} 0\to \Omega^\phi/\pi_0({\mathcal{G}} )^\phi\to C_{{\mathcal{G}}} {}^\phi\to \Pi_{{\mathcal{G}}} \to 0. \end{align} $$

Let $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ . Take $\beta \in \pi _0({\mathcal {G}} )\subset \Omega $ lifting $\bar {\beta }$ . Then

$$\begin{align*}\beta=(1-\phi)\gamma \end{align*}$$

for some $\gamma \in \Omega $ . Note that, since $\phi ({\gamma })-{\gamma }$ is in $\pi _0({\mathcal {G}} )$ , the image $\bar \gamma $ of $\gamma $ in C is $\phi $ -invariant and maps to $\bar {\beta }$ under the connecting homomorphism $C^\phi \to \Pi _{{\mathcal {G}}} $ . Conversely, the image $\bar {\beta }$ of an element $\bar \gamma \in C^\phi $ in $\Pi _{{\mathcal {G}}} $ is described as follows. Let $\gamma \in \Omega $ be a lift of $\bar \gamma $ . Then $\bar {\beta }$ is the class of $\beta =\phi (\gamma )-\gamma \in \pi _0({\mathcal {G}} )$ .

Recall that the quasi-parahoric subgroup $\breve K$ is defined via a point $\bf x$ in the building of $G_{\mathrm {ad}}$ . Write the Iwahori-Weyl group as $\widetilde W=W_a\rtimes \Omega $ and

$$\begin{align*}1\to T(\breve{\mathbb{Q}} {}_p)^0\to N(\breve{\mathbb{Q}} {}_p)\to \widetilde W=W_a\rtimes \Omega\to 1 \end{align*}$$

with $T=Z_G(S_{\breve {\mathbb {Q}} {}_p})$ as in [Reference Haines and RapoportHR08]. Here the choices are made such that $\bf x$ is in the base alcove $\frak {a}_0$ of the apartment of S. Then $T(\breve {\mathbb {Q}} {}_p)^0\subset \breve K^o$ . When we consider $\Omega $ as a subset of $\widetilde W$ , we write the group law in a multiplicative way.

Lemma 4.3.1. Let $\beta \in \pi _0({\mathcal {G}} )$ be of the form $\beta =\phi (\gamma )-\gamma $ , for $\gamma \in \Omega $ . There is a lift $\dot {\gamma }\in N(\breve {\mathbb {Q}} {}_p)$ of $\gamma \in \Omega \subset \widetilde W$ such that

$$\begin{align*}\phi(\dot{\gamma})^{-1}\dot{\gamma}\in \breve K. \end{align*}$$

Proof. Lift $\gamma \in \Omega $ to $\dot {\gamma }\in N(\breve {\mathbb {Q}} {}_p)$ . Then $\phi (\dot {\gamma })^{-1}\dot {\gamma }$ lifts $\beta \in \pi _0({\mathcal {G}} )\subset \Omega $ . Now we use the exact sequence

$$ \begin{align*}1\to T(\breve{\mathbb{Q}} {}_p)^0\to N(\breve{\mathbb{Q}} {}_p)\cap\breve K\to \pi_0({\mathcal{G}} )\to 1. \end{align*} $$

This holds since, denoting by $U=U_{K^o}$ the pro-unipotent radical of $\breve K^o$ , we have $\breve K\subset U\cdot N(\breve {\mathbb {Q}} {}_p)$ , cf. [Reference Haines and RapoportHR08, proof of Prop. 8]. We can lift $\beta $ to $\dot {\beta }\in N(\breve {\mathbb {Q}} {}_p)\cap \breve K$ . Then $\phi (\dot {\gamma })^{-1}\dot {\gamma }\dot \beta ^{-1}\in T(\breve {\mathbb {Q}} {}_p)^0$ . Since $T(\breve {\mathbb {Q}} {}_p)^0\subset \breve K^o$ , the result follows.

For $\beta $ , $\gamma $ , $\dot \gamma $ as above, it follows that

$$\begin{align*}\breve K\dot{\gamma}^{-1}=\breve K\phi(\dot{\gamma})^{-1}. \end{align*}$$

Consider now the conjugates of $\breve K$ , resp. $\breve K^o$ ,

(4.3.6) $$ \begin{align} \breve K_\gamma:=\dot{\gamma} \breve K\dot{\gamma}^{-1}, \quad \breve K^o_\gamma:=\dot{\gamma} \breve K^o\dot{\gamma}^{-1}. \end{align} $$

These subgroups of $G(\breve {\mathbb {Q}} {}_p)$ depend only on $\gamma $ . Indeed, if $\dot \gamma $ is replaced by $\dot \gamma \dot \delta $ , with $\dot \delta \in T(\breve {\mathbb {Q}} {}_p)^0$ , then $\dot {\gamma } \breve K^o\dot {\gamma }^{-1}$ is replaced by $\dot {\gamma } \dot \delta \breve K^o\dot \delta ^{-1} \dot {\gamma }^{-1}=\dot {\gamma } \breve K^o\dot {\gamma }^{-1}$ . We have

$$\begin{align*}\phi(\dot{\gamma} \breve K\dot{\gamma}^{-1})=\dot{\gamma}\breve K\dot{\gamma}^{-1},\quad \phi(\dot{\gamma} \breve K^o\dot{\gamma}^{-1})=\dot{\gamma} \breve K^o\dot{\gamma}^{-1}. \end{align*}$$

Hence, $\breve K_\gamma $ and $\breve K^o_\gamma $ are rational, i.e., correspond to subgroups of $G(\mathbb {Q} {}_p)$ . The parahorics $ K^o_\gamma $ are conjugate to $K^o$ in $G(\breve {\mathbb {Q}} {}_p)$ but not necessarily in $G(\mathbb {Q} {}_p)$ .

In the sequel, we make a fixed choice of $\dot \gamma $ , for each element $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ . We denote the corresponding groups by $\breve K_\beta $ , resp. $\breve {\mathcal {G}} {}_\beta $ , resp. $\breve K^o_\beta $ , resp. $\breve {\mathcal {G}} {}^o_\beta $ , by slightly abusing notation. We consider the map

(4.3.7) $$ \begin{align} \begin{aligned} \pi_\gamma: X_{{\mathcal{G}} {}_\gamma}=LG/L^+\breve K_\gamma=LG/&L^+(\dot{\gamma} \breve K\dot{\gamma}^{-1})\to LG/L^+ \breve K=X_{{\mathcal{G}} },\\ & g(\dot{\gamma} \breve K\dot{\gamma}^{-1})\mapsto g\dot{\gamma} \breve K. \end{aligned} \end{align} $$

Let us check the dependency of $\dot \gamma $ . If $\dot \gamma $ is replaced by $\dot \gamma '=\dot \gamma \dot \delta $ with $\dot \delta \in T(\breve {\mathbb {Q}} {}_p)^0$ , then $g\dot \gamma '\breve K\dot \gamma ^{\prime -1}\in X_{{\mathcal {G}} {}_{\gamma '}}$ is mapped under $\pi _{\gamma '}$ to $g\dot \gamma '\breve K=g\dot \gamma \breve K$ ; hence $\pi _\gamma =\pi _{\gamma '}$ under the identity identification $X_{{\mathcal {G}} {}_\gamma }=X_{{\mathcal {G}} {}_{\gamma '}}$ . Similarly, if $\gamma $ is replaced by $\gamma +\delta $ with $\delta \in \Omega ^\phi $ , then, choosing $\dot \gamma '=\dot \delta \dot \gamma $ with $\dot \delta \in N(\mathbb {Q} {}_p)\cap \breve K$ , we can identify $X_{{\mathcal {G}} {}_\gamma }$ with $X_{{\mathcal {G}} {}_{\gamma '}}$ via $g \breve K_\gamma \mapsto g\dot \delta ^{-1} \breve K_{\gamma '}$ . The first element is mapped under $\pi _\gamma $ to $g\dot \gamma \breve K$ , the second element is mapped under $\pi _{\gamma '}$ to $(g\dot \delta ^{-1})\dot \gamma '\breve K=g\dot \gamma \breve K$ , hence $\pi _\gamma =\pi _{\gamma '}$ . Note that $\dot \delta $ is not unique. But if $\dot \delta $ is replaced by $\dot \delta '=\dot \delta \varepsilon $ with $\varepsilon \in T(\breve {\mathbb {Q}} {}_p)^0$ , then the identification of $X_{{\mathcal {G}} {}_\gamma }$ and $X_{{\mathcal {G}} {}_{\gamma '}}$ is not affected. Finally, if $\beta $ is replaced by $\beta +(\phi -1)\delta $ with $\delta \in \pi _0({\mathcal {G}} )$ , then $\gamma $ is replaced by $\gamma +\delta $ . Then we may replace $\dot \gamma $ by $\dot \gamma '=\dot \gamma \dot \delta $ , where $\dot \delta \in N(\breve {\mathbb {Q}} {}_p)\cap \breve K$ . Since $\dot \delta $ normalises $\breve K^o$ , the spaces $X_{{\mathcal {G}} {}_{\gamma }}$ and $X_{{\mathcal {G}} {}_{\gamma '}}$ are identified compatibly with the maps $\pi _\gamma $ , resp. $\pi _{\gamma '}$ .

Proposition 4.3.4. The above map defines by restriction a map

$$\begin{align*}\pi_\beta: X_{{\mathcal{G}} {}_\beta}(b, \mu)\to X_{{\mathcal{G}}} (b, \mu). \end{align*}$$

Proof. Suppose that $g\breve K_\beta \in X_{{\mathcal {G}} {}_\beta }(b, \mu )$ , i.e.,

$$\begin{align*}g^{-1}b\phi(g)\in \breve K_\beta \mathrm{{Adm}}(\mu) \breve K_\beta=\dot{\gamma} \breve K\dot{\gamma}^{-1} \mathrm{{Adm}}(\mu) \dot{\gamma} \breve K\dot{\gamma}^{-1}. \end{align*}$$

Since $\mathrm {{Adm}}(\mu )$ is stable under the conjugation action of $\Omega $ , we see that

$$ \begin{align*}\breve K\dot{\gamma}^{-1} \mathrm{{Adm}}(\mu) \dot{\gamma} \breve K=\breve K\mathrm{{Adm}}(\mu) \breve K. \end{align*} $$

Hence

$$\begin{align*}g^{-1}b\phi(g)\in \dot{\gamma} \breve K \mathrm{{Adm}}(\mu) \breve K\dot{\gamma}^{-1}, \end{align*}$$

so

$$\begin{align*}\dot{\gamma}^{-1}g^{-1}b\phi(g)\phi(\dot{\gamma})\in \breve K \mathrm{{Adm}}(\mu) \breve K\dot{\gamma}^{-1}\phi(\dot{\gamma})\subset \breve K \mathrm{{Adm}}(\mu) \breve K , \end{align*}$$

since $\breve K\dot {\gamma }^{-1}\phi (\dot {\gamma })=\breve K.$ Hence

$$\begin{align*}(g\dot{\gamma} )^{-1} b\phi(g\dot{\gamma})\in \breve K \mathrm{{Adm}}(\mu) \breve K , \end{align*}$$

i.e., $g\dot {\gamma }\breve K\in X_{{\mathcal {G}}} (b, \mu )$ , as had to be shown.

Composing the above map with $ X_{{\mathcal {G}} {}^o_\beta }(b, \mu )\to X_{{\mathcal {G}} {}_\beta }(b, \mu )$ and letting $\bar {\beta }$ vary, we obtain the map

(4.3.8) $$ \begin{align} \pi: \bigsqcup_{\bar{\beta}\in \Pi_{{\mathcal{G}}} } X_{{\mathcal{G}} {}^o_{\beta}}(b, \mu)\to X_{{\mathcal{G}}} (b, \mu). \end{align} $$

In the sequel, we call a component of $X_{{\mathcal {G}} {}^o_{\beta }}(b, \mu ) $ , resp. of $X_{{\mathcal {G}} }(b, \mu ) $ , the non-empty fibres of the Kottwitz map to $\Omega $ , resp. to $C_{\mathcal {G}} =\Omega /\pi _0({\mathcal {G}} )$ . In other words, we partition the points of $X_{{\mathcal {G}} {}^o_{\beta }}(b, \mu ) $ , resp. of $X_{{\mathcal {G}} }(b, \mu ) $ , according to which connected component of $X_{{\mathcal {G}} {}^o_{\beta }}$ , resp. of $X_{{\mathcal {G}} }$ , they lie in.

Proof. For (i), let $x_1\in X_{{\mathcal {G}} {}^o_{\beta _1}}(b, \mu )$ and $x_2\in X_{{\mathcal {G}} {}^o_{\beta _2}}(b, \mu )$ be such that their images in $C_{{\mathcal {G}}} $ are the same. We deduce that

$$ \begin{align*}\kappa_G(x_1)+\gamma_1= \kappa_G(x_2)+\gamma_2\ \mod \pi_0({\mathcal{G}} ). \end{align*} $$

On the other hand, $\kappa _G(x_i)=c_{b, \mu } -\lambda _i$ with $\lambda _i\in \Omega ^\phi $ , for $i=1, 2$ . We therefore get

$$ \begin{align*}\gamma_1-\gamma_2=\lambda_1-\lambda_2+\mu , \quad \mu\in \pi_0({\mathcal{G}} ). \end{align*} $$

Applying $1-\phi $ to this identity, we obtain

$$ \begin{align*}\beta_1-\beta_2=(1-\phi)\mu , \end{align*} $$

i.e., $\bar {\beta }_1=\bar {\beta }_2$ , as desired.

For (ii), let $g\breve K\in X_{{\mathcal {G}}} (b, \mu )$ . By [Reference Görtz, He and NieGHN24, Lem. 5.10, (iii)],

(4.3.9) $$ \begin{align} \breve K^o \mathrm{{Adm}}(\mu) \breve K =\breve K \mathrm{{Adm}}(\mu) \breve K. \end{align} $$

Hence $g^{-1}b\phi (g)\in \breve K\mathrm {{Adm}}(\mu )\breve K=\breve K^o\mathrm {{Adm}}(\mu )\breve K$ . Write accordingly

$$\begin{align*}g^{-1}b\phi(g)=k^o_1\omega k. \end{align*}$$

Apply $\kappa : G(\breve {\mathbb {Q}} {}_p)\to \Omega $ to get

$$\begin{align*}(\phi-1)\kappa(g)+\kappa(b)=\kappa(k)+\kappa(\mu). \end{align*}$$

Hence we get

(4.3.10) $$ \begin{align} (\phi-1)\kappa(g)=\kappa(k)-(\phi-1)c_{b,\mu}. \end{align} $$

Let $\beta =-\kappa (k)\in \pi _0({\mathcal {G}} )$ and denote by $\bar {\beta }\in \pi _0({\mathcal {G}} )_\phi $ its image. Then the last equation shows that $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ . We write $\beta =(1-\phi )\gamma $ with $\gamma \in \Omega $ and choose a lift $\dot \gamma \in N(\breve {\mathbb {Q}} {}_p)$ as in Lemma 4.3.1. Then $ g\dot {\gamma }^{-1} (\dot {\gamma } K^o\dot {\gamma }^{-1})$ is in $X_{\dot {\gamma } K^o\dot {\gamma }^{-1}}(b, \mu )$ and maps to $gK\in X_K(b, \mu )$ . Indeed, $(g\dot {\gamma }^{-1})^{-1} b\phi ( g\dot {\gamma }^{-1})$ lies in $(\dot {\gamma }K^o\dot {\gamma }^{-1})\mathrm {{Adm}}(\mu )(\dot {\gamma }K^o\dot {\gamma }^{-1}) $ because

$$\begin{align*}(g\dot{\gamma}^{-1})^{-1} b\phi( g\dot{\gamma}^{-1})&=\dot{\gamma} g^{-1}b\phi(g)\phi( \dot{\gamma}^{-1})= \dot{\gamma}k_1^o \omega k \phi( \dot{\gamma}^{-1})\\&= (\dot{\gamma} k^o_1 \dot{\gamma}^{-1})(\dot{\gamma}\omega \dot{\gamma}^{-1}) (\dot{\gamma} k \phi({\dot\gamma}^{-1})). \end{align*}$$

The middle factor lies in $\mathrm {{Adm}}(\mu )$ because conjugation by an element in $\Omega $ preserves $\mathrm {{Adm}}(\mu )$ . The last factor lies in $\dot {\gamma }\breve K\dot {\gamma }^{-1} $ . The result follows because the last factor lies even in $\dot {\gamma }\breve K^o\dot {\gamma }^{-1} $ because $\kappa (\dot {\gamma } k \phi ({\dot \gamma }^{-1}))=(1-\phi )\gamma -\beta =0$ .

Proof. Let $x=\bar c_{b, \mu }+y$ , where $y\in C_{{\mathcal {G}}} {}^\phi $ . Let us first assume that the image of y in $\Pi _{{\mathcal {G}}} $ is trivial, then there exists $\tilde y\in \Omega ^\phi $ mapping to y. Then

$$\begin{align*}c_{b, \mu}+\tilde y\in \Omega \end{align*}$$

is a lift of x which, by (4.3.2), can be lifted to a point of $X_{{\mathcal {G}} {}^o}$ which then maps to a lift of x in $X_{{\mathcal {G}} }$ , as required.

In general, let $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ be the image of x. Let $C^\phi \to \Pi _{{\mathcal {G}}} $ be the natural surjective map from (4.3.5), and let $(C^\phi )_{\bar {\beta }}$ be the fibre of this map over $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ . Via our fixed choice of $\gamma $ for $\bar {\beta }$ , this can be identified with $\Omega ^\phi /\pi _0({\mathcal {G}} {}_\beta )^\phi $ . By (4.3.2), the set of components of the inverse image of $\bar c_{b, \mu }+(C^\phi )_{\bar {\beta }}$ in $X_{{\mathcal {G}} {}^o_{\beta }}(b, \mu )$ can then be identified with $ c_{b, \mu }+\Omega ^\phi $ . The assertion follows.

We introduce the intermediate group $K^o\subset K^\prime \subset K$ , corresponding to the subgroup $\pi _0({\mathcal {G}} )^\phi $ of $\pi _0({\mathcal {G}} )$ . We denote the corresponding group scheme by ${\mathcal {G}} {}^\prime $ . Then we obtain a factorisation

$$ \begin{align*}X_{{\mathcal{G}} {}^o}\to X_{{\mathcal{G}} '}\to X_{{\mathcal{G}}}. \end{align*} $$

Each of the maps induces an isomorphism of a component onto its image. In fact, the first map is the quotient by the finite abelian group $\pi _0({\mathcal {G}} )^\phi $ , the second map is the quotient by the finite abelian group $\pi _0({\mathcal {G}} )/\pi _0({\mathcal {G}} )^\phi $ , and the composed map is the quotient by the finite abelian group $\pi _0({\mathcal {G}} )$ . Here the actions of these covering groups are trivial, in the sense that they only permute connected components.

Similarly, for arbitrary $\bar {\beta }\in \Pi _{{\mathcal {G}}} $ , we introduce the intermediate subgroup $K_\beta ^o\subset K_\beta ^\prime \subset K_\beta $ , corresponding to the subgroup $\pi _0({\mathcal {G}} {}_\beta )^\phi $ of $\pi _0({\mathcal {G}} {}_\beta )$ . We obtain a sequence of maps,

(4.3.11) $$ \begin{align} X_{{\mathcal{G}} {}_{\beta}^o}\to X_{{\mathcal{G}} {}^{\prime}_\beta}\to X_{{\mathcal{G}} {}_\beta}, \end{align} $$

where the first map is the quotient by the finite group $\pi _0({\mathcal {G}} {}_\beta )^\phi $ , the second map is the quotient by the finite abelian group $\pi _0({\mathcal {G}} {}_\beta )/\pi _0({\mathcal {G}} {}_\beta )^\phi $ , and the composed map is the quotient by the finite abelian group $\pi _0({\mathcal {G}} {}_\beta )$ .

Proposition 4.3.7. Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is a quasi-parahoric group scheme for G.

(i) The ADLV $X_{{\mathcal {G}} {}^o_{\beta }}(b, \mu )$ is invariant under the action of $\pi _0({\mathcal {G}} {}_\beta )^\phi $ .

(ii) The map (4.3.8) induces an isomorphism

Proof. (i) Let $h\in \breve K_\beta ^\prime $ . Then $\phi (h)=hk$ , with $k\in \breve K_\beta ^o$ . The invariance follows from

$$ \begin{align*}(gh)^{-1}b \phi(gh)=h^{-1}g^{-1}b\phi(g)\phi(h)=h^{-1}g^{-1}b\phi(g)h k , \end{align*} $$

because the last element lies in $\breve K_\beta ^oh^{-1}\mathrm {{Adm}}(\mu )h \breve K_\beta ^o=\breve K_\beta ^o\mathrm {{Adm}}(\mu ) \breve K_\beta ^o.$

(ii) The surjectivity follows from Proposition 4.3.5, (ii). For the injectivity, we note that by Proposition 4.3.5, (i), this becomes a question one $\bar {\beta }$ at the time. But if the components of $X_{{\mathcal {G}} {}^o_{\beta }}(b, \mu )$ corresponding to $\lambda _1$ and $\lambda _2$ in $\Omega $ go to the same component of $X_{{\mathcal {G}}} (b, \mu )$ , it follows that $\lambda _1-\lambda _2\in \pi _0({\mathcal {G}} )$ . But by (4.3.2), this element lies in $\Omega ^\phi $ , hence also in $ \pi _0({\mathcal {G}} )^\phi $ .

4.4 Integral moduli spaces of local shtukas

We first define a morphism of integral LSV

(4.4.1) $$ \begin{align} \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}_{\beta}, b, \mu}\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}. \end{align} $$

Recall that

$$ \begin{align*}\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}_\beta, b, \mu}(S)=\{\text{isomorphism classes of } (S^\sharp, \mathcal{P} {}_\beta, \phi_{\mathcal{P} {}_\beta}, i_{r, \beta}) \}. \end{align*} $$

Starting with a point $(S^\sharp , \mathcal {P} {}_\beta , \phi _{\mathcal {P} {}_\beta }, i_{r, \beta })$ of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}_\beta , b, \mu }$ , we define a ${\mathcal {G}} $ -torsor $\mathcal {P} $ by twisting the ${\mathcal {G}} {}_\beta $ -action by $\dot \gamma $ ,

$$ \begin{align*}g\star x=( \dot\gamma g \dot\gamma^{-1})\cdot x, \quad g\in{\mathcal{G}}, \end{align*} $$

where $\dot \gamma $ is the fixed choice above. This makes sense since $\dot \gamma g\dot \gamma ^{-1}\in {\mathcal {G}} {}_\beta $ . Let $\xi =\phi (\dot \gamma )^{-1}\dot \gamma $ . Then $\xi \in {\mathcal {G}} (\breve {\mathbb {Z}} {}_p)$ , cf. Lemma 4.3.1. We define the new Frobenius $\phi _{\mathcal {P} }$ as

(4.4.2) $$ \begin{align} \phi_{\mathcal{P}} (x)=\xi\star\phi_{\mathcal{P} {}_\beta}(x). \end{align} $$

This is indeed a morphism of ${\mathcal {G}} $ -torsors: on the one hand

$$ \begin{align*} \begin{aligned} \phi_{\mathcal{P} }(g\star x)&=\xi\star\phi_{\mathcal{P} {}_\beta}(g\star x)= \dot\gamma\xi\dot\gamma^{-1} \phi_{\mathcal{P} {}_\beta}(( \dot\gamma g \dot\gamma^{-1})x)\\&= \dot\gamma(\phi( \dot\gamma)^{-1} \dot\gamma) \dot\gamma^{-1}(\phi( \dot\gamma)\phi(g)\phi( \dot\gamma)^{-1})\phi_{\mathcal{P} {}_\beta}(x)= \dot\gamma\phi(g)\phi( \dot\gamma)^{-1}\phi_{\mathcal{P} {}_\beta}(x). \end{aligned} \end{align*} $$

On the other hand,

$$ \begin{align*} \begin{aligned} \phi(g)\star\phi_{\mathcal{P}} (x)&=( \dot\gamma \phi(g) \dot\gamma^{-1})( \dot\gamma\xi \dot\gamma^{-1})\phi_{\mathcal{P} {}_\beta}(x)\\&= \dot\gamma \phi(g) (\phi( \dot\gamma)^{-1} \dot\gamma) \dot\gamma^{-1}\phi_{\mathcal{P} {}_\beta}(x) = \dot\gamma \phi(g) \phi( \dot\gamma)^{-1}\phi_{\mathcal{P} {}_\beta}(x), \end{aligned} \end{align*} $$

which proves the claim.

We define the framing $i_r$ as

(4.4.3) $$ \begin{align} i_r(x)=i_{r, \beta}( \dot\gamma x). \end{align} $$

This is indeed an isomorphism of ${\mathcal {G}} $ -bundles $i_r: G_{\mathcal {Y} {}_{[r, \infty ]}}\to \mathcal {P} {}_{|\mathcal {Y} {}_{[r, \infty ]}}$ : on the one hand

$$ \begin{align*}i_r(hg)=i_{r, \beta}( \dot\gamma hg)= \dot\gamma h i_{r, \beta}(g). \end{align*} $$

On the other hand,

$$ \begin{align*}h\star i_r(g)=( \dot\gamma h \dot\gamma^{-1}) i_{r, \beta}( \dot\gamma g)=( \dot\gamma h \dot\gamma^{-1}) \dot\gamma i_{r, \beta}( g)= \dot\gamma h i_{r, \beta}(g). \end{align*} $$

The framing $i_r$ is also compatible with the Frobenius: on the one hand,

$$ \begin{align*} \begin{aligned} \phi_{\mathcal{P}} (i_r(g))&=\xi\star\phi_{\mathcal{P} {}_\beta}(i_{r, \beta}( \dot\gamma g))= \dot\gamma(\phi( \dot\gamma)^{-1} \dot\gamma) \dot\gamma^{-1}\phi_{\mathcal{P} {}_\beta}(i_{r, \beta}( \dot\gamma g))\\ &= ( \dot\gamma\phi( \dot\gamma)^{-1})\phi( \dot\gamma) i_{r, \beta}( b\phi(g))= \dot\gamma i_{r, \beta}( b\phi(g)). \end{aligned} \end{align*} $$

On the other hand,

$$ \begin{align*} \begin{aligned} i_r(b\phi(g))=i_{r, \beta}( \dot\gamma b\phi(g))= \dot\gamma i_{r, \beta}( b\phi(g)). \end{aligned} \end{align*} $$

The morphism (4.4.1) is now defined by sending $(S^\sharp , \mathcal {P} {}_\beta , \phi _{\mathcal {P} {}_\beta }, i_{r, \beta })$ to $(S^\sharp , \mathcal {P}, \phi _{\mathcal {P} }, i_{r })$ . We precompose (4.4.1) with the natural morphism $ \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o_{\beta }, b, \mu }\to \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}_\beta , b, \mu }$ to obtain a morphism of integral LSV

$$ \begin{align*}\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}^o_{\beta}, b, \mu}\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu} , \end{align*} $$

and hence a morphism

(4.4.4) $$ \begin{align} \pi: \bigsqcup_{\bar{\beta}\in\Pi_{{\mathcal{G}}} }\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}^o_{\beta}, b, \mu}\to \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}. \end{align} $$

This morphism is compatible, via the bijective map of Proposition 3.3.1, with the morphism (4.3.8) for $\mu ^{-1}$ . Indeed, let $(P_\beta , \phi _{P_\beta }, i_\beta )$ be an object of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}_\beta , b, \mu }(\mathrm{Spd}(k))$ , which gives a ${\mathcal {G}} {}_\beta $ -torsor $P_\beta $ over $W(k)$ , a Frobenius and a trivialisation of $P_\beta $ over $W(k)[1/p]$ . The image of $(P_\beta , \phi _{P_\beta }, i_\beta )$ in $X_{{\mathcal {G}} {}_\beta }(b, \mu ^{-1})$ is given by the unique element $g_\beta \in LG(k)/L^+{\mathcal {G}} {}_\beta (k)$ such that $i_\beta ^{-1}(P_\beta )=g_\beta ^{-1} \cdot {\mathcal {G}} {}_\beta $ . Let $(P, \phi _{P}, i)\in \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }(k)$ be the image of $(P_\beta , \phi _{P_\beta }, i_\beta )$ under (4.4.1). Then $i^{-1}(P)= \dot \gamma ^{-1}i_\beta ^{-1}(P_\beta )$ . Hence the image $g\in LG(k)/L^+{\mathcal {G}} (k)$ of $(P, \phi _{P}, i)$ is equal to $g=g_\beta \gamma $ , as required.

Theorem 4.4.1. Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is a quasi-parahoric group scheme for G. The morphism (4.4.4) induces an isomorphism

where the quotient is for an action of $\pi _0({\mathcal {G}} {}_\beta )^\phi $ which permutes connected components.

Let us first define the action of $\pi _0({\mathcal {G}} {}_\beta )^\phi $ on $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o_{\beta }, b, \mu }$ . We use the exactness of the sequence

$$ \begin{align*}0\to {\mathcal{G}} {}_{\beta}^o(\mathbb{Z} {}_p)\to {\mathcal{G}} {}_\beta(\mathbb{Z} {}_p)\to \pi_0({\mathcal{G}} {}_\beta)^\phi\to 0. \end{align*} $$

Let $\delta \in G(\mathbb {Q} {}_p)$ be in the normaliser of ${\mathcal {G}} {}_\beta ^o$ . Recall that

$$ \begin{align*}\mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} {}_\beta^o, b, \mu}(S)=\{\text{isomorphism classes of } (S^\sharp, \mathcal{P} , \phi_{\mathcal{P}} , i_r) \} , \end{align*} $$

where $\mathcal {P} $ is a ${\mathcal {G}} {}_\beta ^o$ -torsor, etc. We define a new ${\mathcal {G}} {}_\beta ^o$ -torsor $\mathcal {P} '=\mathcal {P} {}_\delta $ by twisting the ${\mathcal {G}} {}_\beta ^o$ -action by $\delta $ ,

$$ \begin{align*}g\star x=(\delta g\delta^{-1})\cdot x. \end{align*} $$

The new Frobenius $\phi _{\mathcal {P} '}$ is taken to be identical to $\phi _{\mathcal {P}} $ . This is indeed a morphism of ${\mathcal {G}} {}_\beta ^o$ -torsors since

$$ \begin{align*}\phi_{\mathcal{P} '}(g\star x)=\phi_{\mathcal{P}} ((\delta g\delta^{-1})\cdot x)=(\delta \phi(g)\delta^{-1})\cdot\phi_{\mathcal{P}} (x)=\phi(g)\star\phi_{\mathcal{P} '}(x), \end{align*} $$

where we used that $\delta \in G(\mathbb {Q} {}_p)$ . Finally, the framing $i^{\prime }_r$ is given by $i^{\prime }_r(g)=i_r(\delta g)$ . This is indeed an isomorphism of ${\mathcal {G}} {}_\beta ^o$ -bundles, since

$$ \begin{align*}i^{\prime}_r(hg)=i_r(\delta hg)=i_r((\delta h\delta^{-1})(\delta g))=(\delta h\delta^{-1})\cdot i_r(\delta g)=h\star i^{\prime}_r(g). \end{align*} $$

The compatibility with the Frobenius follows from

$$ \begin{align*}\phi_{\mathcal{P} '}(i^{\prime}_r(g))=\phi_{\mathcal{P}} (i_r(\delta g))=\delta \phi_{\mathcal{P}} (i_r(g))=\delta i_r(b\phi(g))=i^{\prime}_r(b\phi(g)). \end{align*} $$

Note that if $\delta \in {\mathcal {G}} {}_\beta ^o(\mathbb {Z} {}_p)$ , then the tuple $(S^\sharp , \mathcal {P} ', \phi _{\mathcal {P} '}, i^{\prime }_r)$ is isomorphic to $(S^\sharp , \mathcal {P}, \phi _{\mathcal {P}}, i_r)$ . Indeed, the map $x\mapsto \delta \cdot x$ defines an isomorphism $\alpha :\mathcal {P} \to \mathcal {P} '$ compatible with $\phi _{\mathcal {P}} $ and $\phi _{\mathcal {P} '}$ , and with $i_r$ and $i^{\prime }_r$ . For instance

$$ \begin{align*}\phi_{\mathcal{P} '}(\alpha(x))=\phi_{\mathcal{P}} (\delta\cdot x)=\phi(\delta)\cdot\phi_{\mathcal{P}} (x)=\delta\cdot\phi_{\mathcal{P}} (x)=\alpha(\phi_{\mathcal{P}} (x)). \end{align*} $$

Therefore we obtain an action of the factor group $N_{G(\mathbb {Q} {}_p)}({\mathcal {G}} {}_\beta ^o)/{\mathcal {G}} {}_\beta ^o(\mathbb {Z} {}_p)$ on $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu }$ . Restricting this action to ${\mathcal {G}} {}_\beta (\mathbb {Z} {}_p)/{\mathcal {G}} {}_\beta ^o(\mathbb {Z} {}_p)$ , we obtain the desired action of $\pi _0({\mathcal {G}} )^\phi $ on $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}^o, b, \mu }$ .

The same argument as the one above shows that $\delta \in {\mathcal {G}} {}_\beta (\mathbb {Z} {}_p)$ induces an isomorphism between the images of $(S^\sharp , \mathcal {P}, \phi _{\mathcal {P}, } i_r)$ and $(S^\sharp , \mathcal {P} ', \phi _{\mathcal {P} '}, i^{\prime }_r)$ in $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ under (4.4.1), hence we obtain a morphism $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} {}_\beta ^o, b, \mu }/\pi _0({\mathcal {G}} {}_\beta )^\phi \to \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ . Letting now $\bar {\beta }$ vary, we obtain a morphism from the LHS to the RHS in Theorem 4.4.1.

We define a map of v-sheaves

$$ \begin{align*}\kappa_G\circ \mathrm{sp}\colon \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}\to \underline{C_{{\mathcal{G}} }} , \end{align*} $$

where $\mathrm {sp}\colon |\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }|\to |X_{{\mathcal {G}}} (b, \mu ^{-1})|$ denotes the (continuous) specialisation map under the identification $(\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu })_{\mathrm {red}} =X_{{\mathcal {G}} }(b, \mu ^{-1} )$ , cf. Proposition 3.3.1. For $\tau \in C_{{\mathcal {G}} }$ , we denote by

(4.4.5) $$ \begin{align} \mathcal{M} {}^{\mathrm{int,\tau}}_{{\mathcal{G}} , b, \mu}= \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu}\times_{\underline{C_{{\mathcal{G}} }}}\underline{\{\tau\}}\subset \mathcal{M} {}^{\mathrm{int}}_{{\mathcal{G}} , b, \mu} \end{align} $$

the fibre over $\tau $ . This is an open and closed v-subsheaf of $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ . We call $ \mathcal {M} {}^{\mathrm {int,\tau }}_{{\mathcal {G}}, b, \mu }$ the component of $ \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }$ corresponding to $\tau $ .

Corollary 4.4.2. Let $({\mathcal {G}}, b, \mu )$ be an integral local shtuka datum such that ${\mathcal {G}} $ is a quasi-parahoric group scheme for G and let ${\mathcal {G}} \to {\mathcal {G}} '$ be a morphism extending the identity morphism of G in the generic fibres of quasi-parahoric group schemes for G with the same associated parahoric group scheme. Then for every $\tau \in \Omega _G/\pi _0({\mathcal {G}} )$ , with image ${\tau '\in \Omega _G/\pi _0({\mathcal {G}} ')}$ , the natural morphism $\mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}}, b, \mu }\to \mathcal {M} {}^{\mathrm {int}}_{{\mathcal {G}} ', b, \mu }$ induces an isomorphism

of v-sheaves.