2.1.2

We will use the notion of perfectly proper morphisms of perfect algebraic spaces (see [Reference ZhuZhu17, Definition A.18]). A morphism $f:X \to Y$ of pfp algebraic spaces over k is perfectly proper if and only if it is isomorphic to the perfection of a proper morphism of algebraic spaces of finite presentation over k (see [Reference ZhuZhu17, Lemma A.19]). We will often write perfectly proper algebraic space to mean a perfect algebraic space whose structure map to $\operatorname {Spec} k$ is perfectly proper.

Recall that a morphism $f:X \to Y$ of perfect algebraic spaces is called perfectly smooth of relative dimension d at x, where $x \in X$ , if there is an étale neighbourhood $U \to X$ of x and $V \to Y$ of $f(x)$ , such that $U \to X \to Y$ factors through a map $h:U \to V$ and such that h factors as

(2.1.2)

where $h'$ is étale and where $\operatorname {pr}$ is the projection onto V. It is called perfectly smooth of relative dimension d if it is perfectly smooth of relative dimension d at all points $x \in X$ . This property is preserved under base change, and the composition of a perfectly smooth morphism of relative dimension d with a perfectly smooth morphism of relative dimension $d'$ is perfectly smooth of relative dimension $d+d'$ . A morphism $X \to Y$ is called perfectly smooth if it is perfectly smooth of some dimension at every $x \in X$ . This property is also preserved under base change and composition.

The following lemma is a straightforward consequence of the definition.

We will later use the notion of normality for perfect algebraic spaces. Note that if an algebraic space Y is normal, then its perfection $Y^{\mathrm {perf}}$ is normal. Indeed, since normality is étale local, see [Sta23, Lemma 034F], this can be reduced to the affine case using the fact that Frobenius is affine, and then it follows from the fact that a filtered colimit of normal rings is normal.

Proof. Fix $x \in X$ with image $y \in Y$ . By definition, we know that f is perfectly smooth of relative dimension d at x. Thus, there are étale neighbourhoods $U \to X$ of x and $V \to Y$ of u, such that $U \to X \to Y$ factors through a map $h:U \to V$ and such that h factors as in equation (2.1.2).

Assume that Y is normal, and choose a normal deperfection $Y \to Y_0$ of Y using Lemma 2.1.6. Then, by topological invariance of the étale site, see [Sta23, Theorem 05ZH], there is a unique étale morphism $V_0 \to Y_0$ whose perfection recovers $V \to Y$ . Similarly, there is a unique étale morphism $U_0 \to \mathbb {A}^{d}_{k} \times V_0$ whose perfection recovers $h'$ . The induced map $U_0 \to Y_0$ is smooth because it is a composition of smooth maps. It follows from [Sta23, Lemma 034F] that $U_0$ is normal since $Y_0$ is normal. Hence, U is normal and so X is normal (in a neighbourhood of x) (by [Sta23, Lemma 034F]). This argument works for arbitrary x and thus proves the normality of X.

We will now assume that X is normal and show that Y is normal in a neighbourhood of y (if f is surjective, this thus shows that normality of X implies the normality of Y). Let $Y \to Y_0$ be any deperfection, then we will show that the normalisation $\tilde {Y}_0 \to Y_0$ is a universal homeomorphism. This implies that $\tilde {Y}_0 \to Y_0$ induces an isomorphism on perfections by [Reference Bhatt and ScholzeBS17, Lemma 3.8], and thus Y is the perfection of a normal scheme, and hence normal.

Let $V_0 \to Y_0$ and $U_0 \to \mathbb {A}^{d}_{k} \times V_0$ be as above. Since normalisation commutes with smooth base change, see [Sta23, Lemma 082F], we find that there is a Cartesian diagram (where $\tilde {U}_0 \to U_0$ is the normalisation of $U_0$ )

Since X is normal, we find that U is normal by [Sta23, Lemma 034F]. By [Sta23, Lemma 0BB4], there is a unique commutative diagram

This shows that $\tilde {U}_0 \to U_0$ is injective on k-points, and hence universally injective. Since $\tilde {U}_0 \to U_0$ is also surjective and closed, it follows that $\tilde {U}_0 \to U_0$ is a universal homeomorphism. It follows from [Sta23, Lemma 0CFX] that $\tilde {Y}_0 \to Y_0$ is a universal homeomorphism, as desired.

2.1.8

Let $f:X \to Y$ be a morphism of pfp algebraic spaces. We expect that if there is a perfectly smooth surjective map $g:Z \to X$ , such that $f \circ g$ is perfectly smooth, then f itself is perfectly smooth. However, we do not know how to prove this, hence, we make the auxiliary definitions 2.1.9 and 2.1.16.

This property is preserved under base change, and the composition of a weakly perfectly smooth morphism of relative dimension $d_1$ with a weakly perfectly smooth morphism of relative dimension $d_2$ is a weakly perfectly smooth morphism of relative dimension $d_1+d_2$ . Indeed, suppose we are given morphisms $f_1:X \to Y$ and $f_2:Y \to Z$ together with surjections $g_1:X' \to X$ and $g_2:Y' \to Y$ that are perfectly smooth of relative dimension $e_1$ and $e_2$ , respectively, and such that $X' \to X \to Y$ and $Y' \to Y \to Z$ are perfectly smooth of relative dimension $d_1+e_1$ and $d_2+e_2$ , respectively. Then $X":=X' \times _{Y} Y'$ is perfectly smooth over X of relative dimension $e_1+e_2$ , and $X" \to X \to Z$ is perfectly smooth of relative dimension $e_2+d_2+e_1+d_1$ by writing it as the composition of $X" \to Y'$ and $Y' \to Z$ .

The following lemmas show that the integer d is well-defined.

Proof. Since the dimension can be computed étale locally, we may assume that X and Y are equidimensional pfp schemes. We can, moreover, compute the dimensions of X and Y in terms of the Krull dimensions of their local rings at closed points since X and Y are pfp. So let $x \in X$ be a closed point with image $y \in Y$ , then since f is the perfection of a finite type morphism between Noetherian schemes, it follows from [Sta23, Lemma 00OM] that

(2.1.3) $$ \begin{align} \operatorname{Dim} \mathcal{O}_{X,x} \le \operatorname{Dim} \mathcal{O}_{Y,y} + \operatorname{Dim} \mathcal{O}_{f^{-1}(y),x}. \end{align} $$

Since f is flat (perfectly smooth morphisms are clearly flat, and flatness can be checked after an fpqc cover (in particular a perfectly smooth cover)), it follows that going down holds for $\mathcal {O}_{X,x} \to \mathcal {O}_{Y,y}$ (see [Sta23, Lemma 00HS]). The inequality in (2.1.3) is then an equality by [Sta23, Lemma 00ON] applied to the local rings of a choice of deperfection of f.

The following lemma has a straightforward proof.

A morphism $f:Y \to Z$ is called weakly perfectly smooth if there is a perfectly smooth surjection $g:X \to Y$ , such that $f \circ g$ is perfectly smooth. The following lemma relates this to Definition 2.1.9, the proof is straightforward.

2.1.15

We follow [Sta23, Section 04XB] to define certain properties of morphisms of prestacks on $\mathbf {Aff}_k^{\mathrm {perf}}$ that are representable in morphisms of perfect algebraic spaces. For example, a morphism $f:X \to Y$ of prestacks that is representable in perfect algebraic spaces is called perfectly smooth if it is representable in perfectly smooth morphisms of perfect algebraic spaces. In other words, if for every morphism $T \to Y$ , where T is a perfect algebraic space, the base change $X_T \to T$ is a perfectly smooth morphism of perfect algebraic spaces.

A pfp algebraic stack is a stack Y on $\mathbf {Aff}_k^{\mathrm {perf}}$ for the étale topology with diagonal representable in pfp algebraic spaces that admits a perfectly smooth surjectiveFootnote ⁶ map $f:U \to Y$ from a pfp algebraic space. The main example that we will be interested in is the quotient stackFootnote ⁷ $[X/G]$ of a pfp algebraic space X by a pfp group scheme G. This is a pfp algebraic stack because $X \to [X/G]$ is perfectly smooth since G is perfectly smooth over $\operatorname {Spec} k$ (see Example 2.1.4). We will also need a notion of weak perfect smoothness for morphisms of pfp algebraic stacks that are not necessarily representable.

As before, this property is preserved under base change and composition. If $f:Y \to Z$ is representable, then this is (per definition) equivalent to asking that $f:Y \to Z$ is representable in weakly perfectly smooth morphisms of perfect algebraic spaces.

Lemma 2.1.20. Suppose that X is a pfp algebraic space that is equidimensional of dimension d with an action of a pfp group scheme G, and let Y be a pfp algebraic space together with a weakly perfectly smooth morphism

$$ \begin{align*} f:Y \to \left[X/G \right]. \end{align*} $$

Then Y is equidimensional if and only if f is weakly perfectly smooth of relative dimension n, where $\operatorname {Dim} Y =d+n-\operatorname {Dim} G$ .

Proof. Consider the fibre product diagram

The lemma is now a straightforward consequence of Lemma 2.1.11.

2.2.1

Let $k=\overline {\mathbb {F}}_{p}$ , and let $\breve {\mathbb {Z}}_p=W(k)$ and $\breve {\mathbb {Q}}_p=\breve {\mathbb {Z}}_p[1/p]$ , which come equipped with an automorphism $\sigma $ coming from the absolute Frobenius on k. Let G be a connected reductive group over $\mathbb {Q}_p$ and let $B(G,\mathbb {Q}_p)$ (respectively, $B(G,\breve {\mathbb {Q}}_p)$ ) denote the (extended) Bruhat–Tits building of G over $\mathbb {Q}_p$ (respectively, $\breve {\mathbb {Q}}_p$ ). For a nonempty bounded subset $\Xi \subset B(G,\mathbb {Q}_p)$ which is contained in an apartment, we let $G(\mathbb {Q}_p)_\Xi $ (respectively, $G(\breve {\mathbb {Q}}_p)_{\Xi }$ ) denote the subgroup of $G(\mathbb {Q}_p)$ (respectively, $G(\breve {\mathbb {Q}}_p)$ ) which fixes $\Xi $ pointwise. By the main result of [Reference Bruhat and TitsBT84], there exists a smooth affine group scheme $\tilde {\mathcal {G}}_{\Xi }$ over $\mathbb {Z}_{p}$ with generic fibre G which is uniquely characterised by the property $\tilde {\mathcal {G}}_{\Xi }(\breve {\mathbb {Z}}_p)=G(\breve {\mathbb {Q}}_p)_{\Xi }$ . We call such a group scheme the Bruhat–Tits stabiliser group scheme associated to $\Xi $ . If $\Xi =\{x\}$ is a point, we write $G(\mathbb {Q}_p)_x$ (respectively, $\mathcal {\tilde {G}}_x)$ for $G(\mathbb {Q}_p)_{\{x\}}$ (respectively, $\tilde {\mathcal {G}}_{\{x\}}$ ).

For $\Xi \subset B(G,\mathbb {Q}_p)$ as above, we let $\mathcal {G}_{\Xi }$ denote the ‘connected stabiliser’ (cf. [Reference Bruhat and TitsBT84, Section 4]). We are mainly interested in the case that $\Xi $ is a point or an open facet $\mathfrak {f}$ . In this case, $\mathcal {G}_{\mathfrak {f}}$ (respectively, $\mathcal {G}_x$ ) is the parahoric group scheme associated to $\mathfrak {f}$ (respectively, x).

We may also consider the corresponding objects over $\breve {\mathbb {Q}}_p$ , and we use the same notation in this case. When it is understood which point of $B(G,\mathbb {Q}_p)$ or $B(G,\breve {\mathbb {Q}}_p)$ we are referring to, we simply write $\tilde {\mathcal {G}}$ and $\mathcal {G}$ for the corresponding group schemes.

An important case that we need for applications is when $\tilde {\mathcal {G}}_x=\mathcal {G}_x$ , that is, when the parahoric is equal to the Bruhat–Tits stabiliser. When this happens, we necessarily have $\tilde {\mathcal {G}}_{\mathfrak {f}}=\tilde {\mathcal {G}}_x$ , where $\mathfrak {f}$ is the facet containing x, and $x\in \mathfrak {f}$ is a point ‘in general position’. A parahoric group scheme $\mathcal {G}$ over $\mathbb {Z}_{p}$ (respectively, $\breve {\mathbb {Z}}_p$ ) is called a connected parahoric if there exists $x\in B(G,\mathbb {Q}_p)$ (respectively, $x\in B(G,\breve {\mathbb {Q}}_p)$ ), such that $ \mathcal {G}=\mathcal {G}_x=\tilde {\mathcal {G}}_x$ .

Let $\pi _1(G)$ be the algebraic fundamental group of $G \otimes \overline {\mathbb {Q}}_p$ , equipped with its action of $\operatorname {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$ (see the Introduction of [Reference BorovoiBor98]), and let $I \subset \operatorname {Gal}(\overline {\mathbb {Q}}_p/\mathbb {Q}_p)$ be the inertia group.

2.2.3

Let $S \subset G_{\breve {\mathbb {Q}}_p}$ be a maximal $\breve {\mathbb {Q}}_p$ -split torus defined over $\mathbb {Q}_p$ which exists by [Reference Kaletha and PrasadKP23, Axiom 4.1.27.UR2], and let T be its centraliser. Then T is a maximal torus of G because $G_{\breve {\mathbb {Q}}_p}$ is quasi-split by a theorem of Steinberg (see [Reference Kaletha and PrasadKP23, Theorem 2.3.3]). Choose a $\sigma $ -invariant alcove $\mathfrak {a}$ in the apartment of $B(G,\breve {\mathbb {Q}}_p)$ associated to S. Let N be the normaliser of T in $G_{\breve {\mathbb {Q}}_p}$ . We define the relative Weyl group as

$$ \begin{align*} W_0:=N(\breve{\mathbb{Q}}_p)/T(\breve{\mathbb{Q}}_p) \end{align*} $$

and the Iwahori–Weyl group (or extended affine Weyl group) as

$$ \begin{align*} \tilde{W}:=N(\breve{\mathbb{Q}}_p)/\mathcal{T}(\breve{\mathbb{Z}}_p), \end{align*} $$

where $\mathcal {T}$ over $\breve {\mathbb {Z}}_p$ is the connected Néron model of T. There is a short exact sequence (see [Reference Pappas and RapoportPR08, Definition 7 of the Appendix])

$$ \begin{align*} 0 \to X_{\ast}(T)_I \to \tilde{W} \to W_0 \to 0, \end{align*} $$

where I is the inertia group and $X_{\ast }(T)_I$ denotes the inertia coinvariants of the cocharacter lattice $X_{\ast }(T)$ of T. The map $X_{\ast }(T)_I \to \tilde {W}$ is denoted on elements by $\lambda \mapsto t^{\lambda }$ . Let $\mathbb {S} \subset \tilde {W}$ denote the set of simple reflections in the walls of $\mathfrak {a}$ , and let $\tilde {W}_a$ be the subgroup of $\tilde {W}$ generated by $\mathbb {S}$ , which we will call the affine Weyl group.

Parahoric subgroups $\mathcal {K}$ of $G(\breve {\mathbb {Q}}_p)$ that contain the Iwahori subgroup corresponding to $\mathfrak {a}$ are called standard parahoric subgroups; they correspond to subsets $K \subset \mathbb {S}$ , such that the subgroup $W_K$ generated by K is finite; we will call such subsets types. This identification is Frobenius equivariant in the sense that $\sigma (\mathcal {K})$ corresponds to $\sigma (K)$ . In particular, a subset $K \subset \mathbb {S}$ corresponds to a parahoric subgroup of G if and only if $\sigma (K)=K$ ; note that our fixed Iwahori subgroup corresponds to $\emptyset \subset \mathbb {S}$ . There are parahoric group schemes $\mathcal {G}_K$ over $\breve {\mathbb {Z}}_p$ associated to types K as above, and we have identifications $\sigma ^{\ast } \mathcal {G}_K \simeq \mathcal {G}_{\sigma (K)}$ . In particular, if K is stable under $\sigma $ , then $\mathcal {G}_K$ is defined over $\mathbb {Z}_p$ . The maximal reductive quotient $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ of the special fibre $\overline {\mathcal {G}}_K$ of $\mathcal {G}_K$ is a split reductive group over the residue field k of $\breve {\mathbb {Q}}_p$ , and the image of $\mathcal {G}_{\emptyset }$ in $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ is a Borel subgroup. The set of simple roots of $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ with respect to this Borel subgroup can be identified with K. The following lemma should be compared with [Reference Kisin and PappasKP18, Remark 4.2.14.b)].

2.2.5

A type $K \subset \mathbb {S}$ is called very special if $W_K \subset \tilde {W}$ maps isomorphically onto $W_0$ . Very special types correspond to very special vertices in $\mathfrak {a}$ , see [Reference Kaletha and PrasadKP23, Lemma 1.3.42, Proposition 1.3.43], where they are called extra special vertices. If K is $\sigma $ -stable, then the parahoric subgroup $\mathcal {G}_K(\mathbb {Z}_{p})$ associated to a very special type is called a very special parahoric subgroup. A fact that will be crucial for us is that there exists a $\sigma $ -stable very special type K if G is quasi-split (see [Reference Kaletha and PrasadKP23, Proposition 10.2.1]). Thus, if G is quasi-split, then the standard Iwahori subgroup $\mathcal {G}_{\emptyset }(\mathbb {Z}_{p})$ contains a very special parahoric subgroup.

2.2.6

There is a split short exact sequence (our choice of $\mathfrak {a}$ provides a splitting, see [Reference Pappas and RapoportPR08, Lemma 14 of the appendix])

(2.2.1) $$ \begin{align} 0 \to \tilde{W}_a \to \tilde{W} \to \pi_1(G)_I \to 0. \end{align} $$

The affine Weyl group $\tilde {W}_a$ has the structure of a Coxeter group, and we will use this to define a Bruhat order (denoted by $\le $ ) and a notion of length on $\tilde {W}$ , by splitting (2.2.1) and regarding $\pi _1(G)_I \subset \tilde {W}$ as the subset of length zero elements. We will write $\ell (w)$ for the length of an element of $\tilde {W}$ . Similarly, we define a partial order $\le $ and a length function on $W_K \backslash \tilde {W}/W_K$ by taking minimal length representatives of double cosets.

2.2.7

In this section, we will recall some definitions from [Reference ZhuZhu17, Reference Xiao and ZhuXZ17, Reference Shen, Yu and ZhangSYZ21] and state some results. Let the notation be as in Sections 2.2.1 and 2.2.3, so, in particular, G denotes a connected reductive group over $\mathbb {Q}_p$ . Let $\mathcal {G}_{K}$ be a parahoric group scheme over $\breve {\mathbb {Z}}_p$ corresponding to a $\sigma $ -stable type $K \subset \mathbb {S}$ . For an object R of $\mathbf {Aff}_k^{\mathrm {perf}}$ , we set

$$ \begin{align*} D_R=\operatorname{Spec} W(R), \qquad D_R^{\ast}=\operatorname{Spec} W(R)[1/p], \end{align*} $$

where $W(R)$ denotes the ring of p-typical Witt vectors of R. We define group-valued functors on $\mathbf {Aff}_k^{\mathrm {perf}}$ sending an object R to

$$ \begin{align*} LG(R)&:=G(D_R^{\ast}) \\ L^+ \mathcal{G}_{K}(R)&:= \mathcal{G}_K(D_R) \\ L^m \mathcal{G}_K(R)&:=\mathcal{G}_K\left(W(R)/p^m W(R) \right), \end{align*} $$

which we call the loop group, respectively, the positive loop group, respectively, the m-truncated loop group. It follows from [Reference ZhuZhu17, Section 1.1] that $L^m \mathcal {G}_K$ and $L^+ \mathcal {G}_{K}$ are representable in perfect schemes over k and that $L^+ \mathcal {G}_{K} = \varprojlim _m L^m \mathcal {G}_K$ . Moreover, [Reference ZhuZhu17, Proposition 1.1] tells us that $LG$ is representable by an ind-(perfect scheme), which means that it is isomorphic to an inductive limit of perfect schemes along closed immersions. By [Reference ZhuZhu17, Lemma 1.2. (i)], the natural map $L^+ \mathcal {G}_{K} \to LG$ is a closed immersion.

2.2.8

Fix an algebraic closure $\overline {\mathbb {F}}_{p}$ of $\mathbb {F}_{p}$ , and set $k=\overline {\mathbb {F}}_{p}$ . Let R be a perfect k-algebra, and let $\mathcal {E}$ and $\mathcal {F}$ be $\mathcal {G}_{K}$ -torsors on $D_R$ .Footnote ⁸ Recall from [Reference Xiao and ZhuXZ17, Section 3.1.3] that a modification $\beta : \mathcal {E} \dashrightarrow \mathcal {F}$ is an isomorphism of G-torsors

$$ \begin{align*} \beta: { \left. \mathcal{E} \vphantom{\big|} \right|{}_{D_R^{\ast}} } \to { \left. \mathcal{F} \vphantom{\big|} \right|{}_{D_R^{\ast}} }. \end{align*} $$

It follows from the proof of [Reference ZhuZhu17, Lemma 1.3] that there is an étale cover $\operatorname {Spec} R' \to \operatorname {Spec} R$ , such that $\mathcal {E}$ is trivial after pullback along $\operatorname {Spec} D_{R'} \to \operatorname {Spec} D_{R}$ . Therefore, we can also think of $\mathcal {G}_{K}$ -torsors over $D_R$ as étale $L^+\mathcal {G}_{K}$ -torsors over $\operatorname {Spec} R$ .

We define the (partial) affine flag variety $\operatorname {Gr}_{K}$ to be the functor on $\mathbf {Aff}_k^{\mathrm {perf}}$ sending R to the set of isomorphism classes of modifications

$$ \begin{align*} \alpha:\mathcal{E} \dashrightarrow \mathcal{E}^0, \end{align*} $$

where $\mathcal {E}$ is a $\mathcal {G}_{K}$ -torsor over $D_R$ and where $\mathcal {E}^0$ is the trivial $\mathcal {G}_{K}$ -torsor over $D_R$ . There is a natural action of $LG$ , thought of as the functor

$$ \begin{align*}R \mapsto \operatorname{Aut}({ \left. \mathcal{E}^0 \vphantom{\big|} \right|{}_{D_R^{\ast}} })\end{align*} $$

on $\operatorname {Gr}_{K}$ , by postcomposing $\alpha $ with an automorphism of the restriction to $D_R^{\ast }$ of $\mathcal {E}^0$ , and the orbit of the k-point of $\operatorname {Gr}_{K}$ given by the identity modification $\mathcal {E}^0 \to \mathcal {E}^0$ induces a map $O:LG \to \operatorname {Gr}_{K}$ . The map O induces an identification (that we will implicitly use from now on)

$$ \begin{align*} \operatorname{Gr}_{K}(k) \simeq G(\breve{\mathbb{Q}}_p)/\mathcal{G}_{K}(\breve{\mathbb{Z}}_p). \end{align*} $$

It is a result of [Reference ZhuZhu17, Reference Bhatt and ScholzeBS17] that $\operatorname {Gr}_{K}$ is representable by an inductive limit of perfections of projective k-schemes, with closed immersions as transition maps. In short, $\operatorname {Gr}_{K}$ is ind-perfectly projective. We also define the Hecke stack $\operatorname {Hk}_{K}$ to be the presheaf in groupoids on $\mathbf {Aff}_k^{\mathrm {perf}}$ sending R to the groupoid of modifications $\beta : \mathcal {E} \dashrightarrow \mathcal {F}$ . The natural map $\operatorname {Gr}_{K} \to \operatorname {Hk}_{K}$ is an $L^+ \mathcal {G}_{K}$ -torsor for the étale topology, where $L^+ \mathcal {G}_{K}$ acts on $\operatorname {Gr}_{K}$ via the closed immersion $L^+ \mathcal {G}_{K} \subset LG$ .

2.2.9

Recall from [Reference Shen, Yu and ZhangSYZ21, Definition 4.1.3] that a (local) $\mathcal {G}_K$ -shtuka over a perfect k-algebra R is a pair $(\mathcal {E}, \beta )$ , where $\mathcal {E}$ is a $\mathcal {G}_{K}$ -torsor over $D_R$ and where $\beta $ is a modification $\beta :\sigma ^{\ast } \mathcal {E} \dashrightarrow \mathcal {E}$ . Here, $\sigma :D_R \to D_R$ denotes the Frobenius morphism induced from the absolute Frobenius on R, and we consider the restriction of $\sigma ^{\ast } \mathcal {E}$ to $D_R^{\ast }$ as a G-torsor via the isomorphism $\sigma :\sigma ^{\ast } G \to G$ , coming from the fact that G is defined over $\mathbb {Q}_p$ . A morphism of shtukas $(\mathcal {E}, \beta ) \to (\mathcal {E}', \beta ')$ is an isomorphism $f:\mathcal {E} \to \mathcal {E}'$ of $\mathcal {G}_{K}$ -torsors, such that the following diagram commutes

We will write $\operatorname {Sht}_{G,K}(R)$ for the groupoid of $\mathcal {G}_K$ -shtukas over R and $\operatorname {Sht}_{G,K}$ for the presheaf in groupoids on $\mathbf {Aff}_k^{\mathrm {perf}}$ sending R to $\operatorname {Sht}_{G,K}(R)$ .

Example 2.2.10. Our main examples of shtukas come from p-divisible groups. More precisely, for $\mathcal {G}_K=\operatorname {GL}_{n, \mathbb {Z}_p}$ , a $\mathcal {G}_K$ -shtuka over a perfect ring R is a projective module M of rank n over $W(R)$ together with an isomorphism

$$ \begin{align*} \beta:\sigma^{\ast}M[1/p] \to M[1/p]. \end{align*} $$

If the map $\beta $ satisfies $pM \subset \beta (\sigma ^{\ast }M) \subset M$ , then the pair $(M,\beta )$ is a (contravariant) Dieudonné module. By a result of Gabber, see [Reference LauLau18], there is a p-divisible group over $\operatorname {Spec} R$ with contravariant Dieudonné module $(M, \beta )$ .

2.2.11

For an inclusion of types $J \subset K$ , there is a natural morphism of parahoric group schemes $\mathcal {G}_J \to \mathcal {G}_K$ . The induced morphism on loop groups $L^+\mathcal {G}_{J} \subset L^+\mathcal {G}_{K}$ is a closed immersion, since this induced morphism commutes with the natural closed immersions of source and target to $LG$ (see [Reference ZhuZhu17, Lemma 1.2. (i)]).Footnote ⁹

If J and K are $\sigma $ -stable, then pushing out torsors along $L^+\mathcal {G}_{J} \to L^+\mathcal {G}_{K}$ induces a forgetful map

$$ \begin{align*} \operatorname{Sht}_{G,J} \to \operatorname{Sht}_{G,K}. \end{align*} $$

In this section, we will show that these forgetful maps are representable in perfectly proper algebraic spaces, which is an analogue of [Reference Pappas and RapoportPR08, Proposition 8.7].

Let $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ be the maximal reductive quotient of the $1$ -truncated loop group $L^1 \mathcal {G}_{K}=\overline {\mathcal {G}}_K$ , and let $H_J$ be the image of $\mathcal {G}_{J}$ in $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ ; it is a standard parabolic subgroup of type $J \subset K$ (recall that K can be identified with the set of simple roots of $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ with respect to the Borel B that is the image of $\mathcal {G}_{\emptyset } \to (\overline {\mathcal {G}}_K)^{\mathrm {red}}$ ). Recall that for a perfect group scheme H, we write $\mathbf {B} H$ for the classifying stack of H; in other words, $\mathbf {B}H$ is the groupoid valued functor that sends an object R of $\mathbf {Aff}_k^{\mathrm {perf}}$ to the groupoid of H torsors (in the étale topology) over $\operatorname {Spec} R$ . There is a natural morphism $\operatorname {Spec} k \to \mathbf {B} H$ corresponding to the trivial H-torsor over $\operatorname {Spec} k$ , which induces an isomorphism $\left [ \operatorname {Spec} k /H \right ] \to \mathbf {B}H$ .

Proof. Let R be a perfect k-algebra, and let X be an $L^+ \mathcal {G}_{K}$ torsor over $\operatorname {Spec} R$ represented by a map $\operatorname {Spec} R \to \mathbf {B} L^+ \mathcal {G}_{K}$ . It follows from the definition of quotient stacks that both squares in the following diagram of stacks are Cartesian

By [Reference ZhuZhu17, Lemma 1.3], there is an étale cover $T \to \operatorname {Spec} R$ , such that $X_T$ is isomorphic to the trivial $L^+ \mathcal {G}_{K}$ torsor over T, hence $[X/L^+ \mathcal {G}_{J}]$ is étale locally isomorphic to $\operatorname {Spec} R \times [L^+ \mathcal {G}_{K} / L^+ \mathcal {G}_{J}]$ . Therefore, it suffices to show that $[L^+ \mathcal {G}_{K} / L^+ \mathcal {G}_{J}]$ is representable in a perfectly proper scheme.Footnote ¹¹ We will argue as in the proof of [Reference Pappas and RapoportPR08, Proposition 8.7] that there is an isomorphism

$$ \begin{align*} [L^+ \mathcal{G}_{K} / L^+ \mathcal{G}_{J}] \simeq [(\overline{\mathcal{G}}_K)^{\mathrm{red}}/H_{J}], \end{align*} $$

and the latter is representable in a perfectly proper scheme because it is the perfection of a partial flag variety for $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ . By Lemma 2.3.5 below, the following commutative diagram of perfect group schemes is Cartesian

It now follows formally that $ [L^+ \mathcal {G}_{K} / L^+ \mathcal {G}_{J}] \simeq [(\overline {\mathcal {G}}_K)^{\mathrm {red}}/H_J]$ .

Proof. This follows because the following diagram is Cartesian

(2.2.2)

Indeed, this is a straightforward consequence of the definitions (a $\mathcal {G}_{J}$ -shtuka is the same thing as a $\mathcal {G}_{K}$ -shtuka $(\mathcal {E}, \beta )$ together with an $L^+ \mathcal {G}_{J}$ -torsor $\mathcal {E}'$ and an isomorphism $\alpha :\mathcal {E}' \times _{L^+ \mathcal {G}_{J}} L^+ \mathcal {G}_{K} \simeq \mathcal {E}$ , because the natural map $\mathcal {G}_{J} \to \mathcal {G}_{K}$ is an isomorphism over $\mathbb {Q}_p$ ).

2.2.14 Relative position

It follows from the discussion in [Reference He and ZhouHZ20, Section 3.6] that there is an $L^+ \mathcal {G}_{K}$ -equivariant stratification

$$ \begin{align*} \operatorname{Gr}_{K}=\bigcup_{w \in W_K \backslash \tilde{W}/W_K} \operatorname{Gr}_{K}(w), \end{align*} $$

where each $\operatorname {Gr}_{K}(w)$ is a locally closed subscheme of $\operatorname {Gr}_{K}$ , such that on k-points, we recover the Bruhat–Tits decomposition

$$ \begin{align*} \operatorname{Gr}_{K}(k)=G(\breve{\mathbb{Q}}_p)/\mathcal{G}_{K}(\breve{\mathbb{Z}}_p) = \bigcup_{w \in W_K \backslash \tilde{W}/W_K} \mathcal{G}_{K}(\breve{\mathbb{Z}}_p) \dot{w} \mathcal{G}_{K}(\breve{\mathbb{Z}}_p)/\mathcal{G}_{K}(\breve{\mathbb{Z}}_p) \end{align*} $$

(see [Reference Pappas and RapoportPR08, Proposition 8 of the appendix]). We deduce from this that we get a decomposition

$$ \begin{align*} \left[L^+ \mathcal{G}_{K} \backslash \operatorname{Gr}_{K} \right] =: \operatorname{Hk}_{K} = \bigcup_{w \in W_K \backslash \tilde{W}/W_K} \operatorname{Hk}_{K}(w), \end{align*} $$

where $\operatorname {Hk}_{K}(w) = \left [L^+ \mathcal {G}_{K} \backslash \operatorname {Gr}_{K}(w) \right ]$ . It follows, moreover, from [Reference He and ZhouHZ20, Section 3.6] that the closure of $\operatorname {Gr}_{K}(w)$ is equal to

$$ \begin{align*} \operatorname{Gr}_{K}(\le \! w):=\bigcup_{w' \le w} \operatorname{Gr}_{K}(w'). \end{align*} $$

Furthermore, when $K=\emptyset $ , the Schubert cell $\operatorname {Gr}_{K}(w)$ is equidimensional of dimension equal to the length $\ell (w)$ of w. This latter statement is proved for a certain Demazure resolution $D_w \to \operatorname {Gr}_{K}(\le \! w)$ in [Reference He and ZhouHZ20, Proposition 3.4], and follows for $\operatorname {Gr}_{K}(\le \! w)$ (and hence $\operatorname {Gr}_{K}(w)$ ) since the map $D_w \to \operatorname {Gr}_{K}(\le \! w)$ is birational as explained in the proof of [Reference He and ZhouHZ20, Proposition 3.7].

2.2.15

Let $\{\mu \}$ be a $G(\overline {\mathbb {Q}}_p)$ -conjugacy class of cocharacters of $G_{\overline {\mathbb {Q}}_p}$ . Recall that we fixed a maximal torus T of G in Section 2.2.3. Choose a Borel B of $G_{\breve {\mathbb {Q}}_p}$ containing $T_{\breve {\mathbb {Q}}_p}$ , and let $\overline {\mu }$ be the image in $X_{\ast }(T)_I$ of a B-dominant representative of $\{\mu \}$ . The set of $\{\mu \}$ -admissible elements is defined as

$$ \begin{align*} \operatorname{Adm}(\{\mu\})=\{ w \in \tilde{W} \; : \; w \le t^{x(\overline{\mu})} \text{ for some } x \in W_0 \}. \end{align*} $$

There is a unique element $\tau =\tau _{\mu } \in \operatorname {Adm}(\{\mu \})$ of length zero and, in fact, $\operatorname {Adm}(\{\mu \}) \subset \tilde {W}_a \tau $ . For K a $\sigma $ -stable type, we define $\operatorname {Adm}(\{\mu \})_K$ as the image of $\operatorname {Adm}(\{\mu \})$ under $\tilde {W} \to W_K \backslash \tilde {W} / W_K$ . We write ${}^{K} \!\! \operatorname {Adm}(\{\mu \})$ for $\operatorname {Adm}(\{\mu \}) \cap \! {}^{K} \tilde {W}$ , where $\! {}^{K} \tilde {W} \subset \tilde {W}$ denotes the subset of elements that are of minimal length in their left $W_K$ -coset.

If $\{\mu \}$ is minuscule and K is a $\sigma $ -stable type, then we define the perfect local model attached to $\{\mu \}$ and K to be the perfectly projective closed subscheme of $\operatorname {Gr}_{K}$ given by

$$ \begin{align*} \operatorname{M}^{\mathrm{loc}}_{K, \{\mu\}}:=\bigcup_{w \in \operatorname{Adm}(\{\mu\})_K} \operatorname{Gr}_{K}(w). \end{align*} $$

This definition is motivated by the discussion in [Reference Shen, Yu and ZhangSYZ21, Section 2.1.7] and, in particular, [Reference Shen, Yu and ZhangSYZ21, Corollary 2.1.11]. It follows from the discussion in [Reference Shen, Yu and ZhangSYZ21, Section 2.1.7] that the scheme $\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ is equidimensional of dimension $d=\langle 2 \rho , \overline {\mu } \rangle $ , which is precisely the dimension of the flag variety for G associated to $\mu $ .

2.2.16

Let $\{\mu \}$ be a conjugacy class of cocharacters of $G_{\overline {\mathbb {Q}}_p}$ as above, and let $\operatorname {Adm}(\{\mu \})_K$ be the $\mu $ -admissible set. Recall that the stack $\operatorname {Hk}_{K}$ is the moduli stack of modifications $\mathcal {E} \dashrightarrow \mathcal {F}$ of $L^+ \mathcal {G}_{K}$ -torsors. We define a map $\operatorname {Rel}:\operatorname {Sht}_{G,K} \to \operatorname {Hk}_{K}$ sending $(\mathcal {E}, \beta )$ to $\beta :\sigma ^{\ast } \mathcal {E} \dashrightarrow \mathcal {E}$ . For $w \in W_K \backslash \tilde {W}/W_K$ , we have the locally closed substack $\operatorname {Hk}_{K}(w) \subset \operatorname {Hk}_{K}$ from Section 2.2.14, and its pullback along $\operatorname {Rel}$ defines a locally closed substack

$$ \begin{align*} \operatorname{Sht}_{G,K}(w) \subset \operatorname{Sht}_{G,K}. \end{align*} $$

Following [Reference Shen, Yu and ZhangSYZ21, Definition 4.1.3], we define the stack of shtukas of level $\mathcal {G}_K$ and type $\mu $ to be

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}:=\bigcup_{w \in \operatorname{Adm}(\{\mu\})_K} \operatorname{Sht}_{G,K}(w); \end{align*} $$

it is a closed substack of $\operatorname {Sht}_{G,K}$ by the discussion in Section 2.2.14. If $J \subset K$ is another $\sigma $ -stable type, then the following diagram commutes by definition of $\operatorname {Adm}(\{\mu \})_J$ and $\operatorname {Adm}(\{\mu \})_K$ (but it is generally not Cartesian)

2.3.1

Let $\beta _K: L^+ \mathcal {G}_{K} \to (\overline {\mathcal {G}}_K)^{\mathrm {red}}$ be the natural map, where $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ is the maximal reductive quotient of $\overline {\mathcal {G}}_K=L^1 \mathcal {G}_{K}$ . Define $\operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}:=\ker \beta _K \backslash \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}}$ ; it is a $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ -torsor over $\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ . We then define

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}^{(\infty,1)}:=\left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^+ \mathcal{G}_{K}} \right]. \end{align*} $$

It follows from Lemma 2.1.1 that the twisted conjugation action of $L^+ \mathcal {G}_{K}$ on $\operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}$ factors through the action of $L^m \mathcal {G}_{K}$ for $m \gg 0$ . Indeed, this follows by applying the lemma to the inverse system $\{L^m \mathcal {G}_K \times \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}\}_{m \in \mathbb {Z}}$ and the action map $L^+ \mathcal {G}_{K} \times \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}} \to \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}$ from the inverse limit to the pfp algebraic space $\operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}$ . For such m, we define the stack of $(m)$ -restricted shtukas of type $\{\mu \}$ by

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}^{(m,1)}:=\left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right]. \end{align*} $$

Note that there are natural morphisms

(2.3.1) $$ \begin{align} \operatorname{Sht}_{G,K, \{\mu\}}=\left[\frac{\operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^+ \mathcal{G}_{K}} \right] \to \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^+ \mathcal{G}_{K}} \right] \to \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right]=\operatorname{Sht}_{G,K, \{\mu\}}^{(m,1)} \end{align} $$

induced by the natural map $\operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}} \to \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}$ and the natural map $L^+ \mathcal {G}_{K} \to L^m \mathcal {G}_{K}$ .

Remark 2.3.2. There is a ‘local model diagram’

The left-hand map is an $L^m \mathcal {G}_{K}$ -torsor while the right-hand map is a $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ -torsor. In particular, the stack $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ is an equidimensional pfp algebraic stack.Footnote ¹³ Indeed, $\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ is pfp and equidimensional, and since the right-hand map is perfectly smooth of relative dimension $\operatorname {Dim} (\overline {\mathcal {G}}_K)^{\mathrm {red}}$ , we find that $\operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{K, \{\mu \}}$ is pfp and equidimensional by Lemma 2.1.10.

2.3.3

The goal of this section is to compare $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ and $\operatorname {Sht}_{G,\emptyset , \{\mu \}}^{(m',1)}$ . Unfortunately, there is no natural map between them when $K \not =\emptyset $ . However, we will be able to construct a correspondence between them instead, and study its properties (see Proposition 2.3.4, Lemma 2.3.9 and Section 2.3.10).

Consider the closed immersion $L^+ \mathcal {G}_{\emptyset } \subset L^+ \mathcal {G}_{K}$ , which induces a closed immersion $B \subset (\overline {\mathcal {G}}_K)^{\mathrm {red}}$ , where B is the image of $L^1 \mathcal {G}_{\emptyset }$ in $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ ; let $\gamma :L^+ \mathcal {G}_{\emptyset } \to B$ be the natural surjection. By Lemma 2.1.1, we can choose $m \gg 0$ , such that the action $\operatorname {Ad}_{\sigma } L^+ \mathcal {G}_{K}$ on $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}}$ factors through $L^m \mathcal {G}_{K}$ . As in equation (2.3.1), the natural maps

$$ \begin{align*} \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}} &\to \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}} \\ L^+ \mathcal{G}_{K} &\to L^m \mathcal{G}_{K} \end{align*} $$

induce a natural map

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}=\left[\frac{\operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^+ \mathcal{G}_{K}} \right] \to \left[ \frac{ \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}}{ \operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right]. \end{align*} $$

For m as above, let $H_m$ be the image of $L^+ \mathcal {G}_{\emptyset }$ in $L^m \mathcal {G}_{K}$ . Since $\operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}} \subset \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}}$ , it follows that the action of $L^+ \mathcal {G}_{\emptyset } \subset L^+ \mathcal {G}_{K}$ on $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}}$ factors through $H_m$ . Therefore, there is a natural map

$$ \begin{align*} \operatorname{Sht}_{G,\emptyset, \{\mu\}}=\left[\frac{\operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^+ \mathcal{G}_{\emptyset}} \right] \to \left[ \frac{ \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}}{ \operatorname{Ad}_{\sigma} H_m} \right] \end{align*} $$

induced by $\operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}} \to \ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}}$ and $L^+ \mathcal {G}_{\emptyset } \to H_m$ .

Proposition 2.3.4. If m is an integer, such that the action $\operatorname {Ad}_{\sigma } L^+ \mathcal {G}_{K}$ on $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}}$ factors through $L^m \mathcal {G}_{K}$ , then the diagram

where the right vertical map is induced by the closed immersions and is Cartesian.

We start by proving a lemma.

Lemma 2.3.5. Both squares in the following diagram of perfect group schemes are Cartesian.

(2.3.2)

Lemma 2.3.6. The stacks

$$ \begin{align*} \left[ \frac{ \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}}{ \operatorname{Ad}_{\sigma} H_m} \right] \text{ and } \left[ \frac{ \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}}{ \operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right] \end{align*} $$

are equidimensional of the same dimension.

Proof. To compute the dimensions, we note that it follows from the right Cartesian square in Lemma 2.3.5 that

$$ \begin{align*} \operatorname{Dim} H_m = \operatorname{Dim} L^m \mathcal{G}_{K} - (\operatorname{Dim} (\overline{\mathcal{G}}_K)^{\mathrm{red}} - \operatorname{Dim} B), \end{align*} $$

and thus it suffices to show that

$$ \begin{align*} \operatorname{Dim} \left(\ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}\right) = \operatorname{Dim} \left(\ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}\right) - \operatorname{Dim} (\overline{\mathcal{G}}_K)^{\mathrm{red}} + \operatorname{Dim} B. \end{align*} $$

The map $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}} \to \operatorname {M}^{\mathrm {loc}}_{\emptyset , \{\mu \}}$ is a B-torsor by construction and $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}} \to \operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ is a $(\overline {\mathcal {G}}_K)^{\mathrm {red}}$ -torsor by construction (see (2.3.2)). Therefore, the equality above is equivalent to the equality

$$ \begin{align*} \operatorname{Dim} \operatorname{M}^{\mathrm{loc}}_{K, \{\mu\}} = \operatorname{Dim} \operatorname{M}^{\mathrm{loc}}_{\emptyset, \{\mu\}}, \end{align*} $$

which is true (see Section 2.2.15).

Proof of Proposition 2.3.4.

Consider the following diagram, where the maps are defined as in (2.3.1)

It follows from Lemma 2.3.5 that $P^m:=\operatorname{Ker} \left (L^+ \mathcal {G}_{K} \to L^m \mathcal {G}_{K} \right )$ is contained in $L^+ \mathcal {G}_{\emptyset }$ and that $P^m$ is also equal to the kernel of $L^+ \mathcal {G}_{\emptyset } \to H_m$ .

We deduce that the right horizontal maps are both $P^m$ gerbes. Therefore, the map from the top-left term of the right-most square to the fibre product is a morphism of $P^m$ -gerbes, and thus an isomorphism. Similarly, the middle horizontal maps are both $\ker \gamma $ -torsors. Therefore, the map from the top-left term of the middle square to the fibre product is a morphism of $\ker \gamma $ -torsors and thus an isomorphism. We deduce that the outer square of the diagram is Cartesian.

2.3.7

In this section, we record two more lemmas.

Proof. Fix $m'$ . Recall that

$$ \begin{align*} L^+ \mathcal{G}_{K} &\simeq \varprojlim_m L^m \mathcal{G}_{K} \\ L^+ \mathcal{G}_{\emptyset} &\simeq \varprojlim_m L^m \mathcal{G}_{\emptyset}, \end{align*} $$

and the first of these equalities, moreover, implies that $L^+ \mathcal {G}_{\emptyset } \simeq \varprojlim _m H_m$ . The lemma now follows from Lemma 2.1.1.

It follows from Lemma 2.3.8 that for each $m'$ , there is an $m \gg m'$ , such that the natural map $L^+ \mathcal {G}_{\emptyset } \to L^{m'} \mathcal {G}_{\emptyset }$ factors through the natural map $L^+ \mathcal {G}_{\emptyset } \to H_m$ via a surjection $H_m \to L^{m'} \mathcal {G}_{\emptyset }$ . Note, moreover, that $L^+ \mathcal {G}_{\emptyset } \to L^1 \mathcal {G}_{\emptyset }=\overline {\mathcal {G}}_{\emptyset } \to (\overline {\mathcal {G}}_{\emptyset })^{\mathrm {red}}$ factors through $L^+ \mathcal {G}_{\emptyset } \to L^1 \mathcal {G}_{\emptyset } \to B$ because the maximal reductive quotient of $\overline {\mathcal {G}}_{\emptyset }$ is isomorphic to the maximal reductive quotient of B. Thus, there is a natural map $\ker \gamma \to \ker (L^+ \mathcal {G}_{\emptyset } \to (\overline {\mathcal {G}}_{\emptyset })^{\mathrm {red}})$ which induces a map $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}} \to \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{\emptyset , \{\mu \}}$ . Recall, moreover, that for $m \gg 0$ , the action $\operatorname {Ad}_{\sigma } L^+ \mathcal {G}_{\emptyset }$ on $\operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{\emptyset , \{\mu \}}$ factors through an action of $L^{m'}\mathcal {G}_{\emptyset }$ .

Lemma 2.3.9. Let $m' \gg 0$ be a positive integer, and let $m \gg m'$ satisfy the conclusion of Lemma 2.3.8. Then the map (induced by $\ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{\emptyset , \{\mu \}} \to \operatorname {M}^{\mathrm {loc},1-\mathrm {red}}_{\emptyset , \{\mu \}}$ and $H_m \to L^{m'}\mathcal {G}_{\emptyset }$ )

$$ \begin{align*} \left[\frac{\ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} H_m} \right] \to \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^{m'}\mathcal{G}_{\emptyset}} \right] = \operatorname{Sht}_{G,\emptyset, \{\mu\}}^{(m',1)} \end{align*} $$

is weakly perfectly smooth.

Proof. The natural map

$$ \begin{align*} \left[\frac{\ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} H_m} \right] \to \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} H_m} \right] = \operatorname{Sht}_{G,\emptyset, \{\mu\}}^{(m',1)} \end{align*} $$

is a torsor for $\ker (B \to (\overline {\mathcal {G}}_{\emptyset })^{\mathrm {red}})$ , and hence weakly perfectly smooth. The natural map

$$ \begin{align*} \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} H_m} \right] \to \left[\frac{\operatorname{M}^{\mathrm{loc},1-\mathrm{red}}_{\emptyset, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^{m'}\mathcal{G}_{\emptyset}} \right] \end{align*} $$

is a gerbe for $\ker (H_m \to L^{m'}\mathcal {G}_{\emptyset })$ and is thus weakly perfectly smooth. It follows that the composition is weakly perfectly smooth, and the lemma is proved.

2.3.10

The stack $\left [ \frac { \ker \gamma \backslash \operatorname {M}^{\mathrm {loc},\infty }_{K, \{\mu \}}}{ \operatorname {Ad}_{\sigma } L^m \mathcal {G}_{K}} \right ]$ is not a stack of restricted shtukas in the sense of Shen–Yu–Zhang [Reference Shen, Yu and ZhangSYZ21]. However, it is closely related to the more general stacks of restricted shtukas introduced in [Reference Xiao and ZhuXZ17, Section 5.3]. We define for $n \ge 2$ the quotient

$$ \begin{align*} \operatorname{M}^{\mathrm{loc,n}}_{K, \{\mu\}}:=\ker \left( L^+ \mathcal{G}_{K} \to L^n \mathcal{G}_{K} \right) \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}. \end{align*} $$

Then, by Lemma 2.1.1, for $m \gg n$ , the action $\operatorname {Ad}_{\sigma } L^+ \mathcal {G}_{K}$ on $\operatorname {M}^{\mathrm {loc,n}}_{K, \{\mu \}}$ will factor through $L^m \mathcal {G}_{K}$ , and we define

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}^{(m,n),\mathrm{loc}}:=\left[\frac{\operatorname{M}^{\mathrm{loc,n}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right]. \end{align*} $$

We have added the ‘loc’ in the superscript and the condition that $n \ge 2$ so that these are not confused with the previously introduced stacks of restricted shtukas (since the notation is not compatible).

The proof of Lemma 2.3.8 shows that for $n \gg 0$ , we have an inclusion $\ker \left ( L^+ \mathcal {G}_{K} \to L^n \mathcal {G}_{K} \right ) \subset \ker \gamma $ and thus a natural map

$$ \begin{align*} \operatorname{M}^{\mathrm{loc,n}}_{K, \{\mu\}} & \to \ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}. \end{align*} $$

This induces a morphism (for $m \gg n$ as before)

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}^{(m,n), \mathrm{loc}} = \left[\frac{\operatorname{M}^{\mathrm{loc,n}}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right] \to \left[\frac{\ker \gamma \backslash \operatorname{M}^{\mathrm{loc},\infty}_{K, \{\mu\}}}{\operatorname{Ad}_{\sigma} L^m \mathcal{G}_{K}} \right], \end{align*} $$

which is a torsor for the image of $\ker \gamma $ in $L^n \mathcal {G}_K$ and thus perfectly smooth.

2.3.11 The EKOR stratification

Recall that ${}^{K} \!\! \operatorname {Adm}(\{\mu \})$ is the intersection of $\operatorname {Adm}(\{\mu \})$ with $^{K} \tilde {W}$ , where $^{K} \tilde {W} \subset \tilde {W}$ denotes the subset of elements that are of minimal length in their left $W_K$ -coset. By [Reference Shen, Yu and ZhangSYZ21, Lemma 4.2.4], the underlying topological space of $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ is isomorphic to ${}^{K} \!\! \operatorname {Adm}(\{\mu \})$ equipped with the partial order topology (for the partial order $\preceq $ on ${}^{K} \!\! \operatorname {Adm}(\{\mu \})$ introduced in [Reference Shen, Yu and ZhangSYZ21, page 3123]). They use this to define locally closed substacks $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}\{w\}$ for $w \in {}^{K} \!\! \operatorname {Adm}(\{\mu \})$ , such that the locally closed substack

$$ \begin{align*} \operatorname{Sht}_{G,K, \{\mu\}}^{(m,1)}\{\preceq w\} := \bigcup_{w' \preceq w} \operatorname{Sht}_{G,K, \{\mu\}}^{(m,1)}\{w'\} \end{align*} $$

is closed. This allows us to define the Ekedahl–Kottwitz–Oort–Rapoport (EKOR) stratification on any stack mapping to $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ ; for example, on $\operatorname {Sht}_{G,K, \{\mu \}}$ via (2.3.1) and later on Shimura varieties of Hodge type. Note that if $K=\emptyset $ , then the EKOR stratification agrees with the Kottwitz–Rapoport (KR) stratification from Section 2.2.15 and $\preceq $ agrees with $\le $ . This follows from [Reference Shen, Yu and ZhangSYZ21, Section 1.3.2] and the discussion preceding [Reference Shen, Yu and ZhangSYZ21, Proposition 4.2.5].

2.4.1

Let $\mathcal {E}$ be an $LG$ -torsor over $k'$ , with $k'$ an algebraically closed field of characteristic p, and let $\beta :\sigma ^{\ast } \mathcal {E} \to \mathcal {E}$ be an isomorphism, where $\sigma $ is the absolute Frobenius. After choosing a trivialisation of $\mathcal {E}$ , we see that $\beta $ can be represented by an element $b_{\beta } \in G(W(k')[1/p])$ well-defined up to $\sigma $ -conjugacy. Since the set of $\sigma $ -conjugacy classes in $G(W(k')[1/p])$ does not depend on the choice of algebraically closed field $k'$ , it thus gives us an element $[b_{\beta }] \in B(G)$ .

Let R be a perfect k-algebra, let $\mathcal {E}$ be an $LG$ -torsor over R and let $\beta :\sigma ^{\ast } \mathcal {E} \to \mathcal {E}$ be an isomorphism. If $x \in \operatorname {Spec} R$ and K is an algebraic closure of the residue field $k(x)$ , then we will write $[b_{\beta }(x)] \in B(G)$ for the $\sigma $ -conjugacy class of the pullback of $(\mathcal {E}, \beta )$ along $\operatorname {Spec} K \to \operatorname {Spec} R$ . Then for $[b] \in B(G)$ , the subset (using the partial order introduced above)

$$ \begin{align*} (\operatorname{Spec} R)_{\le [b]}:=\{ x \in \operatorname{Spec} R \: : \: [b_{\beta}(x)] \le [b] \} \end{align*} $$

is closed in $\operatorname {Spec} R$ by [Reference Rapoport and RichartzRR96, Theorem 3.6.(ii)] and

$$ \begin{align*} (\operatorname{Spec} R)_{[b]}:=\{ x \in \operatorname{Spec} R \: : \: [b_{\beta}(x)] = [b] \} \end{align*} $$

is locally closed.

2.4.2

Given $\operatorname {Spec} R \to \operatorname {Sht}_{G,K}$ corresponding to a $\mathcal {G}_K$ -shtuka $(\mathcal {E}', \beta )$ , we can set $\mathcal {E}$ to be the pushout of $\mathcal {E}'$ along $L^+ \mathcal {G}_{K} \to LG$ to obtain a pair $(\mathcal {E}, \beta )$ as above. Then we may form the locally closed subsets $(\operatorname {Spec} R)_{[b]} \subset \operatorname {Spec} R$ as above. This allows us to define a stratification

$$ \begin{align*} \operatorname{Sht}_{G,K}:=\bigcup_{[b] \in B(G)} \operatorname{Sht}_{G,K,[b]}, \end{align*} $$

where $\operatorname {Sht}_{G,K,[b]}$ denotes the locally closed substack of $\operatorname {Sht}_{G,K}$ whose R-points are given by the full subgroupoid

$$ \begin{align*} \operatorname{Sht}_{G,K,[b]}(R) \subset \operatorname{Sht}_{G,K}(R) \end{align*} $$

of maps $\operatorname {Spec} R \to \operatorname {Sht}_{G,K}$ , such that $(\operatorname {Spec} R)_{[b]}=\operatorname {Spec} R$ . We will write ${\operatorname {Sht}_{G,K, \{\mu \},[b]}}$ for the intersection (fibre product over $\operatorname {Sht}_{G,K}$ ) of $\operatorname {Sht}_{G,K, \{\mu \}}$ and $\operatorname {Sht}_{G,K,[b]}$ ; we will see in Corollary 2.4.7 that this is nonempty if and only if $[b] \in B(G, \{\mu \})$ .

2.4.3

Let $K \subset \mathbb {S}$ be a $\sigma $ -stable type, and let $b \in G(\breve {\mathbb {Q}}_p)$ . Then we define the affine Deligne–Lusztig set

$$ \begin{align*} X(\mu,b)_K=\{g \in G(\breve{\mathbb{Q}}_p)/\mathcal{G}_K(\breve{\mathbb{Z}}_p) \; | \; g^{-1} b \sigma(g) \in \bigcup_{w \in \operatorname{Adm}(\{\mu\})_K} \mathcal{G}_K(\breve{\mathbb{Z}}_p) \dot{w} \mathcal{G}_K(\breve{\mathbb{Z}}_p)/ \mathcal{G}_K(\breve{\mathbb{Z}}_p)\}. \end{align*} $$

Let $J_b$ be the algebraic group over $\mathbb {Q}_p$ whose R-points are given by

$$ \begin{align*} J_b(R)=\{g \in G(\breve{\mathbb{Q}}_p \otimes_{\mathbb{Q}_p} R) \: | \; g^{-1} b \sigma (g) = b\}. \end{align*} $$

Then $J_b(\mathbb {Q}_p) \subset G(\breve {\mathbb {Q}}_p)$ acts on $X(\mu ,b)_K$ via left multiplication. By [Reference HeHe16, Theorem 1.1], the set $X(\mu ,b)_K$ is nonempty if and only if $[b] \in B(G, \{\mu \})$ . Moreover, [Reference HeHe16, Theorem 1.1] says that for $J \subset K$ another $\sigma $ -stable type, the natural projection $G(\breve {\mathbb {Q}}_p)/\mathcal {G}_{J}(\breve {\mathbb {Z}}_p) \to G(\breve {\mathbb {Q}}_p)/\mathcal {G}_{K}(\breve {\mathbb {Z}}_p)$ induces a $J_b(\mathbb {Q}_p)$ -equivariant surjection

$$ \begin{align*} X(\mu,b)_J \to X(\mu,b)_K. \end{align*} $$

We will soon see that $X(\mu ,b)_K$ can be identified with the set of k-points of a perfect scheme over k, which we will also denote by $X(\mu ,b)_K$ .

2.4.4

Let K be a $\sigma $ -stable type, let $b \in G(\breve {\mathbb {Q}}_p)$ and consider the functor $X(\mu ,b)_K'$ on $\mathbf {Aff}_k^{\mathrm {perf}}$ sending R to the set of isomorphism classes of commutative diagrams of modifications of $\mathcal {G}_K$ -torsors on $D_R$

(2.4.1)

such that $\beta _1:\sigma ^{\ast } \mathcal {E}_1 \dashrightarrow \mathcal {E}_1$ , considered as an element of $\operatorname {Hk}_{K}(R)$ , lies in $\left (\bigcup _{w \in \operatorname {Adm}(\{\mu \})_K} \operatorname {Hk}_{K}(w)\right )(R)$ . Here, b is the modification of the trivial $G_K$ -torsor $\sigma ^{\ast } \mathcal {E}^0 \simeq \mathcal {E}^0$ given by multiplication by b. We will sometimes refer to $\beta _0$ as a quasi-isogeny of shtukas from $(\mathcal {E}_1, \beta _1) \to (\mathcal {E}^0, b)$ .

Lemma 2.4.5. The morphism $X(\mu ,b)_K' \to \operatorname {Gr}_{K}$ that sends a diagram, as in (2.4.1) to $\beta _0:\mathcal {E}_1 \dashrightarrow \mathcal {E}^0$ , is a closed immersion. Moreover, it identifies

$$ \begin{align*} X(\mu,b)_K'(k) \subset \operatorname{Gr}_{K}(k) = G(\breve{\mathbb{Q}}_p)/\mathcal{G}_K(\breve{\mathbb{Z}}_p) \end{align*} $$

with the affine Deligne–Lusztig set $X(\mu ,b)_K$ from Section 2.4.3.

Proof. Consider the functor $X(b)$ sending R to the set of isomorphism classes of commutative diagrams of modifications of $\mathcal {G}_K$ -torsors on $D_R$

(2.4.2)

as before but now without the condition that $\beta _1 \in \left (\bigcup _{w \in \operatorname {Adm}(\{\mu \})_K} \operatorname {Hk}_{K}(w)\right )(R)$ . As before, [Reference He and ZhouHZ20, the discussion after Remark 3.5] tells us that $X(\mu ,b)_K'$ is a closed subfunctor of $X(b)$ , and the lemma would follow if we could show that the map

$$ \begin{align*} f:X(b) \to \operatorname{Gr}_{K} \end{align*} $$

sending a diagram as in (2.4.2) to $\beta _0:\mathcal {E}_1 \dashrightarrow \mathcal {E}^0$ is an isomorphism. The map f is an isomorphism because the map $g:\operatorname {Gr}_{K} \to X(b)$ sending $\beta _0:\mathcal {E}_1 \dashrightarrow \mathcal {E}^0$ to the diagram

with $\beta _1=\beta _0^{-1} b \sigma ^{\ast } \beta _0$ is an inverse to f. We see that $X(\mu ,b)_K'(k)$ is cut out from $X(b)(k)= G(\breve {\mathbb {Q}}_p)/\mathcal {G}_K(\breve {\mathbb {Z}}_p)$ by the condition that $\beta _1 \in \bigcup _{w \in \operatorname {Adm}(\{\mu \})_K} \operatorname {Hk}_{K}(w)(k)$ , in other words, that

$$ \begin{align*} \beta_0^{-1} b \sigma^{\ast} \beta_0 \in \bigcup_{w \in \operatorname{Adm}(\{\mu\})} \mathcal{G}_K(\breve{\mathbb{Z}}_p) \dot{w} \mathcal{G}_K(\breve{\mathbb{Z}}_p)/\mathcal{G}_K(\breve{\mathbb{Z}}_p). \end{align*} $$

This is precisely the condition cutting out $X(\mu ,b)_K \subset G(\breve {\mathbb {Q}}_p)/\mathcal {G}_K(\breve {\mathbb {Z}}_p)$ , and so we are done.

From now on, we will write $X(\mu ,b)_K$ for $X(\mu ,b)_K'$ by abuse of notation. It follows from [Reference Hamacher and ViehmannHV18, Lemma 1.1] and [Reference Zhou and ZhuZZ20, Corollary 2.5.3] that $X(\mu ,b)_K$ is actually a perfect scheme that is perfectly locally of finite type.

If $b'$ is $\sigma $ -conjugate to b, that is, if $b' = g^{-1} b \sigma (g)$ with $g \in G(\breve {\mathbb {Q}}_p)$ , then $X(\mu ,b)_K \simeq X(\mu ,b')_K$ via the map

We note that this map is nothing more than the action of $g \in LG(k)$ on $X(\mu ,b)_K \subset \operatorname {Gr}_{K}$ via the natural left action of $LG$ on $\operatorname {Gr}_{K}$ . For $b'=b$ , this induces an action of the closed subgroup $F_b \subset LG$ on $X(\mu ,b)_K$ , where $F_b$ is defined as the subfunctor of $LG$ sending a perfect $\mathbb {F}_p$ -algebra R to the group

$$ \begin{align*} F_b(R)=\{g \in LG(R) \; | \; g^{-1} b \sigma(g) = b \}. \end{align*} $$

The k-points of $F_b$ are in bijection with $J_b(\mathbb {Q}_p)$ , where $J_b$ is the algebraic group over $\mathbb {Q}_p$ introduced in Section 2.4.3. Recall the notion of a pro-étale cover of a scheme (see [Reference Bhatt and ScholzeBS15, Definition 1.2]).

Lemma 2.4.6. Consider the morphism $\Theta _b:X(\mu ,b)_K \to {\operatorname {Sht}_{G,K, \{\mu \},[b]}}$ , which sends a diagram as in (2.4.1) to $(\mathcal {E}_1, \beta _1)$ . This morphism is $F_b$ -invariant for the trivial action on the target and induces an isomorphism of groupoids

$$ \begin{align*} {\operatorname{Sht}_{G,K, \{\mu\},[b]}} \simeq \left[F_b \backslash X(\mu,b)_K \right], \end{align*} $$

where the quotient stack is taken in the pro-étale topology. Moreover, $F_b$ is isomorphic to the locally profinite group scheme $\underline {J_b(\mathbb {Q}_p)}$ associated to the topological group $J_b(\mathbb {Q}_p)$ . Footnote ¹⁴

Proof. The morphism $\Theta _b$ is $F_b$ -invariant, since the action of $F_b$ on $X(\mu ,b)_K$ does not change $(\mathcal {E}_1, \beta _1)$ . For a scheme $T \mapsto {\operatorname {Sht}_{G,K, \{\mu \},[b]}}$ , the set $X(\mu ,b)_K(T)$ is the set of quasi-isogenies from $(\mathcal {E}_1, \beta _1)$ to $(\mathcal {E}^0_T, b_T)$ , which is either empty or has a simply transitive action of the group $F_b(T)$ of self quasi-isogenies of $(\mathcal {E}^0_T, b_T)$ . In other words, we have shown that $\Theta _b$ is a pseudo-torsor for $F_b$ . By [Reference Fargues and ScholzeFS21, Theorem I.2.1], for any $\mathcal {G}_{K}$ -shtuka $(\mathcal {E}_1, \beta _1) \in \operatorname {Sht}_{G,K,[b]}(T)$ , the pseudo-torsor of quasi-isogenies to $(\mathcal {E}^0_T, b_T)$ has a section pro-étale locally on T. Thus, we find that map $\Theta _b$ is a pro-étale torsor for $F_b$ . In other words, there is an isomorphism

$$ \begin{align*} {\operatorname{Sht}_{G,K, \{\mu\},[b]}} \simeq \left[F_b \backslash X(\mu,b)_K \right]. \end{align*} $$

It also follows from [Reference Fargues and ScholzeFS21, Theorem I.2.1] that $F_b$ is isomorphic to the locally profinite group scheme $\underline {J_b(\mathbb {Q}_p)}$ associated to $J_b(\mathbb {Q}_p)$ .

3.1.1

Let V be a vector space over $\mathbb {Q}$ of dimension $2g$ equipped with a perfect alternating bilinear form $\psi $ . For a $\mathbb {Q}$ -algebra R, we write $V_R=V\otimes _{\mathbb {Q}}R$ . Let $G_V$ denote the corresponding group of symplectic similitudes, and let $\mathcal {H}_V$ denote the set of homomorphisms $h:\mathbb {S}\rightarrow G_{V,\mathbb {R}}$ corresponding to the Siegel upper and lower half space, where $\mathbb {S}:=\operatorname {Res}_{\mathbb {C}/\mathbb {R}} \mathbb {G}_m$ is the Deligne torus. For the rest of this section, we fix an embedding of Shimura data $\iota :(G,X)\rightarrow (G_V, \mathcal {H}_V)$ . We sometimes write G for $G_{\mathbb {Q}_p}$ when there is no risk of confusion. We will also assume for the rest of this section that the following conditions hold

$$ \begin{align*}G \text{ splits over a tamely ramified extension of}\ \mathbb{Q}_p\ \text{and}\ p\nmid|\pi_1(G^{\mathrm{der}})|. \end{align*} $$

Let $\mathcal {G}$ be a connected parahoric subgroup of G, that is, $\mathcal {G}=\mathcal {G}_x=\mathcal {G}_x^{\circ }$ for some $x\in B(G,\mathbb {Q}_p)$ (see Section 2.2.1). We will follow the notation of Section 2 to write $\mathcal {G}=\mathcal {G}_{K}$ for some $\sigma $ -stable type $K \subset \mathbb {S}$ . By [Reference Kisin and PappasKP18, Section 2.3.15], after replacing $\iota $ by another symplectic embedding, there is a closed immersion $\mathcal {G}_{K} \rightarrow \mathcal {P}$ , where $\mathcal {P}$ is a parahoric group scheme of $G_V$ corresponding to the stabiliser of a lattice $V_{\mathbb {Z}_p}\subset V$ . Upon scaling $V_{\mathbb {Z}_p}$ , we may assume $V_{\mathbb {Z}_p}^\vee \subset V_{\mathbb {Z}_p}$ . This induces a closed immersion (see [Reference Kisin and PappasKP18, Proposition 2.3.7]) of local models

$$ \begin{align*}M^{\mathrm{loc}}_{\mathcal{G}_{K},X}\rightarrow M^{\mathrm{loc}}_{\mathcal{P},\mathcal{H}_V}\otimes\mathcal{O}_{E,v}\end{align*} $$

for every place v of E above p. Here, the local models are as introduced in [Reference Kisin and PappasKP18, Section 2.1].

3.1.2

Let $U_V^p \subset G_V(\mathbb {A}_{f}^{p})$ be a sufficiently small compact open subgroup. Let $V_{\mathbb {Z}_{(p)}}=V_{\mathbb {Z}_p}\cap V$ and write $G_{\mathbb {Z}_{(p)}}$ for the Zariski closure of G in $GL(V_{\mathbb {Z}_{(p)}})$ , then $G_{\mathbb {Z}_{(p)}}\otimes _{\mathbb {Z}_{(p)}}\mathbb {Z}_p\cong \mathcal {G}_{K}$ . The choice of $V_{\mathbb {Z}_{(p)}}$ gives rise to a compact open subgroup $U_{V,p} \subset G_V(\mathbb {Q}_p)$ which gives the Shimura variety $\mathbf {Sh}_{U_V}(G_V,\mathcal {H}_V)$ of level $U_V=U_V^p U_{V,p}$ an interpretation as a moduli space of (weakly polarised) abelian varieties up to prime-to-p isogeny, and hence an integral model $\mathscr {S}_{U_V}(G_V,\mathcal {H}_V)$ over $\mathbb {Z}_{(p)}$ , which is described in [Reference ZhouZho20, Section 6.3].

3.1.3

For the rest of this paper, we fix an algebraic closure $\overline {\mathbb {Q}}$ of E, and for each place v of $\mathbb {Q}$ , an algebraic closure $\overline {\mathbb {Q}}_v$ together with an embedding $\overline {\mathbb {Q}}\rightarrow \overline {\mathbb {Q}}_v$ . Using these embeddings, we get a $G(\overline {\mathbb {Q}}_p)$ -conjugacy class of cocharacter $\{\mu _h\}$ induced from the Hodge cocharacter associated to X.

By [Reference KisinKis10, Lemma 2.1.2], we can choose $U_V^p$ , such that $\iota $ induces a closed immersion

$$ \begin{align*}\mathbf{Sh}_{U}(G,X)\hookrightarrow \mathbf{Sh}_{U_V}(G_V,\mathcal{H}_V)_{E}\end{align*} $$

defined over E. The choice of embedding $E\rightarrow \overline {\mathbb {Q}}_p$ determines a place v of E. Write $\mathcal {O}_{E,{(v)}}$ for the localisation of $\mathcal {O}_{E}$ at v, let $E_v$ be the completion of E at v and $\mathcal {O}_{E,v}$ the ring of integers of $E_v$ . We assume the residue field has $q=p^r$ elements and, as before, k will denote an algebraic closure of $\mathbb {F}_q$ . We define $\mathscr {S}_U(G,X)^-$ to be the Zariski closure of $\mathbf {Sh}_{U}(G,X)$ inside $\mathscr {S}_{U_V}(G_V,\mathcal {H}_V)\otimes _{\mathbb {Z}_{(p)}}\mathcal {O}_{E,(v)}$ , and $\mathscr {S}_{U}(G,X)$ to be its normalisation. By construction, for $U^p_1\subset U^p_2$ compact open subgroups of $G(\mathbb {A}_f^p)$ , there are finite étale transition maps $\mathscr {S}_{U^p_1 U_p}(G,X)\rightarrow \mathscr {S}_{U^p_2 U_p}(G,X)$ , and we write $\mathscr {S}_{U_p}(G,X):=\varprojlim _{U^p}\mathscr {S}_{U^pU_p}(G,X)$ . Under these assumptions, we have the following result:

Theorem 3.1.4 ([Reference Kisin and PappasKP18] Theorems 4.2.2, 4.2.7).

The $\mathcal {O}_{E,(v)}$ scheme $\mathscr {S}_{U_p}(G,X)$ is a flat $G(\mathbb {A}_f^p)$ -equivariant extension of $\mathbf {Sh}_{U_p}(G,X)$ . Moreover, $\mathscr {S}_{U}(G,X)_{\mathcal {O}_{E,v}}$ fits in a local model diagram

where q is a $\mathcal {G}_{K}$ -torsor and $\pi $ is smooth of relative dimension $\operatorname {Dim} G$ .

Note that the main result of [Reference PappasPap22] tells us that the integral model $\mathscr {S}_{U}(G,X)$ does not depend on the choice of Hodge embedding.

3.1.5

By [Reference KisinKis10, Proposition 1.3.2], the subgroup $G_{\mathbb {Z}_{(p)}}$ is the stabiliser of a collection of tensors $s_\alpha \in V_{\mathbb {Z}_{(p)}}^\otimes $ for $\alpha \in \mathscr {A}$ . Let $h:\mathcal {A}\rightarrow \mathscr {S}_{U}(G,X)$ denote the pullback of the universal abelian variety on $\mathscr {S}_{U_V}(G_V,\mathcal {H}_V)$ , and let $V_B:=R^1h_{\mathrm {an}*}\mathbb {Z}_{(p)}$ , where $h_{\mathrm {an}}$ is the map of complex analytic spaces associated to h. We also let $\mathcal {V}=R^1h_*\Omega ^\bullet $ be the relative de Rham cohomology of $\mathcal {A}$ . Using the de Rham isomorphism, the tensors $s_\alpha $ give rise to a collection of Hodge cycles $s_{\alpha ,dR}\in \mathcal {V}_{\mathbb {C}}^\otimes $ , where $\mathcal {V}_{\mathbb {C}}$ is the complex analytic vector bundle associated to $\mathcal {V}$ . By [Reference KisinKis10, Section 2.2], these tensors are defined over E, and, in fact, over $\mathcal {O}_{E,(v)}$ by [Reference Kisin and PappasKP18, Proposition 4.2.6].

Similarly, for a finite prime $\ell \neq p$ , we let $\mathcal {V}_{\ell }=R^1h_{\acute {e}t*}\mathbb {Q}_\ell $ and $\mathcal {V}_p=R^1h_{\eta ,\acute {e}t*}\mathbb {Z}_p$ , where $h_\eta $ is the generic fibre of h. Using the étale-Betti comparison isomorphism, we obtain tensors $s_{\alpha ,\ell }\in \mathcal {V}^\otimes _\ell $ and $s_{\alpha ,p}\in \mathcal {V}_p^\otimes $ . For $*=B, dR,\ell $ and $x\in \mathscr {S}_{U^pU_p}(G,X)(T)$ for some $\mathcal {O}_{E,(v)}$ -scheme T, we write $s_{\alpha ,*,x}$ for the pullback of $s_{\alpha ,*}$ to T via x. Similarly, the image of x under $\mathscr {S}_{U^pU_p}(G,X) \to \mathscr {S}_{U_V}(G_V,\mathcal {H}_V)\otimes _{\mathbb {Z}_{(p)}}\mathcal {O}_{E,(v)}$ gives us a weakly polarised abelian variety up to prime-to-p isogeny $(\mathcal {A}_x,\lambda )$ . Over $\mathscr {S}_{U_p}(G,X)$ , there is a canonical isomorphism of pro-étale $\mathbb {A}_{f}^{p}$ -local systems

(3.1.1) $$ \begin{align} \eta: \mathcal{V}^p\xrightarrow{\sim} V \otimes \mathbb{A}_{f}^{p}, \end{align} $$

which takes $t_{\alpha , \mathbb {A}_{f}^{p}}$ to $t_{\alpha } \otimes 1$ for all $\alpha \in \mathscr {A}$ (see [Reference Kisin, Shin and ZhuKSZ21, Lemma 5.1.9]).

3.1.6

Recall that $\overline {\mathbb {F}}_{p}$ is an algebraic closure of $\mathbb {F}_q$ and $\breve {\mathbb {Q}}_p=W(\overline {\mathbb {F}}_{p})[1/p]$ . Let $\overline {x}\in \mathscr {S}_{U}(G,X)(\overline {\mathbb {F}}_{p})$ and $x\in \mathscr {S}_{U}(G,X)(\mathcal {O}_L)$ a point lifting $\overline {x}$ , where $L/\breve {\mathbb {Q}}_p$ is a finite extension. Let us write $\varphi $ for the Frobenius on $\breve {\mathbb {Q}}_p$ and $\breve {\mathbb {Z}}_p$ .

Let $\mathscr {G}_x$ denote the p-divisible group associated to $\mathcal {A}_x$ and $\mathscr {G}_{x,0}$ its special fibre. Then $T_p\mathscr {G}_x^\vee $ is identified with $H^1_{\acute {e}t}(\mathcal {A}_x,\mathbb {Z}_p)$ , and we obtain $\Gamma _K$ -invariant tensors $s_{\alpha ,\acute {e}t,x}\in T_p\mathscr {G}^{\vee \otimes }$ , whose stabiliser can be identified with $\mathcal {G}_{K}$ . Let $\mathbb {D}_x:=\mathbb {D}(\mathscr {G}_{x,0})$ be the contravariant Dieudonné module associated to the p-divisible group $\mathscr {G}_{x,0}$ . We may apply the constructions of [Reference ZhouZho20, Section 3] to obtain $\varphi $ -invariant tensors $s_{\alpha ,0,x} \in \mathbb {D}_x$ , whose stabiliser group can be identified with $\mathcal {G}_{K} \otimes _{\mathbb {Z}_p} \breve {\mathbb {Z}}_p$ .

This means that we can upgrade the Dieudonné module of $A_x$ to a $\mathcal {G}_K$ -shtuka over $\overline {\mathbb {F}}_{p}$ , and this gives a map (see [Reference ZhouZho20, Proof of Axiom 4 in Section 8])

(3.1.2) $$ \begin{align} \mathscr{S}_{U}(G,X)(\overline{\mathbb{F}}_{p}) \to \operatorname{Sht}_{G,K, \{\mu\}}(\overline{\mathbb{F}}_{p}), \end{align} $$

where $\{\mu \}=\{\sigma (\mu _h^{-1})\}$ . It is a result of Hamacher–Kim ([Reference Hamacher and KimHK19, Proposition 1], see [Reference Shen, Yu and ZhangSYZ21, Proposition 4.4.1]) that there is a morphism $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}$ inducing (3.1.2) on $\overline {\mathbb {F}}_{p}$ -points, where $\operatorname {Sh}_{G,K,U^p}$ is the perfection of the basechange to k of $\mathscr {S}_{U}(G,X)$ .Footnote ¹⁵ It follows from [Reference Pappas and RapoportPR21, Theorem 1.3.4]Footnote ¹⁶ that this morphism does not depend on the choice of Hodge embedding and, moreover, can be upgraded to a $G(\mathbb {A}_{f}^{p})$ -equivariant morphism

$$ \begin{align*} \operatorname{Sh}_{G,K}:=\varprojlim_{U^p} \operatorname{Sh}_{G,K,U^p} \to \operatorname{Sht}_{G,K, \{\mu\}}, \end{align*} $$

where $G(\mathbb {A}_{f}^{p})$ acts trivially on $\operatorname {Sht}_{G,K, \{\mu \}}$ .

It follows from [Reference Shen, Yu and ZhangSYZ21, the discussion after Theorem 4.4.3] that the perfection of the special fibre of $M_{\mathcal {G}_{K},X}^{\mathrm {loc}}$ can be identified with the closed subscheme $\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ of the affine flag variety for $\mathcal {G}_{K}$ introduced in Section 2.2.15. Under this isomorphism, the right action of $L^+ \mathcal {G}_{K}$ on $\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}$ , which factors through $\overline {\mathcal {G}}_K$ , is identified with the $\overline {\mathcal {G}}_K$ actionFootnote ¹⁷ on the perfection of $M_{G,K,\mu }^{\mathrm {loc}}$ . Thus, the local model diagram of Theorem 3.1.4 gives us a (perfectly smooth) morphism

$$ \begin{align*} \lambda_K:\operatorname{Sh}_{G,K,U^p} \to \lbrack \operatorname{M}^{\mathrm{loc}}_{K, \{\mu\}}/\overline{\mathcal{G}}_K \rbrack. \end{align*} $$

3.1.7

Fix $n \ge 2$ , and choose $m \gg 0$ , such that the action $\operatorname {Ad}_{\sigma } L^+ \mathcal {G}_{K}$ on $\operatorname {M}^{\mathrm {loc,n}}_{K, \{\mu \}}$ factors through $L^m \mathcal {G}_{K}$ and such that m satisfies the assumptions of Proposition 2.3.4. Let $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ be the stack of $(m)$ -restricted shtukas of type $\{\mu \}$ from Section 2.3 and also consider the stack $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,n), \mathrm {loc}}$ from Section 2.3.10. If we compose the morphism $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}$ constructed above with the natural map $\operatorname {Sht}_{G,K, \{\mu \}} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ , we obtain a morphism

$$ \begin{align*} \operatorname{Sh}_{G,K,U^p} \to \operatorname{Sht}_{G,K, \{\mu\}}^{(m,1)}. \end{align*} $$

By [Reference Shen, Yu and ZhangSYZ21, Theorem 4.4.3], the perfectly smooth map $\lambda _K:\operatorname {Sh}_{G,K,U^p} \to [\operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}/\overline {\mathcal {G}}_K]$ induced from the local model diagram fits in a commutative diagram

where $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)} \to \lbrack \operatorname {M}^{\mathrm {loc}}_{K, \{\mu \}}/\overline {\mathcal {G}}_K \rbrack $ comes from the diagram in Remark 2.3.2. The map $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ is perfectly smooth by [Reference HoffHof23, Corollary 2.57]. Recall, moreover, that there is a natural map $\operatorname {Sht}_{G,K, \{\mu \}} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,n), \mathrm {loc}}$ which induces a map $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,n), \mathrm {loc}}$ , this is also perfectly smooth by [Reference HoffHof23, Corollary 2.57].

We can use the perfectly smooth map $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ to define the EKOR stratification on $\operatorname {Sh}_{G,K,U^p}$ (see Section 2.3.11). In particular, for $w \in {}^{K} \!\! \operatorname {Adm}(\{\mu \})$ , we will write $\operatorname {Sh}_{G,K,U^p}\{w\}$ for the locally closed EKOR stratum of $\operatorname {Sh}_{G,K,U^p}$ . Since $\operatorname {Sh}_{G,K,U^p} \to \operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ is perfectly smooth and thus open, we find that the closure of $\operatorname {Sh}_{G,K,U^p}\{w\}$ is given by

$$ \begin{align*} \operatorname{Sh}_{G,K,U^p}\{\preceq w \}:=\bigcup_{w' \preceq w} \operatorname{Sh}_{G,K,U^p}\{w'\} \end{align*} $$

because the closure relations hold on $\operatorname {Sht}_{G,K, \{\mu \}}^{(m,1)}$ .

3.1.9 Isogeny classes

Let $x\in \operatorname {Sh}_{G,K,U^p}(\overline {\mathbb {F}}_{p})$ , then attached to x is an abelian variety $\mathcal {A}_x$ over $\overline {\mathbb {F}}_{p}$ . We write $\mathbb {D}_x$ for the contravariant Dieudonné module associated to the p-divisible group $\mathscr {G}_x$ of $\mathcal {A}_x$ ; then $\mathbb {D}_x$ is equipped with a corresponding set of tensors $s_{\alpha ,0,x}$ (see Section 3.1.6). Similarly, for $\ell \neq p$ , the rational $\ell $ -adic Tate module $V_\ell \mathcal {A}_x$ is equipped with tensors $s_{\alpha ,\ell ,x}$ .

Two points $x,x'\in \operatorname {Sh}_{G,K}(\overline {\mathbb {F}}_{p})$ are said to lie in the same isogeny class if there exists a quasi-isogeny $\mathcal {A}_x\rightarrow \mathcal {A}_{x'}$ , such that the induced maps $\mathbb {D}_x[1/p] \rightarrow \mathbb {D}_{x'}[1/p]$ and $V_{\ell }\mathcal {A}_x \rightarrow V_{\ell }\mathcal {A}_{x'}\otimes $ send $s_{\alpha ,0,x'}$ to $s_{\alpha ,0,x}$ and $s_{\alpha ,\ell ,x}$ to $s_{\alpha ,\ell ,x'}$ for all $\ell \neq p$ . We write $\mathscr {I}_x \subset \operatorname {Sh}_{G,K}(\overline {\mathbb {F}}_{p})$ for the isogeny class of x.

For $x\in \operatorname {Sh}_{G,K}(\overline {\mathbb {F}}_{p})$ , we let $I_x$ denote the reductive $\mathbb {Q}$ -group associated to x as in [Reference ZhouZho20, Section 9.2]; it is a subgroup of the algebraic group of self quasi-isogenies of the abelian variety $A_x$ . It comes equipped with a natural map $I_{x,\mathbb {A}_{f}^{p}} \to G_{\mathbb {A}_{f}^{p}}$ coming from the tautological basis of the prime-to-p adelic Tate-module of $A_x$ given by (3.1.1). If we choose an isomorphism $\alpha :\mathbb {D}_x \simeq V_{\mathbb {Z}_p} \otimes _{\mathbb {Z}_{p}} \breve {\mathbb {Z}}_p$ sending $s_{\alpha , 0,x}$ to $s_{\alpha } \otimes 1$ , under which the Frobenius of $\mathbb {D}_x$ corresponds to $b \in G(\breve {\mathbb {Q}}_p)$ , then there is also an induced map $I_{x, \mathbb {Q}_p} \to J_b$ . Note that an isomorphism $\alpha $ as above always exists, by [Reference ZhouZho20, Section 5.6].

3.1.10 Change of parahoric

Now let $J \subset K$ be another $\sigma $ -stable type, let $\mathcal {G}_{J}(\mathbb {Z}_{p})=:U_p' \subset U_p$ and let $U'=U^pU_p'$ . Note that $\mathcal {G}_{J}$ is a connected parahoric since $\mathcal {G}_{K}$ is (see Lemma 2.2.4). We will use $\operatorname {Sh}_{G,J,U^p}$ to denote the perfection of the special fibre of $\mathscr {S}_{U'}(G,X)$ . By [Reference ZhouZho20, Theorem 7.1], there is a (necessarily unique) proper morphism $\pi _{J,K}:\mathscr {S}_{U'}(G,X) \to \mathscr {S}_{U}(G,X)$ which induces the obvious forgetful morphism on generic fibres and induces a $G(\mathbb {A}_{f}^{p})$ -equivariant map

$$ \begin{align*} \varprojlim_{U^p} \mathscr{S}_{U'}(G,X) \to \varprojlim_{U^p} \mathscr{S}_{U}(G,X). \end{align*} $$

We now recall some aspects of the construction of the forgetful map from [Reference ZhouZho20, Section 7.2], which we will need to compare isogeny classes in the source and target. There are facets $\mathfrak {f}, \mathfrak {f}'$ of the extended Bruhat–Tits building $B(G,\mathbb {Q}_p)$ of $G_{\mathbb {Q}_p}$ , such that $U_p$ is the stabiliser of $\mathfrak {f}$ and such that $U_p'$ is the stabiliser of $\mathfrak {f}'$ . Fix a choice of embedding $\theta :B(G,\mathbb {Q}_p) \to B(G_{V},\mathbb {Q}_p)$ as in [Reference Kisin and PappasKP18, Section 1.2], compatible with $G \to G_V$ . Choose facets $\mathfrak {g}, \mathfrak {g}'$ containing $\theta (\mathfrak {f})$ and $\theta (\mathfrak {f}')$ , respectively, and we let $M_p \subset G_V(\mathbb {Q}_p)$ be the stabiliser of $\mathfrak {g}$ and $M_p' \subset G_V(\mathbb {Q}_p)$ be the stabiliser of $\mathfrak {g}'$ . As in [Reference ZhouZho20, Section 8.1], the facets $\mathfrak {g}, \mathfrak {g}'$ correspond to lattice chains $\mathcal {L}$ and $\mathcal {L}'$ in $V_{\mathbb {Q}_p}$ , respectively, with $\mathcal {L}'$ a refinementFootnote ¹⁸ of $\mathcal {L}$ ; note that [Reference ZhouZho20] writes $\mathcal {L}'$ for what we call $\mathcal {L}$ and vice versa.

Then for sufficiently small $M^p \subset G_V(\mathbb {A}_{f}^{p})$ , there are moduli-theoretic integral models $\mathscr {S}_{M^pM_p'}(G_V, \mathcal {H}_V)$ and $\mathscr {S}_{M^pM_p}(G_V, \mathcal {H}_V)$ over $\mathbb {Z}_{(p)}$ . The former is a moduli space of $\mathcal {L}'$ -chains of (weakly polarised) abelian varieties up to prime-to-p isogeny with $M^p$ level structure, as explained in [Reference ZhouZho20, Proof of Axiom 1 in Section 8], and the latter is a moduli space of $\mathcal {L}$ -chains of (weakly polarised) abelian varieties up to prime-to-p isogeny with $M^p$ level structure. There is a natural proper forgetful map

$$ \begin{align*} \pi_{\mathcal{L}',\mathcal{L}}:\mathscr{S}_{M^pM_p'}(G_V, \mathcal{H}_V) \to \mathscr{S}_{M^pM_p}(G_V, \mathcal{H}_V), \end{align*} $$

which sends an $\mathcal {L}'$ -chain of abelian varieties to the underlying $\mathcal {L}$ -chain of abelian varieties.

Taking the direct sum of the lattices in the lattice chain $\mathcal {L}$ (respectively, $\mathcal {L}'$ ), we get a symplectic space $V_{\mathcal {L}}$ (respectively, $V_{\mathcal {L}'}$ ) and a lattice $V_{\mathcal {L},\mathbb {Z}_{p}}$ (respectively, $V_{\mathcal {L}',\mathbb {Z}_{p}}$ ) in $V_{\mathcal {L},\mathbb {Q}_p}$ (respectively, $V_{\mathcal {L}',\mathbb {Q}_p}$ ). Let us denote the stabiliser of $V_{\mathcal {L},\mathbb {Z}_{p}}$ (respectively, $V_{\mathcal {L}',\mathbb {Z}_{p}}$ ) in $G_{V_{\mathcal {L}}}(\mathbb {Q}_p)$ (respectively, $G_{V_{\mathcal {L}'}}(\mathbb {Q}_p)$ ) by $J_p$ (respectively, $J_p'$ ).

Then there are Hodge embeddings $(G_V,\mathcal {H}_V) \to (G_{V_{\mathcal {L}}},\mathcal {H}_{V_{\mathcal {L}}})$ and $(G_V, \mathcal {H}_V) \to (G_{V_{\mathcal {L}'}} \mathcal {H}_{V_{\mathcal {L}'}})$ , which take $M_p$ to $J_p$ and $M_p'$ to $J_p'$ , respectively. These induce finite maps

$$ \begin{align*} \mathscr{S}_{M^pM_p'}(G_V, \mathcal{H}_V) \to \mathscr{S}_{J^{'p}J_p'}(G_{V_{\mathcal{L}'}}, \mathcal{H}_{V_{\mathcal{L}'}}), \qquad \mathscr{S}_{M^pM_p}(G_V, \mathcal{H}_V) \to \mathscr{S}_{J^pJ_p}(G_{V_{\mathcal{L}}},\mathcal{H}_{V_{\mathcal{L}}}) \end{align*} $$

for some $J^p \subset G_{V_{\mathcal {L}}}(\mathbb {A}_{f}^{p})$ and $J^{'p} \subset G_{V_{\mathcal {L}'}}(\mathbb {A}_{f}^{p})$ sufficiently small, which take an $\mathcal {L}'$ -set (respectively, $\mathcal {L}$ -set) of abelian varieties to the product of all the abelian varieties in the $\mathcal {L}'$ -set (respectively, the $\mathcal {L}$ -set), equipped with the product polarisation and level structure. It is explained in [Reference ZhouZho20, Equation 8.1 of Section 8] that our forgetful maps fit in a commutative diagram where all the horizontal maps are finite

(3.1.3)

3.1.11 Change of parahoric and isogeny classes

We set $\operatorname {Sh}_{G,J}:=\varprojlim _{U^p} \operatorname {Sh}_{G,J,U^p}$ , and we let $\pi :\operatorname {Sh}_{G,J} \to \operatorname {Sh}_{G,K}$ denote the $G(\mathbb {A}_{f}^{p})$ -equivariant map induced by $\pi _{J,K}$ . We now defineFootnote ¹⁹ isogeny classes in $\operatorname {Sh}_{G,J}(\overline {\mathbb {F}}_{p})$ using the Hodge embedding $(G,X) \to (G_{V_{\mathcal {L}'}} \mathcal {H}_{V_{\mathcal {L}'}})$ , as in Section 3.1.9. Similarly, we define isogeny classes in $\operatorname {Sh}_{G,K}(\overline {\mathbb {F}}_{p})$ using the Hodge embedding $(G,X) \to (G_{V_{\mathcal {L}}},\mathcal {H}_{V_{\mathcal {L}}})$ . For this, we choose tensors $s_{\beta } \in V_{\mathcal {L}', (p)}^{\otimes }$ cutting out $\mathcal {G}_{J, \mathbb {Z}_{(p)}}$ and tensors $s_{\alpha } \in V_{\mathcal {L}, (p)}^{\otimes }$ cutting out $\mathcal {G}_{K, \mathbb {Z}_{(p)}}$ .