2.1 Some basic Bruhat–Tits theory

In this section we review some basic Bruhat–Tits theory. For more in-depth discussion in this direction we suggest the reader to consult [Reference Bruhat and TitsBT84] or [Reference Kaletha and PrasadKP23] and the references therein.

Throughout this section we fix a discretely valued field K with perfect residue field k. We denote by $K^{\textrm {ur}}$ the maximal unramified extension of K inside of $K^{\textrm {sep}}$ , and by $\Gamma $ the Galois group $\text {Gal}(K^{\textrm {ur}}/K)$ . Denote by the inertia group of K.

The building and its functorialities

Assume G is a reductive group over K. As in [Reference Kaletha and PrasadKP23, Definition 7.6.1] (cf. [Reference Kaletha and PrasadKP23, Definition 9.2.8]), associated to G is a building (in the sense of [Reference Kaletha and PrasadKP23, Definition 1.5.5])

where the right-hand side is as in [Reference Kaletha and PrasadKP23, Definition 7.6.1]. This building comes equipped with an action of $G({K^{\textrm {ur}}})\rtimes \Gamma $ . Moreover, the action of $G({K^{\textrm {ur}}})$ factors through the natural map $G({K^{\textrm {ur}}})\to G^{\text {ad}}({K^{\textrm {ur}}})$ .

Additionally, we have a building

which naturally comes equipped with an action of $G(K)$ which factorizes through the map $G(K)\to G^{\text {ad}}(K)$ . To distinguish these two objects, we call the $\mathscr {B}(G,{K^{\textrm {ur}}})$ the building associated to G over ${K^{\textrm {ur}}}$ and $\mathscr {B}(G,K)$ the building associated to G over K.

Various functorial properties for these buildings have been established in the literature, see [Reference LandvogtLan00] and [Reference Kaletha and PrasadKP23, §14]. However, in this article we require only two basic functoriality statements.

The first concerns quotient maps of reductive groups.

Lemma 2.3 (cf. [Reference Kaletha and PrasadKP23, Proposition 14.1.1]).

Suppose that $f\colon G\to H$ is a faithfully flat map of reductive groups over K. Then, there exists a unique surjective map

$$ \begin{align*} f_\ast\colon \mathscr{B}(G,{K^{\textrm{ur}}})\to \mathscr{B}(H,{K^{\textrm{ur}}}), \end{align*} $$

which is equivariant for the map $G({K^{\textrm {ur}}})\rtimes \Gamma \to H({K^{\textrm {ur}}})\rtimes \Gamma $ . A similar statement holds for the buildings over K.

For the second statement, we first make the following definition. For a field F, a map ${f\colon G\to H}$ of reductive groups over F is an ad-isomorphism if $f(Z(G))\subseteq Z(H)$ , and the induced map ${f^{\text {ad}}\colon G^{\text {ad}}\to H^{\text {ad}}}$ is an isomorphism. We record for later use the following proposition, whose proof is an exercise in assembling well-known facts about reductive groups over fields (e.g., see [Reference MilneMil17]).

The second basic functoriality result is the following.

Lemma 2.5. Suppose $f\colon G \to H$ is an ad-isomorphism of reductive groups over K. Then f induces a bijection of buildings

equivariant for the map $G({K^{\textrm {ur}}})\rtimes \Gamma \to H({K^{\textrm {ur}}})\rtimes \Gamma $ . Moreover, if $g\colon H\to L$ is another ad-isomorphism, then $(g\circ f)_\ast =g_\ast \circ f_\ast $ . A similar statement holds for buildings over K.

Proof. As discussed on [Reference Kaletha and PrasadKP23, p. 264], there are natural $\Gamma $ -equivariant bijections

which are equivariant for the maps $G({K^{\textrm {ur}}})\to G^{\text {ad}}({K^{\textrm {ur}}})$ and $H({K^{\textrm {ur}}})\to H^{\text {ad}}({K^{\textrm {ur}}})$ , respectively. These, together with the $\Gamma $ -equivariant bijection induced by $f^{\text {ad}}$ , gives us a $\Gamma $ -equivariant bijection $\mathscr {B}(G,{K^{\textrm {ur}}})\to \mathscr {B}(H,{K^{\textrm {ur}}})$ . That this is equivariant for the map $G({K^{\textrm {ur}}})\to H({K^{\textrm {ur}}})$ is clear from construction as $G({K^{\textrm {ur}}})$ and $H({K^{\textrm {ur}}})$ act through their images in $G^{\text {ad}}({K^{\textrm {ur}}})$ and $H^{\text {ad}}({K^{\textrm {ur}}})$ , respectively. The proof for the buildings over K follows by passing to the fixed points for $\Gamma $ .

Parahoric groups and group schemes

Let G be a reductive group over K. As in [Reference KottwitzKot97, §7] (see also [Reference Kaletha and PrasadKP23, §11.1]), one may construct a group homomorphism

$$ \begin{align*} \kappa_G\colon G({K^{\textrm{ur}}})\to \pi_1(G)_I, \end{align*} $$

where $\pi _1(G)$ denotes the fundamental group of Borovoi (see [Reference BorovoiBor98, §1]), a finitely generated abelian group, and the subscript $(-)_I$ denotes I-coinvariants. We call $\kappa _G$ the Kottwitz homomorphism associated to G. We let $\kappa _G\otimes 1$ denote the composition of $\kappa _G$ with the map $\pi _1(G)_I\to \pi _1(G)_I\otimes _{\mathbb {Z}}\mathbb {Q}$ .

We denote the kernel of $\kappa _G$ and $\kappa _G\otimes 1$ by $G({K^{\textrm {ur}}})^0$ and $G(K^{\textrm {ur}})^1$ , respectively.Footnote ² The Kottwitz homomorphism is functorial in G, and thus the same holds for the subgroups $G({K^{\textrm {ur}}})^0$ and $G(K^{\textrm {ur}})^1$ . More precisely, if $f\colon G\to H$ is a map of reductive groups over K, then the diagram

commutes.

We then have the following definition of a parahoric group scheme.

As in [Reference Kisin and PappasKP18, §9.2.6], associated to any point x of $\mathscr {B}(G,K)$ is a parahoric group scheme denoted by $\mathcal{G}^\circ _x$ , characterized by Equation (2.1). Moreover, every parahoric group scheme arises in this way. A subgroup of $G(K)$ is called parahoric if it is of the form $\mathcal{G}(\mathcal{O}_K)$ for a parahoric group $\mathcal{O}_K$ -scheme $\mathcal{G}$ .

Parahoric models of tori and R-smooth tori

Suppose that T is a a torus over K. By Lemma 2.5 it’s clear that $\mathscr {B}(T,K)$ is a singleton, and thus that T has a unique parahoric model which we denote by $\mathcal{T}$ . As $\mathscr {B}(T,K)$ is a singleton, $\mathcal{T}(\mathcal{O}_K)$ is equal $T(K)^0$ .

Let $\mathcal{T}^{\textrm {lft}}$ be the Néron model of T (e.g., see [Reference Kaletha and PrasadKP23, §B.7–B.8] and [Reference Bosch, Lütkebohmert and RaynaudBLR90]).

Fix a finite Galois extension $K_1$ of K splitting T, and set $T_1=\textrm {Res}_{K_1/K}T_{K_1}$ which admits a natural embedding $T\to T_1$ .

Consider now a reductive group G over K. As $G_{K^{\textrm {ur}}}$ is quasi-split by Steinberg’s theorem (see [Reference SteinbergSte65, Theorem 1.9]), there is a unique $G(K^{\textrm {ur}})$ -conjugacy class of maximally split maximal tori in $G_{K^{\textrm {ur}}}$ which are precisely the centralizers of maximal split tori (see [Reference BorelBor91, §20]).

As the formation of Néron models commutes with unramified base change, Definition 2.10 agrees with that of Definition 2.9 when G is a torus.

Abelianization of parahoric group schemes

Suppose that G is a reductive group over $\mathcal{O}_K$ , and consider the short exact sequence of reductive groups

$$ \begin{align*} 1\to G^{\mathrm{der}}\xrightarrow{i}G\xrightarrow{f}G^{\text{ab}}\to 1. \end{align*} $$

By Lemma 2.3 and Lemma 2.5 we obtain maps of buildings

where the first map is a bijection and the second is surjective, and the maps are equivariant for $G^{\mathrm {der}}(K)\to G(K)$ and $G(K)\to G^{\text {ab}}(K)$ , respectively. Similar statements hold for the buildings over ${K^{\textrm {ur}}}$ .

Let us fix a point y of $\mathscr {B}(G,K)$ and set $x=i_\ast ^{-1}(y)$ and $z=f_\ast (y)$ . By the functoriality of the Kottwitz homomorphism and the equivariant properties of these maps, we see by [Reference Kaletha and PrasadKP23, Corollary 2.10.10] that i and f extend to maps $i\colon \mathcal{G}_x\to \mathcal{G}_y$ and $f\colon \mathcal{G}_y\to \mathcal{G}_z$ , restricting to maps $i\colon \mathcal{G}_x^\circ \to \mathcal{G}_y^\circ $ and $f\colon \mathcal{G}_y^\circ \to \mathcal{G}_z^\circ $ .

Proposition 2.11. If $G^{\mathrm {der}}$ is R-smooth and the map $\pi _1(G^{\mathrm {der}})_I\to \pi _1(G)_I$ is injective, then we have a short exact sequence

$$ \begin{align*} 1\to \mathcal{G}_x^\circ\to \mathcal{G}_y^\circ\xrightarrow{f} \mathcal{G}_z^\circ\to 1. \end{align*} $$

Proof. First observe that by the setup, we have a diagram

$$ \begin{align*} \mathcal{G}_x^\circ \to \mathcal{G}_y^\circ \to \mathcal{G}_z^\circ. \end{align*} $$

We first claim that $\mathcal{G}_y^\circ (\mathcal{O}_{K^{\textrm {ur}}}) \to \mathcal{G}_z^\circ (\mathcal{O}_{K^{\textrm {ur}}})$ is surjective. Let T be a maximally split maximal torus in $G_{K^{\textrm {ur}}}$ as in [Reference Kaletha and PrasadKP23, §8.3.6]. Then as in loc. cit. there exists a map $\mathcal{T}(\mathcal{O}_{K^{\textrm {ur}}}) \to \mathcal{G}_y^\circ (\mathcal{O}_{K^{\textrm {ur}}})$ , thus it suffices to show that $\mathcal{T}(\mathcal{O}_{K^{\textrm {ur}}}) \to \mathcal{G}_z^\circ (\mathcal{O}_{K^{\textrm {ur}}})$ is surjective. That said, observe that $\ker (T \to G^{\text {ab}}) = T\cap G^{\mathrm {der}}$ is a torus and therefore, by [Reference Kaletha and PrasadKP23, Lemma 2.5.20] the map $T(K^{\textrm {ur}})^0 \to G^{\text {ab}}(K^{\textrm {ur}})^0$ is surjective, so the claim follows.

We now prove that $f\colon \mathcal{G}_y^\circ \to \mathcal{G}_z^\circ $ is faithfully flat. To see that f is surjective, we observe that f is surjective over K, being the map $G\to G^{\text {ab}}$ . Thus, it suffices to show that $f_{k^{\textrm {sep}}}$ contains every point of $\mathcal{G}_z^\circ (k^{\textrm {sep}})$ in its image. As $\mathcal{G}_z^\circ $ is smooth, we know that $\mathcal{G}_z^\circ (\mathcal{O}_{K^{\textrm {ur}}})\to \mathcal{G}_z^\circ (k^{\textrm {sep}})$ is surjective (e.g., see [Reference GrothendieckGro67, Thèoréme 18.5.17]), and, since $\mathcal{G}_y^\circ (\mathcal{O}_{K^{\textrm {ur}}})\to \mathcal{G}_z^\circ (\mathcal{O}_{K^{\textrm {ur}}})$ is surjective, the claim follows. To show that f is flat, we may, by the critére de platitude par fibres (see [Reference GrothendieckGro66, Thèoréme 11.3.10]), show instead that $f_K$ and $f_k$ are flat. To show that $f_k$ is flat, we observe that by [Reference MilneMil17, Theorem 3.34] $f_k$ factorizes as a faithfully flat map g followed by a closed immersion i. But, as $f_k$ is surjective, i must be a homeomorphism on the underlying space which, as $(\mathcal{G}_z^\circ )_k$ is reduced, must be the entire space. Thus, $f_k=g$ is flat. A similar argument holds for $f_K$ .

Finally, since $G^{\mathrm {der}}$ is R-smooth, it follows from [Reference Kisin and ZhouKZ21, Proposition 2.4.8] that the map $\mathcal{G}_x\to \mathcal{G}_y$ is a closed immersion. Next observe that by the injectivity of $\pi _1(G^{\mathrm {der}})_I\to \pi _1(G)_I$ , one may conclude that $G^{\mathrm {der}}(K^{\textrm {ur}})^0=G^{\mathrm {der}}(K^{\textrm {ur}})\cap G(K^{\textrm {ur}})^0$ . Using this, and the fact that the map $\mathcal{G}_x\to \mathcal{G}_y$ is a closed immersion, one may check that $\mathcal{G}_x^\circ \to \mathcal{G}_y^\circ $ is a closed immersion. Indeed, since $\mathcal{G}_x\to \mathcal{G}_y$ is a closed immersion and $\mathcal{G}_y^\circ $ is an open group subscheme of $\mathcal{G}_y$ , $\mathcal{G}_x\cap \mathcal{G}_y^\circ $ is an open group subscheme of $\mathcal{G}_x$ (and thus smooth) and a closed subgroup scheme of $\mathcal{G}_y^\circ $ . We will be done if we can show that $\mathcal{G}_x\cap \mathcal{G}_y^\circ $ and $\mathcal{G}_x^\circ $ are equal. By [Reference Kaletha and PrasadKP23, Corollary 2.10.11] it suffices to check the two have the same $\mathcal{O}_{K^{\textrm {ur}}}$ -points, and this follows from the equality $G^{\mathrm {der}}(K^{\textrm {ur}})^0=G^{\mathrm {der}}(K^{\textrm {ur}})\cap G(K^{\textrm {ur}})^0$ . Finally to show that the closed immersion $\mathcal{G}_x^\circ \to \mathcal{G}_y^\circ $ identifies $\mathcal{G}_x^\circ $ with $\ker (\mathcal{G}_y^\circ \to \mathcal{G}_z^\circ )$ it suffices to check this claim on the generic fiber,Footnote ⁴ where it is clear.

In practice, if we have fixed a parahoric model $\mathcal{G}$ of G, then we write the unique parahoric model of $G^{\text {ab}}$ by $\mathcal{G}^{\text {ab}}$ . Given Proposition 2.11 this notational abuse is not too severe. Similarly, we denote the associated parahoric model of $G^{\textrm {der}}$ by $\mathcal{G}^{\textrm {der}}$ .

Parahoric group schemes and central quotients

Let $f\colon G\to G'$ be a faithfully flat map such that is a torus over K. Let x be a point of $\mathscr {B}(G,K)$ and $x'$ its image under the map the map $f_\ast $ from Lemma 2.3. Let us write $\mathcal{G}=\mathcal{G}_x^\circ $ and $\mathcal{G}'=\mathcal{G}_{x'}^\circ $

As $f_\ast $ is equivariant for the map $G({K^{\textrm {ur}}})\to G'({K^{\textrm {ur}}})$ and the Kottwitz homomorphism is functorial, we deduce that f maps $\mathcal{G}(\mathcal{O}_{K^{\textrm {ur}}})$ into $\mathcal{G}'(\mathcal{O}_{K^{\textrm {ur}}})$ . Thus, by [Reference Kaletha and PrasadKP23, Corollary 2.10.10] f uniquely lifts to a map $\mathcal{G}\to \mathcal{G}'$ . When Z is R-smooth, one can say more.

Proposition 2.12 ([Reference Kisin and ZhouKZ21, Proposition 2.4.13]).

Suppose that Z is an R-smooth torus. Then, there exists a short exact sequence of group $\mathcal{O}_K$ -schemes

$$ \begin{align*} 1\to \mathcal{Z}\to \mathcal{G}\xrightarrow{f} \mathcal{G}'\to 1, \end{align*} $$

where $\mathcal{Z}$ is the Zariski closure of Z in $\mathcal{G}$ . Moreover, $\mathcal{Z}$ is a smooth group $\mathcal{O}_K$ -scheme.

Parahoric group schemes and ad-isomorphisms

Suppose that $i\colon G\to H$ is an injective ad-isomorphism of reductive groups over K. Suppose that $\mathcal{H}$ is a parahoric model of H, and y is a point of $\mathscr {B}(H,K)$ such that $\mathcal{H}$ is isomorphic to $\mathcal{H}_y^\circ $ . By Lemma 2.5 there is a unique point x of $\mathscr {B}(G,K)$ such that $y=i_\ast (x)$ .

The following basic proposition provides a recognition principle for $\mathcal{G}_x^\circ $ .

Proposition 2.13. A model $\mathcal{G}$ of G is isomorphic to $\mathcal{G}_x^\circ $ if it is a smooth affine group $\mathcal{O}_K$ -scheme with connected fibers and

(2.2) $$ \begin{align} \mathcal{G}(\mathcal{O}_{K^{\textrm{ur}}}) = \mathcal{H}(\mathcal{O}_{K^{\textrm{ur}}}) \cap G({K^{\textrm{ur}}}). \end{align} $$

Moreover, i extends to a map of $\mathcal{O}_K$ -group schemes $\mathcal{G} \to \mathcal{H}$ .

Proof. By the definition of parahoric group schemes, and the definition of $\mathcal{G}_x^\circ $ , it suffices to show that

(2.3) $$ \begin{align} \mathcal{G}(\mathcal{O}_{K^{\textrm{ur}}}) = G({K^{\textrm{ur}}})^0 \cap \textrm{Stab}_{G({K^{\textrm{ur}}})}(x). \end{align} $$

Since $i_\ast $ defines a bijection which is equivariant for the map $G({K^{\textrm {ur}}}) \to H({K^{\textrm {ur}}})$ , we deduce that

(2.4) $$ \begin{align} \textrm{Stab}_{G({K^{\textrm{ur}}})}(x) = \textrm{Stab}_{H({K^{\textrm{ur}}})}(y) \cap G({K^{\textrm{ur}}}). \end{align} $$

The inclusion $\mathcal{G}(\mathcal{O}_{K^{\textrm {ur}}}) \subseteq \textrm {Stab}_{G({K^{\textrm {ur}}})}(x)$ then follows immediately from (2.2). Additionally, as $\mathcal{G}$ is a smooth affine group $\mathcal{O}_K$ -scheme with connected fibers, it follows from [Reference Kaletha and PrasadKP23, Proposition 8.3.15] that $\mathcal{G}(\mathcal{O}_{K^{\textrm {ur}}}) \subseteq G({K^{\textrm {ur}}})^0$ . Thus, $\mathcal{G}(\mathcal{O}_{K^{\textrm {ur}}})\subseteq G({K^{\textrm {ur}}})^0\cap \textrm {Stab}_{G({K^{\textrm {ur}}})}(x)$ .

On the other hand, if g belongs to $G({K^{\textrm {ur}}})^0 \cap \textrm {Stab}_{G({K^{\textrm {ur}}})}(x)$ , then $i(g)$ belongs to $\mathcal{H}(\mathcal{O}_{K^{\textrm {ur}}})$ by functoriality of the Kottwitz homomorphism along with the identity (2.4). Hence (2.2) implies that g belongs to $\mathcal{G}(\mathcal{O}_{K^{\textrm {ur}}})$ .

Finally, the fact that i extends to a morphism $i\colon \mathcal{G}\to \mathcal{H}$ follows from [Reference Kaletha and PrasadKP23, Corollary 2.10.10] by (2.2).

2.2 Fiber products of groups

In this subsection we aim to collect some facts concerning fiber products of groups, in various contexts, that will be useful below.

Smoothness and connectedness of fiber products of group schemes

Fix a connected scheme S and consider a Cartesian diagram of group S-schemes

Set $\mathcal{K}$ to be the group S-scheme $\ker (f)$ .

One may replace all instances of “connected” in the assumptions and conclusion of the above statement by “geometrically connected”, “irreducible”, or “geometrically irreducible” by standard theory concerning algebraic groups (see [Reference MilneMil17, Summary 1.36]). We use this observation freely below.

Some fiber products of parahoric group schemes

Suppose now that, as in §2.1, K is a discretely valued field with perfect residue field. Suppose further that we have a Cartesian diagram of group $\mathcal{O}_K$ -schemes

where $\mathcal{A}$ , $\mathcal{B}$ , and $\mathcal{C}$ are parahoric group $\mathcal{O}_K$ -schemes. For simplicity we denote the generic fiber of $\mathcal{A}$ , $\mathcal{B}$ , $\mathcal{C}$ , and $\mathcal{D}$ , by $A,B,C,$ and D, respectively.

Proposition 2.15. Suppose that f is faithfully flat and quasi-compact, both B and C are tori, and $\ker (f)$ is a smooth group $\mathcal{O}_K$ -scheme with connected fibers. Then, $\mathcal{D}$ is a parahoric group $\mathcal{O}_K$ -scheme. Moreover, if $\mathcal{A}=\mathcal{G}_x^\circ $ , then $\mathcal{D}=\mathcal{G}^\circ _{(x,\ast )}$ under the isomorphism

$$ \begin{align*} \mathscr{B}(D,K)\to \mathscr{B}(A\times_{\mathrm{Spec}\,(K)}C,K)\simeq \mathscr{B}(A,K)\times\{\ast\} \end{align*} $$

from Proposition 2.5.

Proof. By Proposition 2.14, we know that $\mathcal{D}$ is a smooth group $\mathcal{O}_K$ -scheme with connected fibers. Moreover, observe that we have a natural short exact sequence of group $\mathcal{O}_K$ -schemes

$$ \begin{align*} 1\to \mathcal{D}\to \mathcal{A}\times_{\mathrm{Spec}\,(\mathcal{O}_K)}\mathcal{C}\to \mathcal{B}\to 1, \end{align*} $$

where the map $\mathcal{A}\times _{\mathrm{Spec}\,(\mathcal{O}_K)}\mathcal{C}\to \mathcal{B}$ is given by sending $(a,c)$ to $f(a)g(c)^{-1}$ . This implies that D is a normal subgroup of the reductive group $A\times _{\mathrm{Spec}\,(K)}C$ and so reductive (see [Reference MilneMil17, Corollary 21.53]).

On the other hand, this exact sequence also implies that ${D}\to {A}\times _{\mathrm{Spec}\,(K)}{C}$ induces an isomorphism on adjoint subgroups as ${B}$ is abelian. Moreover, observe that evidently

$$ \begin{align*} \mathcal{D}(\mathcal{O}_{K^{\textrm{ur}}})=(\mathcal{A}(\mathcal{O}_{K^{\textrm{ur}}})\times \mathcal{C}(\mathcal{O}_{K^{\textrm{ur}}}))\cap D(K^{\textrm{ur}}). \end{align*} $$

The claim then follows directly from Proposition 2.13.

Torsors for fiber products of groups

We next are interested in understanding the relationship between torsors for a fiber product of groups, and torsors for their constituent groups. The correct generality for this discussion is that of topoi. We follow the notation and conventions concerning torsors and topoi as found in [Reference Imai, Kato and YoucisIKY24a, Reference Imai, Kato and YoucisIKY24b, §A.1].

Let us fix a topos $\mathscr {T}$ with final object $\ast $ , and a Cartesian diagram of group objects

Consider the $2$ -fiber product of groupoids

$$ \begin{align*} \mathbf{Tors}_{\mathcal{A}}( \mathscr{T})\times_{f_\ast,\mathbf{Tors}_{\mathcal{B}}( \mathscr{T}),g_\ast}\mathbf{Tors}_{\mathcal{C}}( \mathscr{T}), \end{align*} $$

(e.g., see [SP17, Tag 02X9]). Observe that there is a natural equivalence of functors

$$ \begin{align*} \theta_{\textrm{nat}}\colon f_\ast\circ p_\ast\simeq (f\circ p)_\ast=(g\circ q)_\ast\simeq g_\ast\circ q_\ast, \end{align*} $$

and thus the triple $(p_\ast ,q_\ast ,\theta _{\textrm {nat}})$ determines an object of the $2$ -fiber product. Thus,

(2.5) $$ \begin{align} \Psi\colon \mathbf{Tors}_{\mathcal{D}}(\mathscr{T})\to \mathbf{Tors}_{\mathcal{A}}(\mathscr{T})\times_{f_\ast,\mathbf{Tors}_{\mathcal{B}}(\mathscr{T}),g_\ast}\mathbf{Tors}_{\mathcal{C}}(\mathscr{T}), \end{align} $$

given by $\Psi (\mathcal{P})=(p_\ast \mathcal{P},q_\ast \mathcal{P},\theta _{\textrm {nat}})$ is a morphism of groupoids.

Conversely, suppose that $(\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta )$ is an object of $\mathbf {Tors}_{\mathcal{A}}(\mathscr {T})\times _{f_\ast ,\mathbf {Tors}_{\mathcal{B}}(\mathscr {T}),g_\ast }\mathbf {Tors}_{\mathcal{C}}(\mathscr {T})$ . Let us observe that we have a natural map

$$ \begin{align*} \mathcal{Q}_{\mathcal{A}}\to f_\ast\mathcal{Q}_{\mathcal{A}}=(\mathcal{B}\times\mathcal{Q}_{\mathcal{A}})/\mathcal{A}, \end{align*} $$

induced by the map $(e,\textrm {id})\colon \mathcal{Q}_{\mathcal{A}}\to \mathcal{B}\times \mathcal{Q}_{\mathcal{A}}$ , where e denotes the identity section of $\mathcal{B}$ . We similarly have a map $\mathcal{Q}_{\mathcal{C}}\to g_\ast \mathcal{Q}_{\mathcal{C}}$ . Let us define

where this limit is taken in $\mathscr {T}$ . It is simple to check that there is a unique $\mathcal{D}$ -action on $\mathcal{Q}_{\mathcal{A}}\times _\theta \mathcal{Q}_{\mathcal{C}}$ for which the natural map $\mathcal{Q}_{\mathcal{A}}\times _{\theta }\mathcal{Q}_{\mathcal{C}}\to \mathcal{Q}_{\mathcal{A}}\times \mathcal{Q}_{\mathcal{C}}$ is equivariant for the map $\mathcal{D}\to \mathcal{A}\times \mathcal{C}$ . This $\mathcal{D}$ -object of $\mathscr {T}$ is evidently functorial in the triple $(\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta )$ .

Note that it is not a priori clear that $\mathcal{Q}_{\mathcal{A}}\times _\theta \mathcal{Q}_{\mathcal{C}}$ is a $\mathcal{D}$ -torsor. That said, we do have the following affirmative result in this direction under the assumption that f is an epimorphism.

Proposition 2.16. Suppose that f is an epimorphism. Then, the functor

$$ \begin{align*} \Phi\colon \mathbf{Tors}_{\mathcal{A}}(\mathscr{T})\times_{f_\ast,\mathbf{Tors}_{\mathcal{B}}(\mathscr{T}),g_\ast}\mathbf{Tors}_{\mathcal{C}}(\mathscr{T})\to \mathbf{Tors}_{\mathcal{D}}(\mathscr{T}) \end{align*} $$

given by $\Phi (\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta )=\mathcal{Q}_{\mathcal{A}}\times _\theta \mathcal{Q}_{\mathcal{C}}$ , is a well-defined quasi-inverse to $\Psi $ . Moreover, the equivalences $\Psi $ and $\Phi $ are compatible with pullbacks along a morphism of topoi $\mathscr {T}'\to \mathscr {T}\kern-1pt$ .

Proof. To show that $\mathcal{Q}_{\mathcal{A}}\times _\theta \mathcal{Q}_{\mathcal{C}}$ is a $\mathcal{D}$ -torsor we are free to localize $\mathscr {T}$ . In particular, we may assume that $\mathcal{Q}_{\mathcal{C}}$ is trivializable. Choose a trivialization corresponding to an element x of $\mathcal{Q}_{\mathcal{C}}(\ast )$ . Consider then the induced element of $g_\ast \mathcal{Q}_{\mathcal{C}}(\ast )$ , which we also denote by x, and the resulting element $\theta ^{-1}(x)$ of $f_\ast \mathcal{Q}_{\mathcal{A}}(\ast )$ . As f is an epimorphism we may, after possibly localizing $\mathscr {T}$ further, assume that there exists some element y of $\mathcal{Q}_{\mathcal{A}}(\ast )$ mapping to $\theta ^{-1}(x)$ . These compatible elements define an isomorphism of diagrams

As $\mathcal{A}\times _{\textrm {id}}\mathcal{C}$ is evidently the trivial $\mathcal{D}$ -torsor, the claim follows.

To prove that $\Phi \circ \Psi $ is naturally isomorphic to the identity, it suffices to prove the following claim: for a $\mathcal{D}$ -torsor $\mathcal{P}$ the natural map

$$ \begin{align*} \mathcal{P}\to f_\ast\mathcal{P}\times g_\ast\mathcal{P}, \end{align*} $$

is equivariant for the map $\mathcal{D}\to \mathcal{A}\times \mathcal{C}$ , and factorizes uniquely through a $\mathcal{D}$ -equivariant map $\mathcal{P}\to f_\ast \mathcal{P}\times _\theta g_\ast \mathcal{P}$ . But this claim can be checked after localizing on $\mathscr {T}$ , which reduces us to the trivial case as in the previous paragraph.

To prove that $\Psi \circ \Phi $ is naturally isomorphic to the identity, let us fix an object $(\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta )$ of the $2$ -fiber product. Let us observe that we have a natural projection map

$$ \begin{align*} \mathcal{Q}_{\mathcal{A}}\times_\theta\mathcal{Q}_{\mathcal{C}}\to \mathcal{Q}_{\mathcal{A}} \end{align*} $$

which is equivariant for the map $\mathcal{D}\to \mathcal{A}$ . By the adjunction in [Reference GiraudGir71, Chapitre III, Proposition 1.3.6 (iii)] we obtain a unique $\mathcal{A}$ -equivariant map

$$ \begin{align*} \alpha\colon f_\ast(\mathcal{Q}_{\mathcal{A}}\times_\theta\mathcal{Q}_{\mathcal{C}})\to \mathcal{Q}_{\mathcal{A}}, \end{align*} $$

which is necessarily an isomorphism of $\mathcal{A}$ -torsors. Similarly, we obtain an isomorphism

$$ \begin{align*} \beta\colon g_\ast(\mathcal{Q}_{\mathcal{A}}\times_\theta\mathcal{Q}_{\mathcal{C}})\to \mathcal{Q}_{\mathcal{C}}. \end{align*} $$

It is then easy to check that $(\alpha ,\beta )$ defines a natural morphism

$$ \begin{align*} \Psi(\Phi(\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta))\to (\mathcal{Q}_{\mathcal{A}},\mathcal{Q}_{\mathcal{C}},\theta), \end{align*} $$

where the matching of $\theta _{\textrm {nat}}$ in the source and $\theta $ in the target may be checked locally on $\mathscr {T}$ , from where we may reduce to the trivial case where it is clear. As the $2$ -fiber product is a groupoid, we deduce that $(\alpha ,\beta )$ is a natural isomorphism.

The final compatibility claim is clear by construction.

Shtukas for fiber products of groups

We finally record the natural implication of Proposition 2.16 for shtukas over a pre-adic space in the sense of [Reference Pappas and RapoportPR24, §2.3].

To this end, let us fix a fiber product of group $\mathbb {Z}_p$ -schemes

where $\mathcal{A}$ , $\mathcal{B}$ , and $\mathcal{C}$ are smooth group $\mathbb {Z}_p$ -schemes with connected fibers, f is faithfully flat and quasi-compact, and $\ker (f)$ is a smooth group $\mathbb {Z}_p$ -scheme with connected fibers. We then see by Proposition 2.14 that $\mathcal{D}$ is a smooth group $\mathbb {Z}_p$ -scheme with connected fibers. Further fix a discretely valued algebraic extension E of $\mathbb {Q}_p$ with residue field $k_E$ , and fix a pre-adic space $\mathscr {X}$ over $\mathcal{O}_{E}$ . We then have the following result.

Corollary 2.17. There is an equivalence of categories

Proof. Fix an morphism $\alpha \colon S\to \mathscr {X}^{\lozenge /}$ , where S is an object of $\mathbf {Perf}_{k_E}$ , with corresponding untilt $S^\sharp $ over $\mathcal{O}_{E}$ . Then, it immediately by applying Proposition 2.16 to the étale topoi of $S\dot {\times }\mathbb {Z}_p$ and $S\dot {\times }\mathbb {Z}_p\setminus S^\sharp $ , that there are equivalences

As this equivalence is $2$ -functorial in the topos, the claim then follows by letting S vary.

3.1 Existence of models and their properties

In this subsection we precisely state the existence of integral models at parahoric level and list some of their properties which are most relevant for our purposes.

Shimura varieties

We begin by recalling some definitions and notation from the theory of Shimura varieties. Let $(\mathbf {G},\mathbf {X})$ be a Shimura datum, meaning that ${\mathbf {G}}$ is a reductive group over $\mathbb {Q}$ , and $\mathbf {X}$ is a ${\mathbf {G}}(\mathbb {R})$ -conjugacy class of homomorphisms

$$ \begin{align*} h: \mathbb{S}\to {\mathbf{G}}_{\mathbb{R}}, \end{align*} $$

(where $\mathbb {S} = {\textrm {Res}}_{\mathbb {C}/\mathbb {R}}\,\mathbb {G}_{m/\mathbb {C}}$ is the Deligne torus), satisfying the axioms (SV1)-(SV3) in [Reference MilneMil05, Definition 5.5]. Associated to $\mathbf {X}$ is a unique conjugacy class of coharacters $\mathbb {G}_{m,\mathbb {C}}\to \mathbf {G}_{\mathbb {C}}$ (see [Reference MilneMil05, p. 344]). We denote the field of definition of by ${\mathbf {E}}$ , and call it the reflex field of $(\mathbf {G},\mathbf {X})$ (see [Reference MilneMil05, Definition 12.2]).

We often fix a place v of $\mathbf {E}$ dividing p, and write E for the completion $\mathbf {E}_v$ . Using [Reference KottwitzKot84, Lemma 1.1.3], gives rise to a unique conjugacy class of cocharacters $\mathbb {G}_{m,\overline {E}}\to G_{\overline {E}}$ , which we also denote by . One may check that this conjugacy class has field of definition E, and so we call this the local reflex field.

For any neat compact open subgroup $\mathsf {K} \subseteq {\mathbf {G}}(\mathbb {A}_f)$ (see [Reference MilneMil05, p. 288]), we attach to $(\mathbf {G},\mathbf {X})$ and $\mathsf {K}$ the Shimura variety $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ , which is a quasi-projective variety over $\mathbb {C}$ , whose $\mathbb {C}$ -points are given by the equality

$$ \begin{align*} \textrm{Sh}_{\mathsf{K}}(\mathbf{G},\mathbf{X})(\mathbb{C}) = {\mathbf{G}}(\mathbb{Q})\backslash \mathbf{X} \times {\mathbf{G}}(\mathbb{A}_f) / \mathsf{K}. \end{align*} $$

We write an element of this double quotient as $[x,g]_{\mathsf {K}}$ . As in [Reference DeligneDel79] (cf. [Reference MoonenMoo98]), $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ admits a canonical model over the reflex field ${\mathbf {E}}$ . Hereafter we will use $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ to denote the canonical model over ${\mathbf {E}}$ .

For a containment $\mathsf {K} \subseteq \mathsf {K}'$ of neat compact open subgroups of ${\mathbf {G}}(\mathbb {A}_f)$ and g in ${\mathbf {G}}(\mathbb {A}_f)$ such that $g^{-1}\mathsf {K} g \subseteq \mathsf {K}'$ , there is a unique finite étale morphism of ${\mathbf {E}}$ -schemes

$$ \begin{align*} t_{\mathsf{K},\mathsf{K}'}(g): \textrm{Sh}_{\mathsf{K}}(\mathbf{G},\mathbf{X}) \to \textrm{Sh}_{\mathsf{K}'}(\mathbf{G},\mathbf{X}), \end{align*} $$

which is given on $\mathbb {C}$ -points by the identity

$$ \begin{align*} t_{\mathsf{K},\mathsf{K}'}(g)[x,g']_{\mathsf{K}} = [x,g'g]_{\mathsf{K}'}. \end{align*} $$

We write $\pi _{\mathsf {K},\mathsf {K}'}$ for $t_{\mathsf {K},\mathsf {K}'}(\text {id})$ , and $[g]_{\mathsf { K}}$ for $t_{\mathsf {K}, g^{-1}\mathsf {K} g}(g)$ .

The morphisms $\pi _{\mathsf {K},\mathsf {K}'}$ form a projective system $\{\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})\}_{\mathsf {K}}$ with finite étale transition morphisms. We define

(3.1)

where the limit is taken over all neat compact open subgroups $\mathsf {K}$ of ${\mathbf {G}}(\mathbb {A}_f)$ . Since each $\pi _{\mathsf {K},\mathsf {K}'}$ is affine, the limit in (3.1) exists as an ${\mathbf {E}}$ -scheme (see [SP17, Tag 01YX]). The morphisms $[g]_{\mathsf {K}}$ endow $\textrm {Sh}(\mathbf {G},\mathbf {X})$ with a continuous action of ${\mathbf {G}}(\mathbb {A}_f)$ in the sense of [Reference DeligneDel79, 2.7.1].

We will often fix a prime p and consider neat compact open subgroups $\mathsf {K} = \mathsf {K}_p \mathsf {K}^p$ with $\mathsf {K}_p \subseteq {\mathbf {G}}(\mathbb {Q}_p)$ and $\mathsf {K}^p \subseteq {\mathbf {G}}(\mathbb {A}_f^p)$ . We define

where $\mathsf {K}^{p}$ varies over neat compact open subgroups of ${\mathbf {G}}(\mathbb {A}_f^p)$ . As in the case of $\textrm {Sh}(\mathbf {G},\mathbf {X})$ , we see that $\textrm {Sh}_{\mathsf { K}_p}(\mathbf {G},\mathbf {X})$ is a scheme over ${\mathbf {E}}$ with a continuous action of ${\mathbf {G}}(\mathbb {A}_f^p)$ .

A morphism of Shimura data $\alpha \colon (\mathbf {G}_1,\mathbf {X}_1)\to (\mathbf {G},\mathbf {X})$ is a morphism of group $\mathbb {Q}$ -schemes $\alpha \colon \mathbf {G}_1\to \mathbf {G}$ with the property that $\alpha _{\mathbb {R}}\circ h_1$ belongs to $\mathbf {X}$ for $h_1$ in $\mathbf {X}_1$ . We say that $\alpha $ is an embedding if $\mathbf {G}_1\to \mathbf {G}$ is a closed embedding. From [Reference DeligneDel79, §5], for a morphism of Shimura data $\alpha $ , one has that $\mathbf {E}\subseteq \mathbf {E}_1$ and there is a unique morphism $\textrm {Sh}(\mathbf {G}_1,\mathbf {X}_1)\to \textrm {Sh}(\mathbf {G},\mathbf {X})_{\mathbf {E}_1}$ of $\mathbf {E}_1$ -schemes equivariant for the map $\alpha \colon \mathbf {G}_1(\mathbb {A}_f)\to \mathbf {G}(\mathbb {A}_f)$ and such that if $\alpha (\mathsf {K}_1)\subseteq \mathsf {K}$ then the induced map on the quotients $\alpha _{\mathsf {K}_1,\mathsf {K}}\colon \textrm {Sh}_{\mathsf {K}_1}(\mathbf {G},\mathbf {X})\to \textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})_{E_1}$ is given on $\mathbb {C}$ -points by

$$ \begin{align*} \alpha_{\mathsf{K}_1,\mathsf{K}}\left([x_1,g_1]_{\mathsf{K}_1}\right)=[\alpha\circ x_1,\alpha(g_1)]_{\mathsf{K}}. \end{align*} $$

We say that $\alpha $ is an ad-isomorphism if $\alpha \colon \mathbf {G}_1\to \mathbf {G}$ is an ad-isomorphism and moreover, that $\alpha \colon (\mathbf {G}_1^{\textrm {ad}},\mathbf {X}_1^{\textrm {ad}})\to (\mathbf {G}^{\textrm {ad}},\mathbf {X}^{\textrm {ad}})$ is an isomorphism.

Parahoric Shimura data

We say that a triple $(\mathbf {G},\mathbf {X},\mathcal{G})$ is a parahoric Shimura datum if $(\mathbf {G},\mathbf {X})$ is a Shimura datum, and $\mathcal{G}$ is a parahoric model of . By a morphism $\alpha \colon (\mathbf {G}_1,\mathbf {X}_1,\mathcal{G}_1)\to (\mathbf {G},\mathbf {X},\mathcal{G})$ of parahoric Shimura data we mean a morphism $\alpha \colon (\mathbf {G}_1,\mathbf {X}_1)\to (\mathbf {G},\mathbf {X})$ of Shimura data together with a specified model $\mathcal{G}_1\to \mathcal{G}$ of $G_1\to G$ , which we also denote $\alpha $ . By [Reference Kaletha and PrasadKP23, Corollary 2.10.10], such an $\alpha $ is unique, and it exists if and only if $\alpha \colon G(\breve {\mathbb {Q}}_p)\to G_1(\breve {\mathbb {Q}}_p)$ maps $\mathcal{G}_1(\breve {\mathbb {Z}}_p)$ into $\mathcal{G}(\breve {\mathbb {Z}}_p)$ . We say that $\alpha $ is an embedding or an ad-isomorphism if $\alpha \colon (\mathbf {G}_1,\mathbf {X}_1)\to (\mathbf {G},\mathbf {X})$ is.

Given a parahoric Shimura datum $(\mathbf {G},\mathbf {X},\mathcal{G})$ we often use the following shorthand

Choosing a point x of $\mathscr {B}(G,\mathbb {Q}_p)$ such that $\mathcal{G}=\mathcal{G}_x^\circ $ , we write

which is a a supergroup of $\mathsf {K}_p$ of finite index.

The following lemma will be important for later constructions. Let us denote by $\mathbf {E}^p$ the maximal extension of $\mathbf {E}$ unramified at places dividing p.

Lemma 3.2 ([Reference Kisin and PappasKP18, Corollary 4.3.9]).

The natural map

$$ \begin{align*} \pi_0(\mathrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{\overline{\mathbf{E}}})\to \pi_0(\mathrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{\mathbf{E}^p}) \end{align*} $$

is a bijection.

In other words, this result says that the action of $\text {Gal}(\overline {\mathbf {E}}/\mathbf {E})$ on $\pi _0(\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})_{\overline {\mathbf {E}}})$ factorizes through $\text {Gal}(\mathbf {E}^p/\mathbf {E})$ (cf. [SP17, Tag 038D]).

Some conditions on a Shimura datum

The construction of Kisin–Pappas–Zhou models involves several conditions on a Shimura datum that we presently recall.

We start with the following standard definitions, in increasing level of generality.

We have the following examples of Shimura data of abelian type.Footnote ⁵

We say that a parahoric Shimura datum $(\mathbf {G},\mathbf {X},\mathcal{G})$ is of Hodge type or abelian type if the underlying Shimura datum $(\mathbf {G},\mathbf {X})$ is.

Finally, we set some terminology used later on in the article. Let $(\mathbf {G},\mathbf {X},\mathcal{G})$ be a parahoric Shimura datum of Hodge type. We call a map of Shimura data

$$ \begin{align*} \iota\colon (\mathbf{G},\mathbf{X},\mathcal{G})\to (\textrm{GSp}(\mathbf{V}),\mathfrak{h}^\pm,\textrm{GSp}(\Lambda)), \end{align*} $$

where $\mathbf {V}$ a symplectic $\mathbb {Q}$ -space and $\Lambda \subseteq \mathbf {V}_{\mathbb {Q}_p}$ is a self-dual $\mathbb {Z}_p$ -lattice, a Hodge embedding of parahoric Shimura data if the underling map $\iota \colon \mathbf {G}\to \textrm {GSp}(\mathbf {V})$ is a closed embedding. We say that $\iota $ respects stabilizers if $\widetilde {\mathsf {K}}_p=\textrm {GSp}(\Lambda )\cap \mathbf {G}(\mathbb {Q}_p)$ .

Existence and properties of models

We are now prepared to state the existence of Kisin–Pappas–Zhou models, as well as the major properties we need of these models.

We will recall below, fairly precisely, the construction of the models $\mathscr {S}_{\mathsf {K}}^{\mathfrak {d}}(\mathbf {G},\mathbf {X})$ . The superscript $\mathfrak {d}$ , which is either a parahoric Shimura datum of Hodge type well-adapted to $(\mathbf {G},\mathbf {X},\mathcal{G})$ , or possibly $\varnothing $ when $(\mathbf {G},\mathbf {X},\mathcal{G})$ is of Hodge type. Its role is to emphasize various choices made in the construction of these models.

3.2 The $\mathscr {A}$ -group and the $\mathscr {E}$ -group

Let ${\mathbf {G}}$ be a reductive $\mathbb {Q}$ -group. In this section we associate to ${\mathbf {G}}$ groups $\mathscr {A}({\mathbf {G}})$ and $\mathscr {A}({\mathbf {G}})^\circ $ , and the so-called $\mathscr {E}$ -group to a parahoric Shimura datum, following [Reference DeligneDel79]. We also define analogs of these groups for $\mathbb {Z}_{(p)}$ -valued points following [Reference Kisin and PappasKP18] and [Reference Kisin and ZhouKZ21], and discuss some functorial properties of these groups. These groups play an important role in the construction of the models $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}_p\mathsf {K}^p}(\mathbf {G},\mathbf {X})$ from Theorem 3.9.

The $\ast $ -product

Let us begin by recalling the $\ast $ -product of Deligne (see [Reference DeligneDel79]).

Suppose H and $\Gamma $ are abstract groups equipped with actions of a group $\Delta $ by group homomorphisms. Let

$$ \begin{align*} \psi: \Gamma \to H,\quad \text{and}\quad\varphi: \Gamma \to \Delta \end{align*} $$

be $\Delta $ -equivariant group homomorphisms, where $\Delta $ acts on itself by inner automorphisms, and assume that $\psi $ and $\varphi $ are compatible in the sense that, for all $\gamma $ in $\Gamma $ , we have the identity

$$ \begin{align*} \varphi(\gamma) \cdot h = \psi(\gamma)^{-1}h\psi(\gamma), \end{align*} $$

where $\cdot $ denotes the action of $\Delta $ on H. Then the semi-direct product $H \rtimes \Delta $ admits a quotient

where one easily checks that the subgroup $\{(\psi (\gamma ), \varphi (\gamma )^{-1})\}$ of all the elements of the form $(\psi (\gamma ), \varphi (\gamma )^{-1})$ for $\gamma $ in $\Gamma $ is a normal subgroup.

For the purposes of this section, we will refer to a datum $(H, \Gamma , \Delta , \psi , \varphi )$ as above as a tuple. A morphism of tuples $(H, \Gamma , \Delta , \psi , \varphi ) \to (H', \Gamma ', \Delta ', \psi ', \varphi ')$ is a triple $(f,g,h)$ , consisting of group homomorphisms $f: H\to H'$ and $g: \Gamma \to \Gamma '$ equivariant relative to a group homomorphism $h: \Delta \to \Delta '$ and such that $\psi ' \circ g = f \circ \psi $ and $h\circ \varphi = \varphi ' \circ f$ .

The $\ast $ -product satisfies the following elementary functoriality.

Lemma 3.10. Any morphism of tuples

$$ \begin{align*} (f,g,h)\colon (H, \Gamma, \Delta, \varphi) \to (H', \Gamma', \Delta', \varphi') \end{align*} $$

induces a homomorphism of groups

$$ \begin{align*} H \ast_\Gamma \Delta \to H' \ast_{\Gamma'} \Delta'. \end{align*} $$

For future use, we also remark that if X is a set equipped with a right action of $H \rtimes \Delta $ , then the action descends to an action of $H \ast _\Gamma \Delta $ on the quotient $X / \Gamma $ , where here $\Gamma $ acts on X via

(3.2) $$ \begin{align} x \cdot \gamma = x \cdot (\psi(\gamma), \varphi(\gamma)^{-1}). \end{align} $$

The groups $\mathscr {A}({\mathbf {G}})$ and $\mathscr {A}({\mathbf {G}})^\circ $

Let us now recall some notation from [Reference DeligneDel79].

If ${\mathbf {G}}$ is a reductive group over $\mathbb {Q}$ , then we denote by ${\mathbf {G}}(\mathbb {R})^+$ the connected component of the identity in ${\mathbf {G}}(\mathbb {R})$ with respect to the real topology. For a subgroup $\mathsf {H}$ of ${\mathbf {G}}(\mathbb {R})$ , we denote by $\mathsf {H}_+$ the inverse image of ${\mathbf {G}}^{\text {ad}}(\mathbb {R})^+$ in $\mathsf {H}$ . We write also ${\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+ = {\mathbf {G}}^{\text {ad}}(\mathbb {Q}) \cap {\mathbf {G}}^{\text {ad}}(\mathbb {R})^+$ .

Now suppose $(\mathbf {G},\mathbf {X})$ is a Shimura datum. Let $\mathbf {Z}$ denote the center of ${\mathbf {G}}$ , and let $\mathbf {Z}(\mathbb {Q})^-$ denote the closure of $\mathbf {Z}(\mathbb {Q})$ in ${\mathbf {G}}(\mathbb {A}_f)$ . We define

where ${\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ acts by conjugation on ${\mathbf {G}}(\mathbb {A}_f) / \mathbf {Z}(\mathbb {Q})^-$ and ${\mathbf {G}}(\mathbb {Q})_+ / \mathbf {Z}(\mathbb {Q})\to {\mathbf {G}}(\mathbb {A}_f) / \mathbf {Z}(\mathbb {Q})^-$ and ${\mathbf {G}}(\mathbb {Q})_+ / \mathbf {Z}(\mathbb {Q})\to {\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ are the obvious maps.

By [Reference MilneMil05, Proposition 5.1], the map ${\mathbf {G}}(\mathbb {R})^+ \to {\mathbf {G}}^{\text {ad}}(\mathbb {R})^+$ is surjective with kernel $\mathbf {Z}(\mathbb {R}) \cap {\mathbf {G}}(\mathbb {R})^+$ , which is contained in the center of ${\mathbf {G}}(\mathbb {R})^+$ , so the conjugation action of ${\mathbf {G}}$ on itself induces an action of ${\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ on $\textrm {Sh}(\mathbf {G},\mathbf {X})$ . Combining this with the action of ${\mathbf {G}}(\mathbb {A}_f)$ on $\textrm {Sh}(\mathbf {G},\mathbf {X})$ determines a right action of $\mathscr {A}({\mathbf {G}})$ on $\textrm {Sh}(\mathbf {G},\mathbf {X})$ .

Denote by ${\mathbf {G}}(\mathbb {Q})_+^-$ the closure of ${\mathbf {G}}(\mathbb {Q})_+$ in ${\mathbf {G}}(\mathbb {A}_f)$ . Define

$$ \begin{align*} \mathscr{A}({\mathbf{G}})^\circ = {\mathbf{G}}(\mathbb{Q})_+^- / \mathbf{Z}(\mathbb{Q})^- \ast_{{\mathbf{G}}(\mathbb{Q})_+/\mathbf{Z}(\mathbb{Q})} {\mathbf{G}}^{\text{ad}}(\mathbb{Q})^+, \end{align*} $$

where ${\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ acts by conjugation on ${\mathbf {G}}(\mathbb {Q})_+^- / \mathbf {Z}(\mathbb {Q})^-$ and ${\mathbf {G}}(\mathbb {Q})_+ / \mathbf {Z}(\mathbb {Q})\to {\mathbf {G}}(\mathbb {Q})_+^- / \mathbf {Z}(\mathbb {Q})^- $ and ${\mathbf {G}}(\mathbb {Q})_+ / \mathbf {Z}(\mathbb {Q})\to {\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ are the obvious maps. Evidently $ \mathscr {A}({\mathbf {G}})^\circ $ is a subgroup of $\mathscr {A}({\mathbf {G}})$ . But, the group $\mathscr {A}({\mathbf {G}})^\circ $ depends only on ${\mathbf {G}}^{\mathrm {der}}$ and not on ${\mathbf {G}}$ , since by [Reference DeligneDel79, 2.1.15] it is given by the completion of ${\mathbf {G}}^{\text {ad}}(\mathbb {Q})^+$ with respect to the topology whose open sets are images of congruence subgroups in ${\mathbf {G}}^{\mathrm {der}}(\mathbb {Q})$ .

It is clear that both the associations $\mathbf {G}\mapsto \mathscr {A}(\mathbf {G})$ and $\mathbf {G}\mapsto \mathscr {A}(\mathbf {G})^\circ $ are functorial in maps of reductive groups $f\colon \mathbf {G}\to \mathbf {H}$ such that $f(Z(\mathbf {G}))\subseteq Z(\mathbf {H})$ .

The groups $\mathscr {A}(\boldsymbol{\mathcal{G}})$ and $\mathscr {A}(\boldsymbol{\mathcal{G}})^\circ $

We define also analogs of $\mathscr {A}({\mathbf {G}})$ and $\mathscr {A}({\mathbf {G}})^\circ $ for $\mathbb {Z}_{(p)}$ -valued points, following [Reference Kisin and PappasKP18, §4.5.6] (see also [Reference Kisin and ZhouKZ21, §4.2.8]).

To begin, we record the following well-known Beauville–Lazslo-type lemma (which is a basic case of [SP17, Tag 0F9Q]). For a ring A, let us denote by the category of finite type affine group A-schemes, and by $\mathbf {AlgGrp}^{\textrm {fl}}_A$ and $\mathbf {AlgGrp}^{\textrm {sm}}_A$ the full subcategory of A-flat and A-smooth objects, respectively.

Proposition 3.11. Let R be a Noetherian ring and f a non-zerodivisor of R. Denote the f-adic completion of R by $\widehat {R}$ . Then the functor

is an equivalence, where the target is the $2$ -fiber product (with the projection maps the obvious base change), $\theta _{\textrm {nat}}$ is the natural isomorphism, and .

Suppose now that that $\mathbf {G}$ is a reductive group over $\mathbb {Q}$ and $\mathcal{G}$ is a parahoric model of $\mathbf {G}_{\mathbb {Q}_p}$ . Let $\mathcal{G}^{\text {ad}}$ denote the parahoric $\mathbb {Z}_p$ -model of $G^{\text {ad}}$ as discussed in §2.1. By Proposition 3.11 there exists a unique morphism of smooth group $\mathbb {Z}_{(p)}$ -schemes $\boldsymbol{\mathcal{G}}\to \boldsymbol{\mathcal{G}}^{\textrm {ad}}$ modeling $\mathbf {G}\to \mathbf {G}^{\textrm {ad}}$ and $\mathcal{G}\to \mathcal{G}^{\textrm {ad}}$ . Let $\mathbf {Z}_{\mathbf {G}}$ denote the center of ${\mathbf {G}}$ , and denote by $\boldsymbol{\mathcal{Z}}_{{\mathbf {G}}}$ the closure of $\mathbf {Z}_{\mathbf {G}}$ in $\boldsymbol{\mathcal{G}}$ .

Let $\boldsymbol{\mathcal{Z}}_{{\mathbf {G}}}(\mathbb {Z}_{(p)})^-$ and $\boldsymbol{\mathcal{G}}(\mathbb {Z}_{(p)})_+^-$ denote the closures of $\boldsymbol{\mathcal{Z}}_{{\mathbf {G}}}(\mathbb {Z}_{(p)})$ and $\boldsymbol{\mathcal{G}}(\mathbb {Z}_{(p)})_+$ in ${\mathbf {G}}(\mathbb {A}_f^p)$ , respectively. Following [Reference Kisin and PappasKP18, §4.5.6] and [Reference Kisin and ZhouKZ21, §4.2.8], we set

and

where in the first definition we have that $\boldsymbol{\mathcal{G}}^{\text {ad}}(\mathbb {Z}_{(p)})^+$ acts on ${\mathbf {G}}(\mathbb {A}_f^p)/\boldsymbol{\mathcal{Z}}_{\mathbf {G}}(\mathbb {Z}_{(p)})^-$ by conjugation and

$$ \begin{align*} \boldsymbol{\mathcal{G}}(\mathbb{Z}_{(p)})_+ / \boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)})\to {\mathbf{G}}(\mathbb{A}_f^p)/\boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)})^-\quad\text{and}\quad \boldsymbol{\mathcal{G}}(\mathbb{Z}_{(p)})_+ / \boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)})\to \boldsymbol{\mathcal{G}}^{\text{ad}}(\mathbb{Z}_{(p)})^+, \end{align*} $$

are the obvious maps, and similarly for the second definition.

By [Reference Kisin and PappasKP18, Lemma 4.6.4 (2)], $\mathscr {A}(\boldsymbol{\mathcal{G}})^\circ $ is a subgroup of $\mathscr {A}(\boldsymbol{\mathcal{G}})$ but depends only on $\boldsymbol{\mathcal{G}}^{\mathrm {der}}$ , where $\mathcal{G}^{\mathrm {der}}$ denotes the parahoric model of $\mathbf {G}_{\mathbb {Q}_p}^{\mathrm {der}}$ obtained as in §2.1 and $\boldsymbol{\mathcal{G}}^{\mathrm {der}}$ is obtained from Proposition 3.11.

Lemma 3.12. Let ${\mathbf {G}}$ and $\mathbf {H}$ be reductive groups over $\mathbb {Q}$ and set $G=\mathbf {G}_{\mathbb {Q}_p}$ and $H=\mathbf {H}_{\mathbb {Q}_p}$ . Fix parahoric models $\mathcal{G}$ and $\mathcal{H}$ of G and H corresponding to points x and y in the buildings $\mathscr {B}(G,\mathbb {Q}_p)$ and $\mathscr {B}(H,\mathbb {Q}_p)$ , respectively. Suppose $f: {\mathbf {G}} \to \mathbf {H}$ is a homomorphism of reductive $\mathbb {Q}$ -groups carrying $\mathbf {Z}_{\mathbf {G}}$ into $\mathbf {Z}_{\mathbf {H}}$ , and suppose that

$$ \begin{align*} f_\ast: \mathscr{B}(G,\mathbb{Q}_p^{\textrm{ur}}) \to \mathscr{B}(H, \mathbb{Q}_p^{\textrm{ur}}) \end{align*} $$

is a map of buildings which is equivariant for the map $f: G(\mathbb {Q}_p^{\textrm {ur}}) \to H(\mathbb {Q}_p^{\textrm {ur}})$ and which sends x to y. Then f induces group homomorphisms

$$ \begin{align*} \mathscr{A}(f)\colon \mathscr{A}(\boldsymbol{\mathcal{G}}) \to \mathscr{A}(\boldsymbol{\mathcal{H}}) \quad \text{and}\quad \mathscr{A}(f)^\circ\colon \mathscr{A}(\boldsymbol{\mathcal{G}})^\circ \to \mathscr{A}(\boldsymbol{\mathcal{H}})^\circ. \end{align*} $$

Proof. Since f carries $\mathbf {Z}_{\mathbf {G}}$ into $\mathbf {Z}_{\mathbf {H}}$ , it follows that f induces a homomorphism of groups

$$ \begin{align*} f\colon {\mathbf{G}}(\mathbb{A}_f^p)/ \boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)})^- \to \mathbf{H}(\mathbb{A}_f^p)/ \boldsymbol{\mathcal{Z}}_{\mathbf{H}}(\mathbb{Z}_{(p)})^-. \end{align*} $$

The equivariance of $f_\ast $ implies that $f(\operatorname {Stab}_{G(\mathbb {Q}_p^{\textrm {ur}})}(x)) \subseteq \operatorname {Stab}_{H(\mathbb {Q}_p^{\textrm {ur}})}(y)$ , and hence that ${f(\mathcal{G}(\mathbb {Z}_p^{\textrm {ur}}))\subseteq \mathcal{H}(\mathbb {Z}_p^{\textrm {ur}})}$ . Thus f extends to a morphism $f: \mathcal{G} \to \mathcal{H}$ by [Reference Kaletha and PrasadKP23, Corollary 2.10.10]. We similarly obtain an extension $f^{\text {ad}}: \mathcal{G}^{\text {ad}} \to \mathcal{H}^{\text {ad}}$ of $f^{\text {ad}}:G^{\text {ad}} \to H^{\text {ad}}$ , which induces

$$ \begin{align*} f^{\text{ad}}: \boldsymbol{\mathcal{G}}^{\text{ad}}(\mathbb{Z}_{(p)})^+ \to \boldsymbol{\mathcal{H}}^{\text{ad}}(\mathbb{Z}_{(p)})^+. \end{align*} $$

Moreover, as f extends to a morphism $\mathcal{G} \to \mathcal{H}$ , it follows that f restricts to homomorphisms

$$ \begin{align*} f' \colon \boldsymbol{\mathcal{G}}(\mathbb{Z}_{(p)})_+/\boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)}) \to \boldsymbol{\mathcal{H}}(\mathbb{Z}_{(p)})_+/\boldsymbol{\mathcal{Z}}_{\mathbf{H}}(\mathbb{Z}_{(p)}), \end{align*} $$

and

$$ \begin{align*} f^\circ \colon \boldsymbol{\mathcal{G}}(\mathbb{Z}_{(p)})_+^-/\boldsymbol{\mathcal{Z}}_{\mathbf{G}}(\mathbb{Z}_{(p)})^- \to \boldsymbol{\mathcal{H}}(\mathbb{Z}_{(p)})_+^-/\boldsymbol{\mathcal{Z}}_{\mathbf{H}}(\mathbb{Z}_{(p)})^-. \end{align*} $$

By Lemma 3.10, it remains only to check that $(f, f', f^{\text {ad}})$ and $(f^\circ , f', f^{\text {ad}})$ induce morphisms of tuples. By uniqueness of the extension of $G \to H \to H^{\text {ad}}$ to parahoric models and equivariance of the bijections and from Lemma 2.5, we see that the diagram

commutes. Thus it remains only to show that f is equivariant for the action of $\boldsymbol{\mathcal{G}}^{\text {ad}}(\mathbb {Z}_{(p)})^+$ . But this is clear since the action of $\boldsymbol{\mathcal{G}}^{\text {ad}}(\mathbb {Z}_{(p)})^+ \subseteq \mathbf {G}(\mathbb {Q})$ on $\mathbf {G}(\mathbb {A}_f^p)/\boldsymbol{\mathcal{Z}}_{\mathbf {G}}(\mathbb {Z}_{(p)})^-$ is via conjugation, and $f\colon {\mathbf {G}}(\mathbb {A}_f^p)/ \boldsymbol{\mathcal{Z}}_{\mathbf {G}}(\mathbb {Z}_{(p)})^- \to \mathbf {H}(\mathbb {A}_f^p)/ \boldsymbol{\mathcal{Z}}_{\mathbf {H}}(\mathbb {Z}_{(p)})^-$ is a group homomorphism.

The group $\mathscr {E}(\boldsymbol{\mathcal{G}})$

Suppose now that $(\mathbf {G},\mathbf {X}, \mathcal{G})$ is a parahoric Shimura datum with reflex field $\mathbf {E}$ . Let $\mathsf {K}_p = \mathcal{G}(\mathbb {Z}_p)$ . Fix a connected component $\mathbf {X}^+$ of $\mathbf {X}$ , which determines connected Shimura varieties $\textrm {Sh}(\mathbf {G},\mathbf {X})^+ \subseteq \textrm {Sh}(\mathbf {G},\mathbf {X})$ and $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})^+ \subseteq \textrm {Sh}_{\mathsf { K}_p}(\mathbf {G},\mathbf {X})$ .

By Lemma 3.2, the action of $\text {Gal}(\overline {{\mathbf {E}}}/ {\mathbf {E}})$ on $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})^+$ factors through $\text {Gal}({\mathbf {E}}^p/ {\mathbf {E}})$ . By abuse of notation, we will use $\textrm {Sh}_{\mathsf { K}_p}(\mathbf {G},\mathbf {X})^+$ to refer to the ${\mathbf {E}}^p$ -scheme obtained by descent.

Let $\mathscr {E}(\boldsymbol{\mathcal{G}}) \subseteq \mathscr {A}(\boldsymbol{\mathcal{G}}) \times \text {Gal}({\mathbf {E}}^p / {\mathbf {E}})$ denote the stabilizer of $\textrm {Sh}_{\mathsf { K}_p}(\mathbf {G},\mathbf {X})^+$ in $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})$ . By [Reference Kisin and PappasKP18, Lemma 4.6.6], $\mathscr {E}(\boldsymbol{\mathcal{G}})$ is an extension of $\text {Gal}({\mathbf {E}}^p/{\mathbf {E}})$ by $\mathscr {A}(\boldsymbol{\mathcal{G}})^\circ $ , and there is a canonical isomorphism

where an element of $\mathscr {E}(\boldsymbol{\mathcal{G}})$ acts on $\mathscr {A}(\boldsymbol{\mathcal{G}})$ via conjugation by its image in $\mathscr {A}(\boldsymbol{\mathcal{G}})$ .Footnote ⁶

The $\mathscr {E}$ -group satisfies a functoriality akin to that of the $\mathscr {A}$ -groups.

Lemma 3.13. Let $f: (\mathbf {G}_1, \mathbf {X}_1,\mathcal{G}_1) \to (\mathbf {G}, \mathbf {X}, \mathcal{G})$ be a morphism of parahoric Shimura data. Suppose $\mathcal{G}_1$ and $\mathcal{G}$ correspond to points $x_1$ and x in the buildings $\mathscr {B}(G_1, \mathbb {Q}_p)$ and $\mathscr {B}(G,\mathbb {Q}_p)$ , respectively. Suppose the induced $f: \mathbf {G}_1 \to \mathbf {G}$ carries $\mathbf {Z}_{\mathbf {G}}$ into $\mathbf {Z}_{\mathbf {H}}$ , and suppose that

$$ \begin{align*} f_\ast\colon \mathscr{B}(G_1, \mathbb{Q}_p) \to \mathscr{B}(G,\mathbb{Q}_p) \end{align*} $$

is a map of buildings which is equivariant for the map $f\colon G_1(\mathbb {Q}_p^{\textrm {ur}}) \to G(\mathbb {Q}_p^{\textrm {ur}})$ and which sends $x_1$ to x. Then f induces a group homomorphism

$$ \begin{align*} \mathscr{E}(f): \mathscr{E}(\boldsymbol{\mathcal{G}}_1) \to \mathscr{E}(\boldsymbol{\mathcal{G}}). \end{align*} $$

Suppose now we have two parahoric Shimura data $(\mathbf {G}_1, \mathbf {X}_1, \mathcal{G}_1)$ and $(\mathbf {G}, \mathbf {X}, \mathcal{G})$ , and that there is a central isogeny $\alpha \colon \mathbf {G}_1^{\mathrm {der}} \to \mathbf {G}^{\mathrm {der}}$ which induces an isomorphism of Shimura data

Let $x_1$ in $\mathscr {B}(G_1, \mathbb {Q}_p)$ and x in $\mathscr {B}(G,\mathbb {Q}_p)$ denote points in the buildings of $G_1$ and G, respectively, which correspond to $\mathcal{G}_1$ and $\mathcal{G}$ . Let us further assume that $x_1$ and x correspond to the same point $x_1^{\text {ad}} = x^{\text {ad}}$ in the building $\mathscr {B}(G_1^{\text {ad}}, \mathbb {Q}_p) \simeq \mathscr {B}(G^{\text {ad}}, \mathbb {Q}_p)$ .

Fix a connected component $\mathbf {X}_1^+$ as above, which determines a group $\mathscr {E}(\boldsymbol{\mathcal{G}}_1)$ . By the real approximation theorem, we may assume that the image of $\mathbf {X} \subseteq \mathbf {X}^{\text {ad}}$ contains $\mathbf {X}_1^+$ by replacing $\mathbf {X}$ by its conjugate by some element of $\mathbf {G}^{\text {ad}}(\mathbb {Q})$ . We set $\mathbf {E}' = \mathbf {E}_1\mathbf {E}$ , and define $\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ to be the fiber product

(3.3)

Then $\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ is an extension of $\text {Gal}(\mathbf {E}^{\prime p}/\mathbf {E}')$ by $\mathscr {A}(\boldsymbol{\mathcal{G}}_1)^\circ $ , and by [Reference Kisin and PappasKP18, Lemma 4.6.9], there is a natural map of extensions

(3.4) $$ \begin{align} \mathscr{E}_{\mathbf{E}'}(\boldsymbol{\mathcal{G}}_1) \to \mathscr{E}(\boldsymbol{\mathcal{G}}). \end{align} $$

In particular, it follows that there is an isomorphism

(3.5)

3.3 Construction of Kisin–Pappas–Zhou models

We now briefly recall the construction of the Kisin–Pappas–Zhou models $\mathscr {S}_{\mathsf {K}}^{\mathfrak {d}}(\mathbf {G},\mathbf {X})$ from Theorem 3.9, focusing on the aspects most relevant for our purposes. We encourage the reader to consult the relevant parts of [Reference Kisin and PappasKP18], [Reference Kisin and ZhouKZ21], and [Reference Kisin, Pappas and ZhouKPZ24] for details.

Throughout this section we fix the following data/notation:

We refer to §2.1 and §3.2 for the meaning of these terms, as well as other notation related to Bruhat–Tits theory and Shimura varieties.

The construction of Kisin–Pappas–Zhou models is carried out in a four step process.

Step 1: Global construction for Siegel type

Suppose that $(\textrm {GSp}(\mathbf {V}),\mathfrak {h}^\pm )$ is a Siegel datum. Choose a $\mathbb {Z}_p$ -lattice $\Lambda \subseteq \mathbf {V}_{\mathbb {Q}_p}$ and set

$$ \begin{align*} K_p=\textrm{GSp}(\Lambda)\subseteq \textrm{GSp}(\mathbf{V}\otimes_{\mathbb{Q}} \mathbb{Q}_p). \end{align*} $$

For $\mathsf {K}^p$ a neat compact open subgroups of $\textrm {GSp}(\mathbf {V}\otimes _{\mathbb {Q}}\mathbb {A}^f_p)$ , we set ${\pmb {\mathscr {S}}}_{\mathsf {K}_p\mathsf { K}^p}(\textrm {GSp}(\mathbf {V}),\mathfrak {h}^\pm )$ to be the $\mathbb {Z}_{(p)}$ -scheme given by Mumford’s moduli of principally polarized abelian varieties with $\mathsf {K}^p$ -level structure (see [Reference DeligneDel79, §4]). These form a $\textrm {GSp}(\mathbf {V}\otimes _{\mathbb {Q}} \mathbb {A}_f^p)$ -system of $\mathbb {Z}_{(p)}$ -schemes.

Step 2: Construction for Hodge type data

Let us now assume that $(\mathbf {G},\mathbf {X},\mathcal{G})$ is a parahoric Hodge type datum and let us fix a parahoric Hodge embedding

$$ \begin{align*} \iota\colon (\mathbf{G},\mathbf{X},\mathcal{G})\to (\textrm{GSp}(\mathbf{V}),\mathfrak{h}^\pm,\textrm{GSp}(\Lambda)), \end{align*} $$

which respects stabilizers. As in [Reference KisinKis10, Lemma 2.10] we may choose a neat compact open subgroup $\mathsf {K}_1^p\subseteq \mathbf {GSp}(\mathbf {V}\otimes _{\mathbb {Q}}\mathbb {A}_f^p)$ such that $\iota (\widetilde {\mathsf {K}}_p\mathsf {K}^p)\subseteq \textrm {GSp}(\Lambda )\mathsf {K}^p_1$ , and such that the map

(3.6) $$ \begin{align} \textrm{Sh}_{\widetilde{\mathsf{K}}_p\mathsf{K}^p}(\mathbf{G},\mathbf{X})\to \textrm{Sh}_{\textrm{GSp}(\Lambda)\mathsf{K}_1^p}(\textrm{GSp}(\mathbf{V}),\mathfrak{h}^\pm)_{\mathbf{E}}, \end{align} $$

is a closed embedding.

We then set

• ⋄ to be the normalization of the Zariski closure of $\textrm {Sh}_{\widetilde {\mathsf {K}}_p\mathsf {K}^p}(\mathbf {G},\mathbf {X})$ in the scheme ${\pmb {\mathscr {S}}}_{\textrm {GSp}(\Lambda )\mathsf {K}_1^p}(\textrm {GSp}(\mathbf {V}),\mathfrak {h}^\pm )_{\mathcal{O}_{(v)}}$ , via the embedding in (3.6),

• ⋄ to be the normalization (in the sense of [SP17, Tag 0BAK]) of relative to finite map

  $$ \begin{align*} \textrm{Sh}_{\mathsf{K}_p\mathsf{K}^p}(\mathbf{G},\mathbf{X})\to \textrm{Sh}_{\widetilde{\mathsf{K}}_p\mathsf{K}^p}(\mathbf{G},\mathbf{X}). \end{align*} $$

As Zariski closure and normalization are functorial constructions, we see that form a projective system. We then further set

The continuous $\mathbf {G}(\mathbb {A}_f^p)$ -action of $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})_E$ extends to a continuous action on with the property that

compatibly in $\mathsf {K}^p$ . We finally set

which is a scheme over $\mathcal{O}_v$ with $\mathbf {G}(\mathbb {A}_f^p)$ -action and a scheme over $\mathcal{O}_v$ , respectively.

A priori, these schemes depend on the choice of parahoric Hodge embedding which respects stabilizers. That said, the following shows that this not the case, thus justifying the omission of the parahoric Hodge embedding from the notation.

While the action of $\mathbf {G}(\mathbb {A}_f^p)$ extends to actions on and , it is not a priori clear that these can be extended further to actions of $\mathscr {A}(\boldsymbol{\mathcal{G}})$ . That said, this does hold true when $Z(G_1)$ is R-smooth, as explained in [Reference Kisin, Pappas and ZhouKPZ24, Proof of Proposition 7.1.14].Footnote ⁷

Step 3: Global construction relative to a well-adapted Hodge type datum

Continue to assume $p>2$ , and suppose that $(\mathbf {G},\mathbf {X},\mathcal{G})$ is a parahoric Shimura datum of abelian type. The key proposition which allows us to bootstrap from the previous cases is the following.

Throughout the remainder of this section, we fix $(\mathbf {G}_1, \mathbf {X}_1, \mathcal{G}_1)$ as in the proposition above. Choose connected components $\mathbf {X}^+ \subseteq \mathbf {X}$ and $\mathbf {X}_1^+\subseteq \mathbf {X}_1$ which correspond in $\mathbf {X}^{\textrm {ad}}\simeq \mathbf {X}_1^{\textrm {ad}}$ . Using Lemma 3.2, we obtain connected components $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})^+$ and $\textrm {Sh}_{\mathsf {K}_{p,1}}(\mathbf {G}_1,\mathbf {X}_1)^+$ defined over $\mathbf {E}^p$ and $\mathbf {E}_1^p$ , respectively. We also obtain subgroups

$$ \begin{align*} \mathscr{E}(\boldsymbol{\mathcal{G}})\subseteq \mathscr{A}(\boldsymbol{\mathcal{G}})\times\text{Gal}(\mathbf{E}^p/\mathbf{E}),\qquad \mathscr{E}(\boldsymbol{\mathcal{G}}_1)\subseteq \mathscr{A}(\boldsymbol{\mathcal{G}}_1)\times\text{Gal}(\mathbf{E}_1^p/\mathbf{E}), \end{align*} $$

and the group

$$ \begin{align*} \mathscr{E}_{\mathbf{E}'}(\boldsymbol{\mathcal{G}}_1)=\mathscr{E}(\boldsymbol{\mathcal{G}}_1)\times_{\text{Gal}(\mathbf{E}_1^p/\boldsymbol{E})}\text{Gal}(\mathbf{E}^{\prime p}/\boldsymbol{E}), \end{align*} $$

as defined in §3.2.

We have a bijection

(3.7)

as follows from combining the following general results.

The bijection (3.7) shows that $\textrm {Sh}_{\mathsf {K}_{p,1}}(\mathbf {G}_1,\mathbf {X}_1)^+$ corresponds uniquely to a connected component of . Set $\mathcal{O}^{\prime p}_{(v)}=\mathcal{O}_{\mathbf {E}^{\prime p} } \otimes _{\mathcal{O}_{\mathbf {E}}}\mathcal{O}_{(v)}$ . Consider then

(3.8)

As observed in Step 2, we have that the right $\mathscr {A}(\boldsymbol{\mathcal{G}}_1)$ -action extends to and by the bijection (3.7) we deduce that $\mathscr {E}(\boldsymbol{\mathcal{G}}_1)$ stabilizes . We then get an induced action of $\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ on . We then finally let $\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ act on the right of (3.8) by setting

(3.9) $$ \begin{align} (s,a)\cdot e=(se,\overline{e}^{-1}a\overline{e}), \end{align} $$

where $\overline {e}$ denotes the image of e under the homomorphisms

(3.10) $$ \begin{align} \mathscr{E}_{\mathbf{E}'}(\boldsymbol{\mathcal{G}}_1)\to \mathscr{E}(\boldsymbol{\mathcal{G}})\to\mathscr{A}(\boldsymbol{\mathcal{G}}), \end{align} $$

where the first map is as in (3.4), and the second map is projection.

We then consider $\mathscr {A}(\boldsymbol{\mathcal{G}})\rtimes \mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ , where $\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ acts again via the homomorphism from (3.10). Let the semi-direct product $\mathscr {A}(\boldsymbol{\mathcal{G}})\rtimes \mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ act on (3.8) by having $\mathscr {A}(\boldsymbol{\mathcal{G}})$ act by right multiplication on the $\mathscr {A}(\boldsymbol{\mathcal{G}})$ -factor of (3.8), i.e.,

(3.11) $$ \begin{align} (s,a)\cdot a' = (s, aa'). \end{align} $$

As in (3.2) we obtain an action of $\mathscr {A}(\boldsymbol{\mathcal{G}})\ast _{\mathscr {A}(\boldsymbol{\mathcal{G}}_1)^\circ }\mathscr {E}_{\mathbf {E}'}(\boldsymbol{\mathcal{G}}_1)$ on

(3.12)

with notation as in loc. cit. Since

$$ \begin{align*} \mathscr{A}(\boldsymbol{\mathcal{G}})\ast_{\mathscr{A}(\boldsymbol{\mathcal{G}}_1)^\circ}\mathscr{E}_{\mathbf{E}'}(\boldsymbol{\mathcal{G}}_1)\simeq \mathscr{A}(\boldsymbol{\mathcal{G}})\times\text{Gal}(\mathbf{E}^{\prime p}/\mathbf{E}), \end{align*} $$

(see (3.5)), we obtain an action of $\mathscr {A}(\boldsymbol{\mathcal{G}})\times \text {Gal}(\mathbf {E}^{\prime p}/\mathbf {E})$ , and hence of $\mathbf {G}(\mathbb {A}_f^p)\times \text {Gal}(\mathbf {E}^{\prime p}/\mathbf {E})$ , on (3.12).

Now, let $J\subset G(\mathbb {Q}_p)$ denote a set of coset representatives for the image of the natural map

$$ \begin{align*} \mathscr{A}(\boldsymbol{\mathcal{G}}_1)^\circ\backslash \mathscr{A}(\boldsymbol{\mathcal{G}})\to \mathscr{A}(\mathbf{G}_1)^\circ\backslash \mathscr{A}(\mathbf{G})/K_p, \end{align*} $$

where the map $\mathscr {A}(\boldsymbol{\mathcal{G}}_1)^\circ \to \mathscr {A}(\boldsymbol{\mathcal{G}})$ is as in Lemma 3.12. Then J is finite. Let $\mathfrak {d} = (\mathbf {G}_1, \mathbf {X}_1, \mathcal{G}_1)$ , and consider

which we endow with the factor-wise (relative to the J-indexed factors) action of the group $\mathbf {G}(\mathbb {A}_f^p)\times \text {Gal}(\mathbf {E}^{\prime p}/\mathbf {E})$ . This action is continuous, and so by Galois descent gives rise to a unique scheme ${\pmb {\mathscr {S}}}_{\mathsf {K}_p}^{\mathfrak {d}}(\mathbf {G},\mathbf {X})_{\mathcal{O}_{(v)}'}$ with continuous action of $\mathbf {G}(\mathbb {A}_f^p)$ , with $\mathcal{O}_{(v)}'=\mathcal{O}_{\mathbf {E}'}\otimes _{\mathcal{O}_{\mathbf {E}}}\mathcal{O}_{(v)}$ .Footnote ⁸

Step 4: The construction in the local setting

Consider

$$ \begin{align*} ({\pmb{\mathscr{S}}}_{\mathsf{K}_p}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}^{\prime}_{(v)}})_{\mathcal{O}_E}. \end{align*} $$

That $\mathbf {E}'$ is completely split over $\mathbf {E}$ at every prime over p implies, in particular, that this decomposes as several copies of the same $\mathcal{O}_v$ -scheme with a continuous action of $\mathbf {G}(\mathbb {A}_f^p)$ , indexed by by the place of $\mathbf {E}'$ lying over v. Choose any of these copies, and call it $\mathscr {S}_{\mathsf {K}_p}^{\mathfrak {d}}(\mathbf {G},\mathbf {X})$ . Observe that we have a decomposition

(3.13)

where is any copy of pulled back to $\mathcal{O}_E^{\textrm {ur}}$ . We then set

for any neat compact open subgroup $\mathsf {K}^p$ of $\mathbf {G}(\mathbb {A}_f^p)$ . That these quotients exist and are quasi-projective can be deduced from Remark 3.14 in conjunction with [SP17, Tag 01ZY] and [SP17, Tag 07S6].

Remark 3.19. Let us briefly explain how to view (3.13) as an integral model for $\textrm {Sh}_{\mathsf {K}_p}(\mathbf {G},\mathbf {X})_{E^{\textrm {ur}}}$ . By [Reference Kisin and PappasKP18, Proposition 4.6.13], there is a canonical decomposition

(3.14) $$ \begin{align} \textrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{E^{\textrm{ur}}}= \bigsqcup_{j \in J}\left([\textrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_1,\mathbf{X}_1)_{E^{\textrm{ur}}}^+\times \mathscr{A}(\boldsymbol{\mathcal{G}})j]/\mathscr{A}(\boldsymbol{\mathcal{G}}_1)^\circ\right), \end{align} $$

where (among other definitions) this $\textrm {Sh}_{\mathsf {K}_{p,1}}(\mathbf {G}_1,\mathbf {X}_1)_{E^{\textrm {ur}}}^+$ is the generic fiber of . For clarity, denote the jth component of the right-hand side of (3.14) by $\mathscr {X}_j$ . The isomorphism between the generic fiber of the right-hand side of (3.13) and the right-hand side of (3.14) is determined by

where the first map is the tautological one, and the second is given by multiplication by j viewed as an element of $\mathscr {A}(\mathbf {G})$ .

3.4 Some properties and functoriality results

In this subsection we establish some compatibilities satisfied by the Kisin–Pappas–Zhou integral models, and we prove the minimal amount of functoriality needed to prove our main result (see Theorem 4.10). This main result will then establish functoriality in general (see Theorem 4.14).

Kisin–Pappas–Zhou integral models of toral type

To utilize the results of [Reference DanielsDan25], it is useful to explicitly describe Kisin–Pappas–Zhou integral models of tori.

Recall from Example 3.7 that a Shimura datum $(\mathbf {T},\{h\})$ is of toral type if $\mathbf {T}$ is a torus. In this case, for every compact open subgroup $\mathsf {K} = \mathsf {K}_p\mathsf { K}^p \subseteq \mathbf {T}(\mathbb {A}_f^p)$ , the Shimura variety $\textrm {Sh}_{\mathsf {K}}(\mathbf {T},\{h\})$ is zero-dimensional, i.e.,

(3.15) $$ \begin{align} \textrm{Sh}_{\mathsf{K}}(\mathbf{T}, \{h\})_E = \bigsqcup_{i \in I} \mathrm{Spec}\,(E_i) \end{align} $$

for some finite index set I. In fact, each field $E_i$ is the same field, and is an unramified extension of $\mathbb {Q}_p$ (see [Reference DanielsDan25, Lemma 4.2] and the proof of [Reference DanielsDan25, Lemma 4.8]).

Set $T=\mathbf {T}_{\mathbb {Q}_p}$ . As discussed in §2.1, there is a unique parahoric model $\mathcal{T}$ of T, given by the connected component $(\mathcal{T}^{\textrm {lft}})^\circ $ of the Néron model $\mathcal{T}^{\textrm {lft}}$ of T.

Proposition 3.20. Suppose $(\mathbf {T},\{h\}, \mathcal{T})$ is parahoric Shimura datum of toral type, and let $\mathfrak {d} = (\mathbf {G}_1, \mathbf {X}_1, \mathcal{G}_1)$ be a parahoric Shimura datum of Hodge type which is well-adapted to $(\mathbf {T}, \{h\}, \mathcal{T})$ and for which $Z(G_1)$ is R-smooth. Then,

$$ \begin{align*} \mathscr{S}_{\mathsf{K}}^{\mathfrak{d}}(\mathbf{T}, \{h\}) = \bigsqcup_{i \in I} \mathrm{Spec}\,(\mathcal{O}_{E_i}), \end{align*} $$

for any neat compact open subgroup $\mathsf {K}^p\subseteq \mathbf {T}(\mathbb {A}_f^p)$ and with $\mathsf {K} = \mathsf {K}_p\mathsf {K}^p$ . The same holds true for if $(\mathbf {T}, \{h\})$ is of Hodge type.

Proof. First, observe that as $(\mathbf {G}_1, \mathbf {X}_1)$ is adapted to the datum $(\mathbf {T}, \{h\})$ , we have an isogeny $\mathbf {G}_1^{\mathrm {der}} \to \mathbf {T}^{\mathrm {der}} = \{e\}$ , and hence $\mathbf {G}_1^{\mathrm {der}}$ is trivial, since it is connected. Thus, $\mathbf {G}_1$ is a torus.Footnote ⁹ Consider an arbitrary neat compact open subgroup $\mathsf {K}^p_1\subseteq \mathbf {G}_1(\mathbb {A}_f^p)$ .

Choose a parahoric Hodge embedding $\iota \colon (\mathbf {G}_1, \mathbf {X}_1, \mathcal{G}_1) \to (\mathbf {GSp}(\mathbf {V}),\mathfrak {h}^\pm ,\textrm {GSp}(\Lambda ))$ which respects stabilizers, and a neat compact open subgroup $\mathsf {L}_1^p \subseteq \textrm {GSp}(\mathbf {V}\otimes _{\mathbb {Q}} \mathbb {A}_f^p)$ such that $\iota (\mathsf { K}_{1,p}\mathsf {K}^p_1) \subseteq \textrm {GSp}(\Lambda )\mathsf {L}_1^p$ , inducing a closed embedding

(3.16) $$ \begin{align} \textrm{Sh}_{\mathsf{K}_{1,p}\mathsf{K}^p_1}(\mathbf{G}_1, \mathbf{X}_1) \to \textrm{Sh}_{\textrm{GSp}(\Lambda)\mathsf{ L}_1^p}(\mathbf{GSp}(\mathbf{V}),\mathfrak{h}^\pm)_{\mathbf{E}_1}. \end{align} $$

Note that ${\pmb {\mathscr {S}}}_{\textrm {GSp}(\Lambda )\mathsf { L}_1^p}(\mathbf {GSp}(\mathbf {V}),\mathfrak {h}^\pm )$ is flat, separated, and finite over $\mathbb {Z}_{(p)}$ . Thus, considering (3.15), and using notation from Step 2 of §3.3, it follows from Lemma 3.21 below that and thus are disjoint unions of semi-local localizations of $\mathcal{O}_{\mathbf {F}}$ for some finite extension $\mathbf {F}$ of $\mathbf {E}$ .

This implies that, using notation now from Step 3 of §3.3, that is either the spectrum of a field or of a semi-local Dedekind domain. Thus, from the construction in Step 4 of §3.3 we deduce that for each neat compact open $\mathsf {K}^p\subseteq \mathbf {G}(\mathbb {A}_f^p)$ one has that $\mathscr {S}_{\mathsf {K}_p\mathsf {K}^p}^{\mathfrak {d}}(\mathbf {G},\mathbf {X})$ is either the disjoint union of spectra of extensions of E, or the disjoint union of spectra of the rings of integers of finite extensions of E. That said, the former is impossible as it is then clear that such a model cannot satisfy the extension property from (2) of Theorem 3.9, and so the latter must hold. The fist claim follows. Moreover, the second claim follows from similar arguments.

Following [SP17, Tag 035E], if X is a scheme with the property that every quasi-compact open has finitely many connected components, then we denote the normalization of X by $X^\nu $ .

Given Proposition 3.20, we see that the definition of $\mathscr {S}_{\mathsf {K}}^{\mathfrak {d}} (\mathbf {T},\{h\})$ is independent of the choice of $\mathfrak {d}$ . We denote by $\mathscr {S}_{\mathsf {K}}(\mathbf {T},\{h\})$ the common object.

Functoriality for ad-isomorphisms

Suppose that $\alpha \colon (\mathbf {G}_1,\mathbf {X}_1,\mathcal{G}_1)\to (\mathbf {G},\mathbf {X},\mathcal{G})$ is an ad-isomorphism of parahoric abelian-type Shimura data. Take a parahoric Shimura datum $\mathfrak {d}=(\mathbf {G}_2,\mathbf {X}_2,\mathcal{G}_2)$ of Hodge type well-adapted to $(\mathbf {G}_1,\mathbf {X}_1,\mathcal{G}_1)$ for which $Z(G_2)$ is R-smooth. Since $\alpha $ is an ad-isomorphism, one sees that $\mathfrak {d}$ is also well-adapted to $(\mathbf {G},\mathbf {X},\mathcal{G})$ in the obvious way.

Proposition 3.22. There exists a unique quotient-finite étale morphism

$$ \begin{align*} \alpha\colon \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_{p,1}}(\mathbf{G}_1,\mathbf{X}_1)\to \mathscr{S}_{\mathsf{K}_p}^{\mathfrak{d}}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_1}}, \end{align*} $$

which is equivariant for the map $\alpha \colon \mathbf {G}_1(\mathbb {A}_f^p)\to \mathbf {G}(\mathbb {A}_f^p)$ and models

$$ \begin{align*} \alpha\colon \textrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_1,\mathbf{X}_1)\to \textrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{E_1}. \end{align*} $$

Here by quotient-finite étale we mean that if $\mathsf {K}_1^p\subseteq \mathbf {G}_1(\mathbb {A}_f^p)$ and $\mathsf {K}^p\subseteq \mathbf {G}(\mathbb {A}_f^p)$ are neat compact open subgroups such that $\alpha (\mathsf {K}_1^p)\subseteq \mathsf {K}^p$ , then the induced map

$$ \begin{align*} \alpha\colon \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}^p_1}(\mathbf{G}_1,\mathbf{X}_1)\to \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}^p}(\mathbf{G},\mathbf{X})_{\mathcal{O}_{E_1}}, \end{align*} $$

is a finite étale morphism.

Proof of Proposition 3.22.

The fact that such a map is unique is clear by the separatedness of $\alpha $ and the flatness of $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}^p_1}(\mathbf {G}_1,\mathbf {X}_1)$ , which guarantees that its generic fiber is Zariski dense (e.g., see [Reference Görtz and WedhornGW20, Proposition 14.14]).

To see the existence, we note that by definition we have

and

Now, by Lemma 3.12 and Lemma 3.13 we get morphisms

$$ \begin{align*} \alpha\colon \mathscr{A}(\boldsymbol{\mathcal{G}}_1)\to \mathscr{A}(\boldsymbol{\mathcal{G}}),\quad \alpha\colon \mathscr{A}(\boldsymbol{\mathcal{G}}_1)^\circ\to \mathscr{A}(\boldsymbol{\mathcal{G}}),\ \text{ and } \ \alpha\colon \mathscr{E}(\boldsymbol{\mathcal{G}}_1)\to \mathscr{E}(\boldsymbol{\mathcal{G}}). \end{align*} $$

Because $\alpha (\mathsf {K}_{p,1})\subseteq \mathsf {K}_p$ , we have a map of sets $\alpha \colon J_1\to J$ . Since the isomorphism from (3.5) is natural in these operations, we deduce the existence of a map

equivariant for $\alpha \colon \mathbf {G}_1(\mathbb {A}_f^p)\to \mathbf {G}(\mathbb {A}_f^p)$ . That the generic fiber of this map agrees with

$$ \begin{align*} \alpha\colon \textrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_1,\mathbf{X}_1)_{E_1}\to \textrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{E_1}. \end{align*} $$

follows as in [Reference Kisin and PappasKP18, Lemma 4.6.13].

Finally, we show that $\alpha $ is quotient-finite étale. Let $\mathscr {S}_1^+$ denote a connected component of $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}_{p,1}}(\mathbf {G}_1, \mathbf {X}_1)$ mapping to a connected component $\mathscr {S}^+$ of $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}_p}(\mathbf {G}, \mathbf {X})_{\mathcal{O}_{E_1}}$ . By construction, we have

where

$$ \begin{align*} \Delta_1 = \ker(\mathscr{A}(\boldsymbol{\mathcal{G}}_2)^\circ \to \mathscr{A}({\boldsymbol{\mathcal{G}}})) \quad\text{and}\quad\Delta = \ker(\mathscr{A}(\boldsymbol{\mathcal{G}}_2)^\circ \to \mathscr{A}(\boldsymbol{\mathcal{G}})), \end{align*} $$

(see also the arguments of [Reference Kisin and PappasKP18, Corollary 4.6.15]). Thus $\mathscr {S}^+$ is the quotient of $\mathscr {S}_1^+$ by the finite group $\Delta / \Delta _1$ , and the result follows.

In particular, the proposition implies functoriality for morphisms $(\mathbf {T}_1, \{h_1\}, \mathcal{T}_1) \to (\mathbf {T}, \{h\}, \mathcal{T})$ of parahoric Shimura data of toral type.

Functoriality for abelianizations

Suppose now that $(\mathbf {G},\mathbf {X},\mathcal{G})$ is a parahoric Shimura datum of abelian type. Consider the parahoric Shimura datum $(\mathbf {G}^{\text {ab}}, \mathbf {X}^{\text {ab}}, \mathcal{G}^{\text {ab}})$ , where $\mathbf {X}^{\text {ab}}$ is the result of post-composing any element h of $\mathbf {X}$ with the canonical map $\delta : \mathbf {G} \to \mathbf {G}^{\text {ab}}$ . We have the obvious associated morphism of parahoric Shimura data

$$ \begin{align*} \alpha\colon (\mathbf{G}, \mathbf{X}, \mathcal{G}) \to (\mathbf{G}^{\text{ab}}, \mathbf{X}^{\text{ab}}, \mathcal{G}^{\text{ab}}). \end{align*} $$

Write $\mathsf {K}_p^{\text {ab}} = \mathcal{G}^{\text {ab}}(\mathbb {Z}_p)$ .

Proposition 3.23. There exists a unique

$$ \begin{align*} \alpha\colon \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})\to \mathscr{S}_{\mathsf{K}_p^{\mathrm{ab}}}(\mathbf{G}^{\mathrm{ab}},\mathbf{X}^{\mathrm{ab}})_{\mathcal{O}_{E}}, \end{align*} $$

which is equivariant for the map $\alpha \colon \mathbf {G}_1(\mathbb {A}_f^p)\to \mathbf {G}(\mathbb {A}_f^p)$ and models

$$ \begin{align*} \alpha\colon \textrm{Sh}_{\mathsf{K}_{p,1}}(\mathbf{G}_1,\mathbf{X}_1)_{E_1}\to \textrm{Sh}_{\mathsf{K}_p}(\mathbf{G},\mathbf{X})_{E_1}. \end{align*} $$

Lemma 3.24. Suppose that S is a scheme and U is an open subscheme of S which is schematically dense and for which the inclusion $U \hookrightarrow S$ is quasi-compact. Given a diagram

with X a normal scheme, $S'$ a scheme, f quasi-compact, separated, and flat, and g quasi-compact, flat, and integral, there exists a unique arrow dashed arrow as in the diagram above which makes the top square Cartesian.

Proof. By [SP17, Tag 081H], $X_U \subseteq X$ is schematically dense, so the uniqueness follows from [SP17, Tag 01RH]. In turn, by uniqueness, we may localize on S and X to prove existence. Using that g is affine, we may then assume without loss of generality that $S = \mathrm{Spec}\,(A)$ , $S' = \mathrm{Spec}\,(A')$ and $A \to A'$ is integral. Moreover, localizing on X we may further assume $X = \mathrm{Spec}\,(B)$ with B a normal domain. We then have the following diagram of rings

The center and right-hand vertical arrows are injections by flatness of f and g along with [SP17, Tag 081H] and [SP17, Tag 01RE].

We claim that $\alpha (A') \subseteq B$ . Indeed, let $a' \in A'$ . Then by integrality of $A \to A'$ , there exists a monic polynomial $p(T)$ in $A[T]$ such that $p(a') = 0$ . But then $\alpha (a')$ in $\mathcal{O}(X_U)$ satisfies the monic polynomial $f(p)(T)$ in $B[T]$ . Thus $\alpha (a')$ must lie in B by normality of B.

It follows that we have a map $\mathrm{Spec}\,(\alpha ): \mathrm{Spec}\,(B) \to \mathrm{Spec}\, (A')$ . We claim that the diagram

is Cartesian. For this it is enough to show that $\mathrm{Spec}\,(\alpha ) \big |_{X_U} = \alpha $ . Since $U \hookrightarrow X$ is quasi-compact, U is quasi-compact, and therefore the same is true of $S_U'$ and $X_U$ by quasi-compactness of f and g. Since $\mathrm{Spec}\,(\alpha )$ and $\alpha $ induce the same map $\mathcal{O}(S^{\prime }_U) \to \mathcal{O}(X_U)$ , the result now follows from [SP17, Tag 01P9].

Two constructions for Hodge-type data

We record here the following basic compatibility between the constructions made in §3.3. Let $\mathfrak {d}=(\mathbf {G},\mathbf {X},\mathcal{G})$ be a parahoric Shimura datum of Hodge type and assume $Z(G)$ is an R-smooth torus.

Proposition 3.25. There is a canonical $\mathbf {G}(\mathbb {A}^p_f)$ -equivariant identification

extending the identity in the generic fiber.

Proof. Choose a $\mathsf {K}^p \subset G(\mathbb {A}_f^p)$ and let $\mathsf {K} = \mathsf {K}_p\mathsf {K}^p$ . Implementing Zarhin’s trick, we may choose a parahoric Hodge embedding $\iota \colon (\mathbf {G}, \mathbf {X}, \mathcal{G}) \to (\textrm {GSp}(\mathbf {V}), \mathfrak {h}^\pm , \textrm {GSp}(\Lambda ))$ such that $\textrm {GSp}(\Lambda )$ is reductive. Moreover, as in §3.3, we may choose a level $\mathsf {L}_p\mathsf {L}^p \subset $ such that $\iota $ maps $\mathsf {K}$ into $\mathsf {L}$ , such that there is a morphism

(3.17)

which is the normalization of a closed embedding.

Next observe that the statement of the proposition holds for $(\textrm {GSp}(\mathbf {V}), \mathfrak {h}^\pm , \textrm {GSp}(\Lambda ))$ by [Reference KisinKis10, Theorem 3.4.10] (which establishes that the source and target are both integral canonical models in the sense of op. cit.). Moreover, for any connected component , there is a tautological map from to some connected component $\mathscr {S}_{\mathsf {L}}(\mathbf {GSp}(\mathbf {V}),\mathfrak {h}^\pm )^+$ of $\mathscr {S}_{\mathsf {L}}(\mathbf {GSp}(\mathbf {V}),\mathfrak {h}^\pm )$ . Applying the functorialities of the $\mathscr {A}$ -groups as established in §3.2, and the Siegel case of the proposition, we obtain a map

(3.18) $$ \begin{align} \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}}(\mathbf{G},\mathbf{X}) \to \mathscr{S}_{\mathsf{L}}(\mathbf{GSp}(\mathbf{V}),\mathfrak{h}^\pm). \end{align} $$

As $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ is dense in the integral model $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ , the map (3.18) carries $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ into the closure of $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ inside of $\mathscr {S}_{\mathsf {L}}(\mathbf {GSp}(\mathbf {V}),\mathfrak {h}^\pm )$ . Moreover, since $\mathscr {S}^{\mathfrak {d}}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ is normal, the map (3.18) factors through (3.17). Since $\mathcal{G} \to \textrm {GSp}(\Lambda )$ is a closed embedding, one deduces that (3.17) is quasi-finite, and thus so is the map . As both source and target are normal, and this map induces an isomorphism on the dense open $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})$ , we deduce it is an isomorphism by Zariski’s Main Theorem.

This procedure is clearly compatible with varying $\mathsf {K}^p$ , so we obtain the desired result by passing to the limit.

Kisin–Pappas–Zhou integral models for some products

Let $\mathfrak {d} = (\mathbf {G}, \mathbf {X}, \mathcal{G})$ be a parahoric Shimura datum of Hodge type such that $Z(G)$ is an R-smooth torus, and that $(\mathbf {T}, \{h\}, \mathcal{T})$ is a parahoric Shimura datum of toral type. Write $\mathbf {E}'$ for the compositum of the reflex fields of the two Shimura data, which is the reflex field of $(\mathbf {G}\times \mathbf {T}, \mathbf {X}\times \{h\})$ , and set $\mathsf {M}_p = \mathcal{T}(\mathbb {Z}_p)$ .

Proposition 3.26. There is a canonical $\mathbf {G}(\mathbb {A}^p_f)\times \mathbf {T}(\mathbb {A}_f^p)$ -equivariant identification

Proof. The projection map $\mathbf {G}\times \mathbf {T} \to \mathbf {G}$ is an ad-isomorphism, so by Proposition 3.22, we have a quotient-finite map

(3.19) $$ \begin{align} \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_p\times \mathsf{M}_p}(\mathbf{G}\times \mathbf{T}, \mathbf{X}\times\{h\})\to \mathscr{S}_{\mathsf{K}_p}^{\mathfrak{d}}(\mathbf{G}, \mathbf{X})_{\mathcal{O}_{E'}} \end{align} $$

extending the morphism on the generic fibers which is equivariant for the projection $(\mathbf {G}\times \mathbf {T})(\mathbb {A}_f^p) \to \mathbf {G}(\mathbb {A}_f^p)$ . On the other hand, the map of reductive groups $\mathbf {G} \times \mathbf {T} \to \mathbf {T}$ factors through $(\mathbf {G} \times \mathbf {T})^{\text {ab}}$ , which is a torus. Hence Proposition 3.22 and Proposition 3.23 combine to furnish us with a map

$$ \begin{align*} \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_p\times \mathsf{M}_p}(\mathbf{G}\times \mathbf{T}, \mathbf{X}\times\{h\}) \to \mathscr{S}_{\mathsf{M}_p}(\mathbf{T}, \{h\})_{\mathcal{O}_{E'}} \end{align*} $$

extending the morphism on generic fibers, which is equivariant for $(\mathbf {G} \times \mathbf {T})(\mathbb {A}_f^p) \to \mathbf {T}(\mathbb {A}_f^p)$ . From these, we obtain the map

(3.20) $$ \begin{align} \mathscr{S}^{\mathfrak{d}}_{\mathsf{K}_p\times \mathsf{M}_p}(\mathbf{G}\times \mathbf{T}, \mathbf{X}\times\{h\}) \to \mathscr{S}_{\mathsf{K}_p}^{\mathfrak{d}}(\mathbf{G}, \mathbf{X})_{\mathcal{O}_{E'}} \times \mathscr{S}_{\mathsf{M}_p}(\mathbf{T}, \{h\})_{\mathcal{O}_{E'}}, \end{align} $$

which is equivariant for . The map (3.20) is an isomorphism on generic fibers, and is quasi-finite at each finite level because (3.19) is quotient-finite. It follows that (3.20) is an isomorphism by the argument at the end of the proof of Proposition 3.25.

4.1 The Pappas–Rapoport conjecture

We now recall the formulation of the Pappas–Rapoport conjecture as given in [Reference Pappas and RapoportPR24] and extended in [Reference DanielsDan25].

The group $\mathcal{G}^c$

In contrast to the case of Shimura varieties of Hodge type discussed in [Reference Pappas and RapoportPR24], to obtain shtukas on arbitrary Shimura varieties of abelian type, one must consider a modification $\mathcal{G}^c$ of the parahoric group $\mathcal{G}$ .

Let $\mathbf {T}$ be a multiplicative group over $\mathbb {Q}$ . We denote by $\mathbf {T}_{\textrm {ac}}$ the anti-cuspidal part of $\mathbf {T}$ as in [Reference Kisin, Shin and ZhuKSZ21, Definition 1.5.4]. More precisely, if $\mathbf {T}_a$ denotes the largest anisotropic subtorus of $\mathbf {T}$ , then $\mathbf {T}_{\textrm {ac}}$ is the smallest subtorus of $\mathbf {T}_a$ whose base change to $\mathbb {R}$ contains the maximal split subtorus of $(\mathbf {T}_a)_{\mathbb {R}}$ .

We write ${\mathbf {G}}^c$ for the quotient ${\mathbf {G}}^c / \mathbf {Z}_{\textrm {ac}}$ , and if $\mathsf {H}$ is a subgroup of ${\mathbf {G}}(\mathbb {A}_f)$ , we write $\mathsf {H}^c$ for the image of $\mathsf {H}$ under ${\mathbf {G}}(\mathbb {A}_f) \to {\mathbf {G}}^c(\mathbb {A}_f)$ . We also denote by $G^c$ the base change of ${\mathbf {G}}^c$ to $\mathbb {Q}_p$ . The following lemma will be used below.

We set $\mathcal{G}^c$ to be the parahoric model of $\mathbf {G}$ induced by $\mathcal{G}$ in the sense of §2.1. By construction, we see that $G \to G^c$ extends to a morphism of group $\mathbb {Z}_p$ -schemes $\mathcal{G} \to \mathcal{G}^c$ . For a conjugacy class of cocharacters of $G_{\overline {\mathbb {Q}}_p}$ , we denote by the image of this conjugacy class under $G_{\overline {\mathbb {Q}}_p}\to G^c_{\overline {\mathbb {Q}}_p}$ .

The shtuka over $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})_E$ .

We now recall the existence of a “universal shtuka” which lives over the Shimura variety $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})_E$ , as in [Reference Pappas and RapoportPR24, §4.1] and [Reference DanielsDan25, §4.2].

For a normal compact open subgroup $\mathsf {K}_p^{\prime } \subseteq \mathsf {K}_p$ , we temporarily write $\mathsf { K}'$ for the product $\mathsf {K}_p^{\prime }\mathsf {K}^p$ . Then the transition morphism

(4.1) $$ \begin{align} \pi_{\mathsf{K}',\mathsf{K}}\colon \textrm{Sh}_{\mathsf{K}'}(\mathbf{G},\mathbf{X})_E \to \textrm{Sh}_{\mathsf{K}}(\mathbf{G},\mathbf{X})_E \end{align} $$

is a finite étale Galois cover with Galois group $\mathsf {K}_p/(\mathsf {K}_p^{\prime }\mathbf {Z}(\mathbb {\mathbb {Q}})^{-}_{\mathsf {K}})$ , where $\mathbf {Z}(\mathbb {\mathbb {Q}})^{-}_{\mathsf {K}}$ is the closure of $\mathbf {Z}(\mathbb {Q})\cap \mathsf {K}$ in $\mathsf {K}$ (see [Reference Kisin, Shin and ZhuKSZ21, §1.5.8]). We consider the infinite-level Shimura variety

which exists as an E-scheme as each $\pi _{\mathsf {K}',\mathsf {K}}$ is affine (see [SP17, Tag 01YX]), and forms a $\mathsf {K}_p/\mathbf {Z}(\mathbb {Q})_{\mathsf {K}}^{-}$ -torsor for the pro-etale site (in the sense of [Reference Bhatt and ScholzeBS15]) on $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})_E$ .

The map $\mathsf {K}_p\to \mathcal{G}^c(\mathbb {Z}_p)$ factorizes through $\mathsf {K}_p/\mathbf {Z}(\mathbb {Q})_{\mathsf {K}}^{-}$ (see [Reference Kisin, Shin and ZhuKSZ21, §1.5.8]). Thus, as in [Reference DanielsDan25, §4.2] or [Reference Imai, Kato and YoucisIKY24b, §2.1.5 and §4.3], from the $\mathsf {K}_p/\mathbf {Z}(\mathbb {Q})_{\mathsf {K}}^{-}$ -torsor

$$ \begin{align*} \textrm{Sh}_{\mathsf{K}^p}({\mathbf{G}},X)_E \to \textrm{Sh}_{\mathsf{K}_p\mathsf{ K}^p}({\mathbf{G}},X)_E, \end{align*} $$

one constructs a $\mathcal{G}^c(\mathbb {Z}_p)$ torsor $\mathbb {P}_{\mathsf {K}}$ on the pro-étale site (see [Reference ScholzeSch13]) of $\textrm {Sh}_{\mathsf {K}}({\mathbf {G}},X)^{\text {an}}_E$ .

Proposition 4.2 ([Reference Pappas and RapoportPR24, Proposition 4.1.4] and [Reference DanielsDan25, Proposition 4.4]).

There exists a $\mathcal{G}^c$ -shtuka $\mathscr {P}_{\mathsf {K},E}$ over $\textrm {Sh}_{\mathsf {K}}(\mathbf {G},\mathbf {X})_E^\lozenge \to \mathrm {Spd}(E)$ with one leg bounded by which is associated to $\mathbb {P}_{\mathsf {K}}$ in the sense of [Reference Pappas and RapoportPR24, §2.5]. Moreover, if $\mathsf {K}'=\mathsf {K}_p{\mathsf {K}^p}'$ and g is in $\mathbf {G}(\mathbb {A}_f^p)$ is such that $g^{-1}\mathsf {K}^p g\subseteq {\mathsf {K}^p}'$ , there exist compatible isomorphisms

(4.2)

Integral local Shimura varieties

Let us define (see [Reference Scholze and WeinsteinSW20, Definition 24.1.1]), a parahoric local Shimura datum to be a triple $(\mathcal{G},b,\mu )$ where

• ⋄ $\mathcal{G}$ is a parahoric group $\mathbb {Z}_p$ -scheme with generic fiber G,

• ⋄ is a conjugacy class of minuscule cocharacters of $G_{\overline {\mathbb {Q}}_p}$ ,

• ⋄ and b is an element of $G(\breve {\mathbb {Q}}_p)$ inducing an element of (see [Reference Rapoport and ViehmannRV14, Definition 2.3]).

The reflex field of $(\mathcal{G},b,\mu )$ , usually denoted E, is the field of definition of . A morphism of parahoric local Shimura datum is is a morphism of $\mathbb {Z}_p$ -group schemes $f\colon \mathcal{G}_1\to \mathcal{G}$ carrying to and $b_1$ to b. If such a morphism exists, then $E_1\supseteq E$ . We say that f is an ad-isomorphism if $f\colon G_1\to G$ is an ad-isomorphism.

Given a parahoric local Shimura datum with reflex field E, we obtain a presheaf

assigning to any perfectoid space S in characteristic p the set of isomorphism classes of tuples $(S^\sharp , \mathscr {P}, \phi _{\mathscr {P}}, i_r)$ , where:

• ⋄ $S^\sharp $ is the untilt of S over $\mathcal{O}_{\breve {E}}$ associated with $S \to \text {Spd}(\mathcal{O}_{\breve {E}})$ ,

• ⋄ $(\mathscr {P},\phi _{\mathscr {P}})$ is a $\mathcal{G}$ -shtuka on S with one leg along $S^\sharp $ which is bounded by (see [Reference Pappas and RapoportPR24, Definition 2.4.3]),

• ⋄ and $i_r$ is an isomorphism of $\mathcal{G}$ -torsors for large enough r, under which $\phi _{\mathscr {P}}$ is identified with $b \times \textrm {Frob}_S$ (see [Reference Scholze and WeinsteinSW20, §25.1]).