\frac{1}{[L_p:\mathbb{Q}_p]} \sum_{\tau \in \gal(L_p/\qp)} \tau(\mu).
\end{align}

§.§.§ Isocrystals and the Dieudonné gerb

If $G/\mathbb{Q}_p$ is a connected reductive group and $f:\mathfrak{D} \to \mathfrak{G}_G$ is a morphism of $\qpur/\qp$ gerbs, then there is an induced element $b_{f} \in B(G)$, corresponding to evaluating $f$ on the distinguished element $d_{\sigma}$ (c.f. <cit.> 3.3.3). The Newton cocharacter of this element $b_f$ is equal to $-f^{\Delta}$, where $f^{\Delta}:\mathbb{D} \to G$ is the morphism of algebraic groups underlying $f$. When $T/\mathbb{Q}$ is a torus with a cocharacter $\mu$ defined over a Galois extension $L$ as above, then the morphism $\mathfrak{G}_v \to \mathfrak{Q} \to \mathfrak{G}_G$ induced by $\psi_{\mu}$ induces a morphism

\begin{align}
f_{\mu}:\mathfrak{D} \to \mathfrak{G}_T
\end{align}

of $\qpur/\qp$ gerbs. The element of $B(T)=X_{\ast}(T)_{\Gamma}$ associated to $f_{\mu}$ is then given by the image of $-\mu$ under $X_{\ast}(T) \to X_{\ast}(T)_{\Gamma}$ (this can be proved by reducing to the case that $T=\mathbb{G}_m$ as in Section 2 of <cit.>).

§.§.§ Strictly monoidal categories

Recall from Section 3.2 of KisinPoints the notion of crossed module $\tilde{H} \to H$ with main example given by $\operatorname{G}^{\text{sc}} \to G$, which we will sometimes write as $\tilde{G} \to G$. Given a crossed module there is a strict monoidal category $H/\tilde{H}$ and we define $\mathfrak{G}_{G/\tilde{G}}$ to be strict monoidal category corresponding to the crossed module $\tilde{G}(\overline{k}) \to \mathfrak{G}_G$. If $G^{\text{der}}$ is simply connected, then this is isomorphic to $\mathfrak{G}_{G^{\text{ab}}}$ and we encourage the reader to take this as as the main example.

§.§ The Langlands-Rapoport conjecture

Let $(G,X)$ be a Shimura datum with reflex field $E$, let $\mathcal{G}/\mathbb{Z}_{(p)}$ be a parahoric model of $G$, and let $\{\mu\}$ be the associated conjugacy class of cocharacters of $G$, note that this notation conflicts with the previous section. We let $\nu=\sigma(\mu')$ where $\mu'$ is the dominant representative in $\{\mu^{-1}\}$. Kisin defines a morphism (c.f. 3.3.1 of <cit.>)

\begin{align}
\phi_{\mutildeab}:\mathfrak{Q} \to \mathfrak{G}_{G/\tilde{G}},
\end{align}

and morphisms $\xi_v:\mathfrak{G}_v \to \mathfrak{Q}(v)$ for all $v \not=p$ (the trivial morphism when $v \not=\infty$)

A morphism $\phi:\mathfrak{Q} \to \mathfrak{G}_{G}$ is called admissible (w.r.t $X$) if

* The composite (denoted by $\phi_{\tildeab})$

\begin{equation}
\begin{tikzcd}
\mathfrak{Q} \arrow{r}{\phi} & \mathfrak{G}_G \arrow{r} & \mathfrak{G}_{G/\tilde{G}}
\end{tikzcd}
\end{equation}

is conjugate isomorphic to $\psi_{\mu_{\tildeab}}$ (see Section 3.2.1 of KisinPoints for the definition of conjugate isomorphic functors of strictly monoidal categories).

* For $v \not=p$, the composite

\begin{equation}
\begin{tikzcd}
\mathfrak{G}_v \arrow{r}{\zeta_v} & \mathfrak{Q}(v) \arrow{r}{\phi(v)} & \mathfrak{G}_{\tilde{G}}
\end{tikzcd}
\end{equation}

is conjugate to the morphism $\xi_v:\mathfrak{G}_v \to \mathfrak{G}_G(v)$.

* The image $\phi_b$ of $\phi$ in $B(G)$ defined by the composition

\begin{equation}
\begin{tikzcd}
\theta:\mathfrak{G}_p \arrow{r}{\nu(p)} & \mathfrak{Q}(p) \arrow{r}{\phi(p)} & \mathfrak{G}_G(p)
\end{tikzcd}
\end{equation}

lies in $B(G, \mu)$.

Our definition is equivalent to the definition in Section 3.3.6 of <cit.> when $G_{\mathbb{Q}_p}$ is quasi-split and splits over an unramified extension by Theorem <ref>, and equivalent to Definition 9.1 of <cit.> when $G^{\text{der}}$ is simply connected. We now define

\begin{align}
X^p(\phi):=\{(g_v)_{v \not=p, \infty} \in G(\afpbar) \; : \; \operatorname{Int}(g) \circ \xi_v = \phi(v) \circ \zeta_{v}\},
\end{align}

which is nonempty by A2 and in fact a $G(\afp)$ torsor. Write $\theta=\phi(p) \circ \zeta_p$ and define $X_p(\phi)$ to be the set of $g \in G(\qpbar)/\mathcal{G}_K(\zpur)$ such that $g \theta g^{-1}$ is induced by a morphism $\theta_g^{\text{ur}}:\mathfrak{D} \to \mathfrak{G}_{G}^{\text{ur}}$ of $\qpur/\qp$ gerbs satisfying $\theta_{g}^{\text{ur}}=b_{g} \rtimes \sigma$ with $b_{g} \in B(G,\mu)$. This has an action of a $p^r$-Frobenius, where $r=[E_0:\mathbb{Q}_p]$ with $E_0$ the maximal unramified extension of $E_p$, as follows: We define

\begin{align}
\Phi(g)=g b_{g} \sigma(b_g) \cdots \sigma^{r-1}(b_g)
\end{align}

and note that

\begin{align}
\theta_{\Phi(g)}^{\text{ur}}=(1 \rtimes \sigma^r) \theta_g^{\text{ur}} (1 \rtimes \sigma^{r})^{-1},
\end{align}

so that $b_{\Phi(g)}=\sigma(b_g)$, which is still an element of $B(G, \mu^{-1})$. Lemma 2.1 of <cit.> tells us that there is a $g_0 \in G(\qpbar)$ such that $g_0^{-1} \theta g_0$ is conjugate to a map of $\qpur/\qp$ gerbs $\theta:\mathfrak{D} \to \mathfrak{G}_G^{\text{ur}}$. It follows as in Lemma 3.3.4 of <cit.> that the map $g \mapsto g_0 g$ induces a bijection

\begin{align}
\xmub \simeq X_p(\phi),
\end{align}

where we consider both of them as subsets of $G(\qpbar)/\mathcal{G}_K(\zpur)$, and this bijection is compatible with the action of $\Phi$. Define

\begin{align} \label{eq:inverselimitset}
S(\phi)=\varprojlim_{U^p} I_{\phi}(\mathbb{Q}) \backslash X(\phi)/U^p.
\end{align}

where $X(\phi)=X^p(\phi) \times X_p(\phi)$, where the action of $I_{\phi}(\mathbb{Q}) \subset G(\overline{\mathbb{Q}})$ is by left multiplication on $X^p(\phi) \times X_p(\phi) \subset G(\afpbar) \times G(\qpbar)/\mathcal{G}_K(\zpur)$. Note that (<ref>) is not necessarily in bijection with $I_{\phi}(\mathbb{Q}) \backslash X(\phi)$ when Milne's axiom SV5 does not hold, i.e. when $Z_G(\mathbb{Q})$ is not discrete in $Z_G(\mathbb{A}_f)$. However this is not an issue in the Hodge type case, as this axiom will hold automatically. We are now ready to state the conjecture of Langlands and Rapoport, in a version that generalises both Conjecture 9.2 of <cit.> and Conjecture 3.3.7 of <cit.>:
Let $(G,X)$ and $\mathcal{G}$ be as above and let $U_p=\mathcal{G}(\mathbb{Z}_p)$. Consider the tower of Shimura varieties $\{\mathbf{Sh}_{G, U^p U_p}\}_{U^p}$ over the reflex field $E$ with its action of $G(\afp) \times Z_G(\mathbb{Q}_p)$, where $U^p$ varies over compact open subgroups of $G(\afp)$. Then this tower has a $G(\afp) \times Z_G(\mathbb{Q}_p)$-equivariant extension to a tower of flat schemes $\{\mathscr{S}_{G, U^p U_p}\}_{U^p}$ over $\mathcal{O}_{E_{(v)}}$. Moreover, there is a bijection

\begin{align}
\varprojlim_{U^p} \mathscr{S}_{U^p U_p}(\overline{\mathbb{F}}_p) \simeq \coprod_{[\phi]} S(\phi),
\end{align}

compatible with the action of $G(\afp) \times Z_G(\mathbb{Q}_p)$ and the operator $\Phi$, which acts on the left hand side as the geometric $p^r$-Frobenius. Here $\phi$ runs over conjugacy classes of admissible morphisms $\mathfrak{Q} \to \mathfrak{G}_G$.

§.§ Special morphisms

In Section 3.5 of <cit.>, Kisin constructs for every CM extension $\mathbb{Q} \subset L \subset \qbar$ a torus $P^L$ with character group identified with the group of Weil numbers (modulo roots of unity) inside $L^{\times}$. For $L \subset L'$ there is a morphism $P^{L'} \to P^L$ and we let $P$ be the inverse limit, which comes equipped with a map $Q \to P$. Pushing out $\mathfrak{Q}$ along this map gives rise to the pseudo motivic groupoid $\mathfrak{P}$, which is also a pro-Galois gerb. An admissible morphism $\mathfrak{Q} \to \mathfrak{G}_G$ factors through $\mathfrak{P}$ if $Z^0_G$ satisfies the Serre condition (Lemma 3.5.7 of <cit.>), this is automatic if $(G,X)$ is of Hodge type.

If we assume the Tate conjecture for smooth projective varieties over finite fields, then the category of representations of $\mathfrak{P}$, i.e. the category of morphisms $\mathfrak{P} \to \mathfrak{G}_{\operatorname{GL}_n}$, is equivalent to the category of (numerical) pure motives over $\ovfp$, see MilneMotivesFiniteField. This makes sense because motives over finite fields are conjecturally determined by $q$-Weil numbers, which are the characters of $P$.

Take a CM field $L$ as in the previous paragraph, and let $n$ be a sufficiently divisible natural number. Then there is an element $\delta_n \in P^L(\mathbb{Q})$ such for a $q=p^m$-Weil Number $\pi$, evaluating the character $\chi_{\pi}$ of $P^L$ associated to $\pi$ on $\delta_n$ gives

\begin{align}
\chi_{\pi}(\delta_n)=\pi^{n/m}.
\end{align}

Moreover these elements satisfy $\delta_{n'}=\delta_{n}^{n'/n}$ if $n \mid n'$ and the are preserved by the maps $P^{L'} \to P^L$ for $L \subset L' \subset \qbar$. Given a morphism

\begin{align}
\phi: \mathfrak{P} \to \mathfrak{G}_{\operatorname{GL}_n},
\end{align}

conjecturally corresponding to a motive over $\ovfp$ defined over $\mathbb{F}_q$, the image of $\delta_n$ in $\operatorname{GL}_n(\overline{\mathbb{Q}_{\ell}})$ should be thought of as $\operatorname{Frob}_{p^{m/n}}$ acting on the $\ell$-adic realisation of our motive. Lemma 5.5 of LanglandsRapoport tells us that the collection of elements $\{\delta_n\}$ is Zariski-dense in $P^L$.

Let $(G,X)$ be a Shimura datum and let $T \subset G$ be a torus of $G$ over $\mathbb{Q}$ together with an $h:\mathbb{S} \to G_{\mathbb{R}}$ in $X$ that factors through $T_{\mathbb{R}}$. This gives us a cocharacter $\mu$ of $T$ and hence a morphism $\psi_{\mu}:\mathfrak{Q} \to \mathfrak{G}_G$. It follows as in Lemma 3.5.8 of <cit.> that the composition of $\psi_{\mu}$ with $\mathfrak{G}_T \to \mathfrak{G}_G$ is an admissible morphism (to prove A3, use remark 5.2 of <cit.>.) An admissible morphism $\phi:\mathfrak{Q} \to \mathfrak{G}$ is called special if it is conjugate to a morphism induced by $T \subset G$ as above.

Let $\phi:\mathfrak{Q} \to \mathfrak{G}_G$ be an admissible morphism, and suppose that $G_{\mathbb{Q}_p}$ is quasi-split, then $\phi$ is special.

It follows as in the proof of Theorem 3.5.11 of <cit.> that we may reduce to the case that $G^{\text{der}}$ is simply connected and that $Z^0_G$ satisfies the Serre condition, so that $\phi$ factors through $\mathfrak{P}^L$ for some CM field $L$. At this point Kisin invokes Satz 5.3 of <cit.>, which proves the result under the assumption that $G_{\mathbb{Q}_p}$ is quasi-split and split over an unramified extension, we will indicate how the proof of loc. cit. generalises.

Let $\delta_n$ be the distinguished elements of $P^L(\mathbb{Q})$ constructed on page 31 of <cit.> (c.f. Corollary 3.5.4 of <cit.>), they are Zariski dense in $P^L$ by Lemma 5.5 of <cit.>. It follows from Lemma 5.4 of <cit.> that after conjugating $\phi$, there is a maximal torus $T \subset G$ such that $\phi(\delta_n) \in T(\mathbb{Q})$ and such that $T_{\mathbb{R}}^{\text{ad}}$ is anisotropic. As in their proof, it then suffices to prove that this torus transfer to $G$, which they reduce to showing that it transfer to $G$ locally at $p$. Our assumptions imply that $J_{b_{\phi}}$ is an inner form of a $\mathbb{Q}_p$-Levi of $G_{\mathbb{Q}_p}$ (c.f. 1.1.4 of <cit.>), and since $I'_{\qp} \subset J_{b_{\phi}}$ has the same rank, our torus transfers to $G_{\mathbb{Q}_p}$ because $G_{\mathbb{Q}_p}$ is quasi-split.

We can now invoke Corollary 1.1.17 of <cit.> to give us a cocharacter $\mu \in X_{\ast}(T) \cap \{\mu\}$ satisfying (5.l) of <cit.>, generalising the cocharacter constructed in the proof of Lemma 5.11 of <cit.>. The rest of the proof of LanglandsRapoport works without further change.

§.§ Connected components and a refined conjecture

Fix a morphism $\phi:\mathfrak{Q} \to \mathfrak{G}_{G/\tilde{G}}$. In Section 3.6 Kisin defines sets $X^p(\phi)$ and $X^p(\phi)$ with product $X(\phi)$, analogous to our definitions for morphisms $\phi:\mathfrak{Q} \to \mathfrak{G}$. Our definition of $X^p(\phi)$ is the same as his, but we have to slightly modify his definition of $X_p(\phi)$ for general groups by recalling that the Kottwitz map lands in $\pi_1(G)_I$ instead of $\pi_1(G)$. Then for an admissible $\phi_0:\mathfrak{Q} \to \mathfrak{G}_{G^{\text{ad}}}$ we define

\begin{align}
\tilde{\pi}(G, \phi_0) &:= \coprod_{\phi^{\text{ad}}=\phi_0} X(\phi_{\tildeab}) \\
\pi(G, \phi_0)&:=\varprojlim_{U^p} \tilde{\pi}(G, \phi_0)/U^p \gqbarsharpplus,
\end{align}

where $\phi_{\tildeab}$ is the composition $\mathfrak{Q} \to \mathfrak{G}_G \to \mathfrak{G}_{\tilde{G}/G}$ and where $G(\qbar)^{\sharp}_{+}$ is the inverse image of $G^{\text{ad}}(\mathbb{Q})_{+}$ in $G(\qbar)$ acting on $\tilde{\pi}(G, \phi_0)$ by right multiplication. Define

\begin{align}
\pi(G):=G(\mathbb{Q})_{+}^{-} \backslash G(\mathbb{A}_f) / \mathcal{G}(\zp),
\end{align}

where the bar denotes closure, then arguing as in Lemma 3.6.2 of <cit.> we can show that $\pi(G, \phi_0)$ is a $\pi(G)$-torsor, keeping in mind that for any parahoric $\mathcal{G}(\zp) \subset G(\qp)$ we have

\begin{align}
\pi_1(G)_I^{\sigma} \simeq G(\mathbb{Q}_p)/(\tilde{G}(\mathbb{Q}_p) \mathcal{G}(\zp))
\end{align}

by Lemma 5.18.(i) of <cit.>.

Let $T/\mathbb{Q}$ be a torus and let $\mu_T$ be a cocharacter of $T$, then there is an induced morphism $\psi_{\mu_{T}}:\mathfrak{Q} \to \mathfrak{G}_T$ and in section 3.6.6 of <cit.> Kisin defines sets $X^p(\psi_{\mu_{T}}), X_p(\psi_{\mu_{T}})$ with product $X(\psi_{\mu_{T}})$. He then defines

\begin{align}
S(\psi_{\mu_{T}})=T(\mathbb{Q})^{-} \backslash X(\psi_{\mu_{T}}),
\end{align}

where $T(\mathbb{Q})^{-}$ denotes the closure of $T(\mathbb{Q})$ in $T(\mathbb{A}_f)$ and shows (Proposition 3.6.7 of <cit.>) that $X(\phi_{\mu_T})$ is a $T(\afp)/\mathcal{T}(\zp)$-torsor and that there is a canonical isomorphism

\begin{align}
S(\psi_{\mu_{T}}) \simeq T(\mathbb{Q})^{-} \backslash T(\mathbb{A}_f)/\mathcal{T}(\zp).
\end{align}

Consider the category $\mathscr{SH}_p$ whose objects consist of pairs $(\mathcal{G}, X)$, where $\mathcal{G}/\mathbb{Z}_{(p)}$ is a smooth affine group scheme with $G=\mathcal{G}_{\mathbb{Q}}$ connected reductive and $\mathcal{G}_{\zp}$ parahoric, and where $(G,X)$ is a Shimura datum. Morphisms in this category are given by morphisms of group schemes $\mathcal{G} \to \mathcal{G}'$ over $\mathbb{Z}_{(p)}$, such that $(G, X) \to (G', X')$ is a morphism of Shimura data. Given an adjoint Shimura datum $(H,Y)$, we will write $\mathscr{SH}_p(H,Y)$ for the subcategory of objects $(\mathcal{G},X)$ such that $(G^{\text{ad}}, X^{\text{ad}})$ is isomorphic to $(H,Y)$. For $(\mathcal{G},X)$ an object of our category, we set $U_p=\mathcal{G}(\zp) \subset G(\mathbb{Q}_p)$ and write

\begin{align}
\pi(\mathcal{G}, X)=\varprojlim_{U^p} \pi_0(\mathbf{Sh}_{U^p U_p}(\mathbb{C})),
\end{align}

which is a $\pi(G)$-torsor. Let $T/\mathbb{Q}$ be a torus together with a morphism $h_T:\mathbb{S} \to T_{\mathbb{R}}$ with associated cocharacter $\muht=\mu_T$. If we are given a morphism $i:(T, h_T) \to (G,X)$, then there is an induced morphism

\begin{align}
T(\mathbb{Q})^{-} \backslash T(\mathbb{A}_f) / \mathcal{T}(\zp) \to \pi_0(G,X)
\end{align}

and similarly there is an induced morphism

\begin{align}
S(\psi_{\mu_T}) \to \pi_0(G, \phi_0),
\end{align}

where $\phi_0=(i \circ \psi_{\mu_T})_0$. The proof of Proposition 3.6.10 of <cit.> shows that if $\phi_0$ is special, then there is a unique isomorphism of $\pi(G)$-torsors

\begin{align} \label{Eq:Vartheta}
\vartheta_G:\pi(G, \phi_0) \simeq \pi(G,X)
\end{align}

that is functorial with respect to morphisms in $\mathscr{SH}_p(G^{\text{ad}}, X^{\text{ad}})$ and is compatible with the maps $S(\phi_{\mu_T}) \to \pi_0(G, \phi_0)$ induced by special points.

§.§ The refined conjecture

In this section we will compare admissible morphisms for $(G,X)$ and admissible morphisms for the adjoint Shimura datum $(G^{\text{ad}}, X^{\text{ad}})$, following Section 3.4 of <cit.>. We will fix an admissible morphism $\phi_0:\mathfrak{Q} \to \mathfrak{G}_{G^{\text{ad}}}$ throughout this section. From now on we will assume that $G_{\mathbb{Q}_p}$ is quasi-split, which implies that every admissible morphism is special.

We define sets $S_{\tau}(G, \phi_0)$ for $\tau \in I_{\phi_0}^{\text{ad}}(\mathbb{A}_f)$ as in Section 3.4.15 of <cit.>; it follows as in Corollary 3.4.16 of loc. cit. that $S_{\tau}(G, \phi_0)$ is a disjoint union of $S_{\tau}(\phi)$ for admissible morphisms $\phi$ lifting $\phi_0$, where the subscript $\tau$ means that the action of $I_{\phi}(\mathbb{Q})$ on $X(\phi)$ is twisted by $\tau$. Our parahoric model $\mathcal{G}$ defines parahoric models $\mathcal{G}^{\text{der}}$ and $\mathcal{G}^{\text{ad}}$ of $G^{\text{der}}$ and $G^{\text{ad}}$, as in Section 4.6.1 of <cit.>. Let $\zpbar$ be the ring of integers of $\qpbar$ and write $\gzpbarsharpplus$ for the preimage of $\mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}$ in $G(\qbar)$. The following lemma is proven as in KisinPoints.

[Lemma 3.7.2 of <cit.>]
There is an action of $\gzpbarsharpplus$ on

\begin{align}
\coprod_{\phi^{\text{ad}}=\phi_0} X(\phi),
\end{align}

which induces an of $\mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}$ on $S_{\tau}(G, \phi_0)$.

We now have an action of $\mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}$ and of $G(\afp)$ on the sets $S_{\tau}(G,\phi_0)$, which are compatible with the morphisms $\mathcal{G}(\mathbb{Z}_{(p)})_{+} \subset G(\mathbb{Q})_{+} \to G(\afp)$ and $\mathcal{G}(\mathbb{Z}_{(p)})_{+} \to \mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}$, this will induce an action of

\begin{align}
\azp:=G(\afp) \ast_{\mathcal{G}(\mathbb{Z}_{(p)})_{+}} \mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}
\end{align}

on $S_{\tau}(G, \phi_0)$. For the definition of $\ast$, see Section 3.7.3 of <cit.>. Intuitively, this is just a group that captures the action of $G(\afp)$ and $\mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+}$, taking into account that $\mathcal{G}(\mathbb{Z}_{(p)})_{+}$ maps to both of them. We also introduce

\begin{align}
\mathcal{A}(G)&=G(\mathbb{A}_f) \ast_{G(\mathbb{Q})_{+}} G^{\text{ad}}(\mathbb{Q})^{+} \\
\mathcal{A}(G)^{\circ}&=G(\mathbb{Q})_{+}^{-} \ast_{G(\mathbb{Q})_{+}} G^{\text{ad}}(\mathbb{Q})^{+} \\
\azpn&:=\mathcal{G}(\mathbb{Z}_{(p)})_{+}^{-} \ast_{\mathcal{G}(\mathbb{Z}_{(p)})_{+}} \mathcal{G}^{\text{ad}}(\mathbb{Z}_{(p)})^{+},
\end{align}

where the superscript $-$ denotes closure. Let us point out that there are natural maps

\begin{align}
\mathcal{A}(G) &\twoheadrightarrow \pi(G,X) \\
\azp &\to \pi(G, X)
\end{align}

with $\mathcal{A}(G)^{\circ}$ contained in the kernel of the first map and $\azpn$ equal to the kernel of the second map. There are natural projections $X(\phi) \to X(\phi_{\tildeab})$ which induce an $\azp$-equivariant map (c.f. Lemma 3.7.4 of <cit.>)

\begin{align}
c_G:S_{\tau}(G, \phi_0) \to \pi(G, \phi_0).
\end{align}

This map is surjective, but the proof of loc. cit. does not generalise. Instead, it will follow from the fact that $X_p(\phi) \to X_p(\phi_{\tildeab})$ is surjective, which is Lemma 6.1 of <cit.>. We now state the refined conjecture:

Let $(G,X)$ be a Shimura datum and let $\mathcal{G}/\mathbb{Z}_{(p)}$ be a model of $G$ over $\mathbb{Z}_{(p)}$ such that its base change to $\mathbb{Z}_p$ is parahoric and let $U_p=\mathcal{G}(\zp)$. Consider the tower of Shimura varieties $\{\mathbf{Sh}_{G, U^p U_p}\}_{U^p}$ over the reflex field $E$ with its action of $G(\afp) \times Z_G(\mathbb{Q}_p)$, where $U^p$ varies over compact open subgroups of $G(\afp)$. Then this tower has a $G(\afp) \times Z_G(\mathbb{Q}_p)$-equivariant extension to a tower of flat schemes $\{\mathscr{S}_{G, U^p U_p}\}_{U^p}$ over $\mathcal{O}_{E_{(v)}}$. Moreover, the action of $\azp$ on the generic fibre extends to the integral model. Furthermore there is an $\azp \times Z_G(\mathbb{Q}_p)$-equivariant bijection fitting into a commutative diagram

\begin{equation}
\begin{tikzcd}
\coprod_{[\phi_0]} S(G, \phi_0)  \arrow{d}  \arrow{r}{\sim} & \varprojlim_{U^p} \mathscr{S}_{U^p U_p}(\overline{\mathbb{F}}_p) \arrow{d}  \\
\coprod_{[\phi_0]} \pi(G, \phi_0)  \arrow{r}{\vartheta_G} & \pi(G, X).
\end{tikzcd}
\end{equation}

compatible with the action of the operator $\Phi$, which acts on the left hand side as the geometric $p^r$-Frobenius. Here $[\phi_0]$ runs over conjugacy classes of admissible morphisms $\mathfrak{Q} \to \mathfrak{G}_{G^{\text{ad}}}$. We remind the reader that the set $S(G, \phi_0)$ is the same as the set $S(G, \phi_0)_{\tau}$ for $\tau=1$.

Conjecture <ref> follows immediately from Conjecture <ref>, using Lemma 3.4.16 of <cit.>. It follows as in Remark 3.7.10 of KisinPoints that proving an $\azp$-equivariant bijection is enough if
$Z_G(\mathbb{Q}) \cdot Z_{\mathcal{G}}(\mathbb{Z}_p)=Z_G(\mathbb{Q}_p)$. Indeed, in this case it follows as in loc. cit. that $Z_G(\mathbb{Q}_p)$ acts trivially on both sides on the conjectured isomorphism. In general, it is unclear to us how to construct an action of $Z_G(\mathbb{Q}_p)$ on Kisin-Pappas integral models of Shimura varieties.

§.§ Connected components II

In this section we will build a theory of `connected Shimura varieties' for the sets $S_{\tau}(G, \phi_0)$, following Section 3.8 of <cit.>. Let $h:(G,X) \to (G_2,X)$ be a surjective morphism of Shimura data that induces an isomorphism on derived groups. If $\mathcal{G}$ is a parahoric model of $G$, then $h$ defines a parahoric model $\mathcal{G}_2$ of $G_2$ as in 1.1.3 of <cit.>. Let $X^+ \subset X^{\text{ad}}$ be a connected component, and consider the full subcategory $\mathscr{SH}_p(X^+)$ of $\mathscr{SH}_p$ consisting of objects $(H,Y)$ such that $X^+ \subset Y$. Then for an object $(G,X)$ of $\mathscr{SH}_p(X^+)$ we can consider the map

\begin{align}
\pi_0(X) \to \pi(G,X) \xrightarrow{\vartheta_G^{-1}} \pi(G, \phi_0),
\end{align}

the image of $X^+$ under this map is a point $y \in \pi(G, \phi_0)$. Here $\vartheta$ is the map  (<ref>). Let $h:(\mathcal{G},X) \to (\mathcal{G}_2,X)$ be a surjective morphism in $\mathscr{SH}_p$ that induces an isomorphism on derived groups. Define $S_{\tau}(G, \phi_0)^+$ and $S_{\tau}(G_2, \phi_0)^+$ to be the inverse image of $y$ respectively $y_2$ in $\pi(G, \phi_0)$ respectively $\pi(G_2, \phi_0)$.
[c.f. Lemma 3.8.2 of <cit.>]
The natural map $S_{\tau}(G, \phi_0)^+ \to S_{\tau}(G_2, \phi_0)^+$ is a bijection.

Consider the commutative diagram

\begin{equation}
\begin{tikzcd}
S_{\tau}(G, \phi_0) \arrow{d} \arrow{r}{h} & S_{\tau}(G_2, \phi_0) \arrow{d} \\
\pi(G, \phi_0) \arrow{r} & \pi(G_2, \phi_0),
\end{tikzcd}
\end{equation}

which is equivariant for the action of $\azp$ via $\azp \to \mathcal{A}(\mathcal{G}_{2})$. The bottom horizontal map can be identified (using our choice of $y, y_2$) with the map $\pi(G) \to \pi(G_2)$. It suffices to show that $h^{-1}(S_{\tau}(G_2, \phi_0)^{+}) \to S_{\tau}(G_2, \phi_0)^{+}$ is surjective, and that the fibers map bijectively to the fibers of $\pi(G) \to \pi(G_2)$. Both of these statements can be proven as in the proof of Lemma 3.8.2 of <cit.> (but the stronger statement that $\ker(\azp \to \azptwo)$ acts transitively on the fibers does not follow, because 3.7.5 of <cit.> does not hold).

Consider the action of $\langle \Phi \rangle \subset \gal(\qpur/\mathbb{Q}_{p^r})$, where $\mathbb{Q}_{p^r}$ is the maximal unramified extension contained in $E_p$. Then $\langle \Phi \rangle$ acts on $S_{\tau}(G, \phi_0)$ for any $\tau$, and we let

\begin{align}
\mathscr{E}_p^r(\mathcal{G}, \phi_0) \subset \azp \times \langle \Phi \rangle
\end{align}

be the stabiliser of $S_{\tau}(G, \phi_0)^{+}$. In then follows as in the proof of Lemma 3.8.5 of <cit.> that the group $\mathscr{E}(\mathcal{G}, \phi_0)$ is an extension of $\langle \Phi \rangle$ by $\azpn$ and depends only on $\mathcal{G}^{\text{der}}, X^+$ and the integer $r$. Here we have to keep in mind that we only know injectivity (rather than bijectivity) of

\begin{align}
\azpn \backslash \azp \simeq \mathcal{G}(\mathbb{Z}_{(p)})_{+}^{-} \backslash G(\afp) \to G(\mathbb{Q})^{-}_{+} \backslash G(\mathbb{A}_f)/ \mathcal{G}(\zp) \simeq \pi(G),
\end{align}

which is enough for the conclusion.
[Lemma 3.8.8 of <cit.>]
There is a natural isomorphism

\begin{align}
\azp \ast_{\azpn} \mathscr{E}_p^r(\mathcal{G}, \phi_0) \simeq \azp \times \langle \Phi \rangle.
\end{align}

Moreover, there is a natural isomorphism

\begin{align}
S_{\tau}(G, \phi_0) \simeq \coprod_{s \in S}[\azp \times S_{\tau}(G, \phi_0)^+ s]/\azpn,
\end{align}

equivariant for the action of $\azp \times \langle \Phi \rangle$, where $\azp$ acts on the right hand via left multiplication on itself and $S \subset \pi(G)$ is a set of coset representatives for the inclusion

\begin{align}
\azpn \backslash \azp \to \pi(G).
\end{align}

The action of $\mathcal{A}(\mG)$ on $S_{\tau}(G, \phi_0)$ gives us a map

\begin{align}
[\azp \times S_{\tau}(G, \phi_0)^+]/\azpn \xhookrightarrow{} S_{\tau}(G, \phi_0),
\end{align}

which is injective because $\azpn$ is the stabiliser of $S_{\tau}(G, \phi_0)^+$ in $\azp$. This map is not necessarily surjective, because the right hand side surjects onto $\pi(G, \phi_0)$ and the left hand side might not. Using the point $y$, we can identify the image of the left hand side in $\pi(G, \phi_0)$ with

\begin{align}
\azp/\azpn \subset \pi(G)
\end{align}

and the result follows.

Let $f:(G,X) \to (G_2, X)$ be a surjective map with kernel $Z \subset Z_G$ and let $\mathcal{G} \to \mathcal{G}_2$ be the induced map on parahoric models. Suppose that there is an isomorphism $Z_{\mathbb{Q}} \simeq R_{L/\mathbb{Q}} \mathbb{G}_m$.

* Then
\begin{align} \label{3.8.10.1}
S_{\tau}(G_2, \phi_0) \simeq S_{\tau}(G, \phi_0)/Z(\afp) \simeq [\azptwo \times S_{\tau}(G, \phi_0)]/\azp.
\end{align}

* There is a natural isomorphism

\begin{align}
\mathscr{E}^r_p(\mathcal{G}_2) \simeq \azpntwo\ast_{\azp} \mathscr{E}^r_p(\mathcal{G}),
\end{align}

* The natural map of sets with $\mathscr{E}^r_p(
\mathcal{G}_2)$ action

\begin{align}
S_{\tau}(G_2, \phi_0)^+ \simeq [\azpntwo\times S_{\tau}(\mathcal{G}, \phi_0)^+]/\azpn,
\end{align}

is an isomorphism if $L/\mathbb{Q}$ is Galois.

* There is an $\mathcal{A}(\mG_2) \times \langle \Phi \rangle$ equivariant isomorphism

\begin{align}
S_{\tau}(G_2, \phi_0)^{+} \simeq \coprod_{j \in J}[\mathcal{A}(\mG_2) \times S_{\tau}(G, \phi_0)^{+}j]/\azpn,
\end{align}

where $J \subset \pi(G_2)$ runs over a set of coset representatives for the inclusion

\begin{align}
\azpn \backslash \mathcal{A}(\mG_2) \xhookrightarrow{} \pi(G_2).
\end{align}

The proof of the first part follows as in the proof of Lemma 3.8.10 of <cit.>, except that we need to modify the argument showing surjectivity of the map on affine Deligne-Lusztig varieties. The map $\operatorname{Gr}_{\mG} \to \operatorname{Gr}_{\mathcal{G}_2}$ induces an isomorphism $Y \simeq Y_2$ between a connected component $Y$ in the source and a connected component $Y_2$ in the target (c.f. Section 4.2 of <cit.>). Moreover, it induces an isomorphism

\begin{align}
X_{\mu}(b)_{G,K} \cap Y \to X_{\mu_2}(b_2)_{G_2, \mathcal{G}} \cap Y_2,
\end{align}

and so it suffices to check that connected components of $\operatorname{Gr}_{\mathcal{G}_2}$ that intersect with $X_{\mu_2}(b_2)_{G_2, \mathcal{G}}$ are in the image of connected components of $\grg$ that intersect $X_{\mu}(b)_{G,K}$. In other words, we want to show that

\begin{align}
\pi_1(G)_I^{\sigma} \to \pi_1(G_2)_I^{\sigma}
\end{align}

is surjective, which follows by considering the following diagram:

\begin{equation}
\begin{tikzcd}
G(\qp) \arrow[r, two heads] \arrow[d, two heads] & G_2(\qp) \arrow[d, two heads] \\
\pi_1(G)_I^{\sigma} \arrow{r} & \pi_1(G_2)_I^{\sigma}
\end{tikzcd}
\end{equation}

Indeed, the top horizontal arrow is surjective because $Z$ is an induced torus, and the vertical arrrows are surjective by the result of Section 7.7 of <cit.>. The isomorphism

\begin{align}
S_\tau(G, \phi_0)/Z(\afp) \simeq [\azptwo \times S_{\tau}(G, \phi_0)]/\azp
\end{align}

similarly follows as in the proof of Lemma 3.8.10 of <cit.>. The proof (2) is the same as the proof of (2) in loc. cit. and moreover gives us an $\mathscr{E}^r_p(\mathcal{G}_2^{\text{der}})$-equivariant map

\begin{align}
[\azptwo^{\circ} \times S_{\tau}(G, \phi_0)^{+}]/\azpn \to S_{\tau}(G_2, \phi_0)^+.
\end{align}

To prove (3), it suffices to prove that this map is a bijection. It is injective by the second isomorphism of (<ref>) and so it suffices to prove surjectivity. From (i) we get that the natural map

\begin{align}
f^{-1}(S_{\tau}(G_2, \phi_0)^+) \to S_{\tau}(G_2, \phi_0)^+
\end{align}

is surjective, and so it suffices to show that $f^{-1}(S_{\tau}(G_2, \phi_0)^+)$ surjects onto the preimage of $y_2$ in $\pi(G, \phi_0)$. If we identify $\pi(G, \phi_0) \to \pi(G_2, \phi_0)$ with $\pi(G) \to \pi(G_2)$ using $y$ and $y_2$, it comes down to showing that

\begin{align} \label{Eq:Approximation}
Z(\afp) \to \ker(\pi(G) \to \pi(G_2)) \simeq Z(\mathbb{Q})^{-} \backslash Z(\mathbb{A}_f) / \mathcal{Z}(\mathbb{Z}_p)
\end{align}

is surjective, where $\mathcal{Z}$ is the kernel of $\mathcal{G} \to \mathcal{G}_2$. It follows from Proposition 2.4.12 of <cit.> that $\mathcal{Z}$ is in fact the connected Néron model of $Z$, and that $\mathcal{G} \to \mathcal{G}_2$ is surjective (here we use that $Z$ is an induced torus). Moreover Remark 8.3 of <cit.> shows that $Z(\mathbb{Q}) \mathcal{Z}(\mathbb{Z}_p)=\mathbb{Z}(\mathbb{Q}_p)$ (using the fact that $L/\mathbb{Q}$ is Galois). It follows that (<ref>) is surjective. Part (4) of the lemma follows from Lemma <ref> and the fact that $\azpn \to \azpntwo$ is surjective because $Z$ is an induced torus.

The following analogue of Corollary 3.8.12 of <cit.> now follows:

Suppose that $\mathcal{G}^{\text{der}} \to \mathcal{G}_2^{\text{der}}$ is a central isogeny which induces an isomorphism of adjoint Shimura data. Then there is an isomorphism of sets with $\azptwo \times \langle \Phi \rangle$ action

\begin{align}
S_{\tau}(G_2, \phi_0) \simeq \coprod_{j \in J} [\azptwo \times S_{\tau}(G_, \phi_0)^{+} j]/\azpn,
\end{align}

where $J$ ranges over a set of coset representatives for

\begin{align}
\azpn \backslash \azptwo \xhookrightarrow{} \pi(G_2)
\end{align}

The proof is the same as the proof of Corollary 3.8.12 of KisinPoints.

There is a bijection

\begin{align}
\pi(G_2) \simeq \mathcal{A}(G_2)^{\circ} \backslash \mathcal{A}(G_2) \mathcal{G}_2(\zp) \simeq \mathcal{A}(G)^{\circ} \backslash \mathcal{A}(G_2) \mathcal{G}_2(\zp),
\end{align}

which will be useful later when we compare with Lemma 4.6.13 of <cit.>.

§ MAIN RESULTS FOR ABELIAN TYPE SHIMURA VARIETIES

In this section we will discuss Kottwitz triples, prove our main theorem in the Hodge type case and end by deducing our main theorems in the abelian type case.

§.§ Kottwitz triples

Let $(G,X)$ be a Shimura datum of Hodge type and let $\mathcal{G}/\mathbb{Z}_{(p)}$ be a parahoric model of $G$ of type $K$. In Section 4.3 of <cit.>, Kisin defines a notion of (refined) Kottwitz triple and also defines an equivalence relation $\sim$ on Kottwitz triples. If $\tilde{\mathfrak{t}}=(\gamma_0, (\gamma_l)_{l \not=p}, \delta, \iota)$ is a refined Kottwitz triple, then there is a set

\begin{align}
S(\tilde{\mathfrak{t}})=I(\mathbb{Q}) \backslash X_{\mu}(\delta)_K \times G(\afp)
\end{align}

with an action of $Z_G(\mathbb{Q}_p) \times \langle \Phi \rangle \times G(\afp)$. Recall from Kisin that if $\mathfrak{t}' \sim \mathfrak{t}$ then we can transport the refinement $\iota$ to a refinement $\iota'$ of $\mathfrak{t}'$ and obtain an equivariant bijection

\begin{align}
S(\tilde{\mathfrak{t}}) \simeq S(\tilde{\mathfrak{t}'}).
\end{align}

Let $\phi:\mathfrak{Q} \to \mathfrak{G}_G$ be an admissible morphism, then $\phi$ factors through $\phi:\mathfrak{P}^L \to \mathfrak{G}_G$ because $Z^0_G$ satisfies the Serre condition. Moreover, there is a morphism of $\qpur/\qp$ gerbs $\theta:\mathfrak{D} \to \mathfrak{G}_G^{\text{ur}}$ such that its inflation to a map  $\qpbar/\qp$-gerbs is conjugate to $\phi$. We then define a Kottwitz triple $\mathfrak{t}(\phi)$ by

\begin{align}
\gamma_0&=\phi(\delta_n)=\gamma_l \\
\delta&=\theta(d_{\sigma}^n).
\end{align}

It follows from Section 4.5 of <cit.> that this is indeed a Kottwitz triple with $I \simeq I_{\phi}$. Moreover, Lemma 4.5.2 of loc. cit. tells us that there is a $\langle \Phi \rangle \times G(\afp)$-equivariant isomorphism
$$X(\phi) \simeq X_{\mu}(\delta)_K \times G(\afp).$$
Proposition 4.5.7 of loc. cit. tells us that if the fiber of

\begin{align} \label{Eq:Torsor}
\{\text{ Admissible } \phi:\mathfrak{Q} \to \mathfrak{G}_G \}/\text{conjugacy} \to  \{\text{triples}\}/\sim
\end{align}

over a triple $\mathfrak{t}$ is nonempty, then it is a $\Sha_{G}(\mathbb{Q},I)$-torsor (see <cit.> 4.4.7 and 4.4.9 for the definition of this Tate-Shafarevich group). If $T$ is a torus over $\mathbb{Q}$ together with a cocharacter $\mu$, then there is a morphism $\psi_{\mu}:\mathfrak{Q} \to \mathfrak{G}_T$. If $T$ satisfies the Serre condition, then this factors through $\phi:\mathfrak{P}^L \to\mathfrak{G}_T$ and we can define a Kottwitz triple $\mathfrak{t}(T, \mu)$ for $T$ by

\begin{align}
\gamma_0&=\phi(\delta_n)=\gamma_l \\
\delta&=\theta(d_{\sigma}^n).
\end{align}

Since $\phi^{\Delta}:P^L \to T$ is defined over $\mathbb{Q}$, it follows that $\gamma_0 \in T(\mathbb{Q})$ for $n \gg 0$. Now suppose that $i:T \subset G$ such that $\mu=\mu_{h_T}$ for some $h_T:\mathbb{S} \to G_{\mathbb{R}}$ factoring through $T_{\mathbb{R}}$, then $T$ satisfies the Serre condition by 4.3.9 of <cit.>. Furthermore there is an equivalence of Kottwitz triples

\begin{align}
i_{\ast}(\mathfrak{t}(T, \mu) \sim \mathfrak{t}(i \circ \psi_{\mu}),
\end{align}

where $i_{\ast}(\gamma_0, (\gamma_l)_{l\not=p}, \delta)=(i(\gamma_0), (i(\gamma_l))_{l \not=p}, i(\delta))$.

§.§ Mod p-points on Shimura varieties of Hodge type

Let $(G,X)$ be a Shimura variety of Hodge type and let $p>2$ be a prime such that $G_{\mathbb{Q}_p}$ is quasi-split and splits over a tamely ramified extension, such that $p$ does not divide $\# \pi_1(G^{\text{der}})$ and such that all parahorics of $G$ are connected. Let $U^p \subset G(\afp)$ be a sufficiently small compact open subgroup and let $U=U^pU_p$ with $U_p=\mathcal{G}(\zp)$, where $\mathcal{G}$ is a parahoric model of $G$. We let $\mathscr{S}_{U}$ be the Kisin-Pappas integral model of the Shimura variety $\mathbf{Sh}_{U^p U_P}$.

* Suppose that $\mathcal{G}$ is a very special parahoric, then there is an $G(\afp) \times \langle \Psi \rangle$-equivariant bijection

\begin{align}
\mathscr{S}_{G, U^p U_p}(\ovfp) \simeq \coprod_{\phi} S_{\tau(\phi_0)}(\phi),
\end{align}

where $\phi$ runs over conjugacy classes of admissible morphisms $\mathfrak{Q} \to \mathfrak{G}_G$.

* Suppose that $G^{\text{ad}}$ is $\mathbb{Q}$-simple, that $J_{b}$ has no compact type factors and that $\mathbf{Sh}_{U}$ is proper or that Conjecture <ref> holds. Then the same conclusion as in (i) holds.

We would like to point out that the first part of the theorem (for very special parahorics) is due to Rong Zhou; it follows almost immediately from Appendix <ref> and <cit.>.

§.§ Proof of Theorem <ref>

In this section we will prove Theorem <ref>, following Section 4 of <cit.>. Because the arguments are so similar to the arguments in loc. cit., we will not give many details. The strategy of the proof can be summed up by the following diagram (we use the Kisin-Pappas integral models for Shimura varieties of abelian type)

\begin{equation}
\begin{tikzcd}
\{\text{Admissible morphisms} \}/\text{conjugacy} \arrow{dr} \\
\{\text{special points}\} \arrow{r} \arrow[u, two heads] \arrow[d, two heads]& \{\text{Kottwitz triples}\}/\sim \\
\{\text{Isogeny classes}\} \arrow[ur, densely dotted].
\end{tikzcd}
\end{equation}

It follows from Theorem <ref> that all admissible morphisms are special and it follows from part (2) of Theorem <ref> that all isogeny classes come from special points. To go from isogeny classes to Kottwitz triples, we choose a special point landing in the isogeny class and then take the Kottwitz triple associated to that special point, which will be independent of the choice of special point up to equivalence as in 4.4.6 of <cit.>. Moreover, it follows as in loc. cit. that we can twist isogeny classes by elements of $\Sha_{G}^{\infty}(\mathbb{Q},I)$ and that this turns

\begin{align}
\{\text{Isogeny classes}\} \to \{\text{Kottwitz triples}\}/\sim
\end{align}

into an $\Sha_{G}(\mathbb{Q},I)$-torsor (Proposition 4.4.13 of loc. cit.) over its image. We can now conclude that there is a bijection between isogeny classes and admissible morphisms, keeping in mind (<ref>), so it suffices to deal with the uniformisation of isogeny classes. Part (1) of Theorem <ref> combined with Theorem <ref> tells us that under our assumptions isogeny classes $\mathscr{I} \subset \mathscr{S}_U(\ovfp)$ have the following shape

\begin{align}
\mathscr{I} \simeq I(\mathbb{Q}) \backslash \xmub \times G(\afp) / U^p,
\end{align}

and moreover this identification is $\langle \Phi \rangle \times G(\afp)$-equivariant. Recall that there is an action of $\azp$ on $\mathscr{S}_U(\ovfp)$. It follows as in the proof of Proposition 4.4.14 of <cit.> that the stabiliser of $\mathscr{I}$ is given by $\azp^I \subset \azp$, where

\begin{align}
(\azp)^I= G(\afp) \ast_{\mathcal{G}(\mathbb{Z}_{p})_{+}} \mathcal{G}^{\text{ad}}(\mathbb{Z}_p)_{+}^I
\end{align}

and where $\mathcal{G}^{\text{ad}}(\mathbb{Z}_p)_{+}^I$ is the kernel of

\begin{align}
\mathcal{G}^{\text{ad}}(\mathbb{Z}_p)_{+} \to H^1(\mathbb{Q}, Z_G) \to H^1(\mathbb{Q},I).
\end{align}

Similarly, it follows as in Lemma 4.3.5 of loc. cit. that the stabiliser of $S_{\tau}(\phi) \subset S_{\tau}(G, \phi_0)$ under the action of $\azp$ is given by $(\azp)^I$. Let $(T, h_T,i)$ be a special point mapping to $\mathscr{I}$ and let $\phi$ be an admissible morphism conjugate to $i \circ \psi_{\mu_{h_{T}}}$. We we now write

\begin{align}
\mathscr{I}_0=\bigcup_{h \in G^{\text{ad}}(\mathbb{Q})^{+}} \mathscr{I}^{[h]},
\end{align}

where $[h]$ is the class of $h$ in $\Sha_{G}^{\infty}(\mathbb{Q},I)$. As in <cit.>, the theorem can be deduced from the following proposition:
[c.f. Proposition 4.6.2 of <cit.>]
Let $\phi$ be as above, then there is an $\langle \Phi \rangle \times \azp$-equivariant bijection $\xi:\mathscr{I}_0 \simeq S_{\tau}(G, \phi_0)$ for some $\tau=\tau(\phi_0)$ fitting in a commutative diagram

\begin{equation}
\begin{tikzcd}
\mathscr{I}_0 \arrow{d}{c_G} \arrow{r}{\xi} & S_{\tau}(G, \phi_0) \arrow{d} \\
\pi(G, X) \arrow{r}{\vartheta_G} & \pi(G, \phi_0).
\end{tikzcd}
\end{equation}

Moreover, each $\mathscr{I}^{[h]}$ is taken isomorphically to $S_{\tau}(\phi^{[h]})$.

The proof is the same as the proof in loc. cit., except that we need the fact that $J_b(\mathbb{Q}_p) \to \pi_1(G)_I^{\sigma}$ is surjective. In the unramified case, this is Corollary 2.5.12 of CKV and we adapt their proof: Let $T$ be the centraliser of a maximal split torus of $G_{\mathbb{Q}_p}$, this is a maximal torus since $G_{\mathbb{Q}_p}$ is quasi-split. The short exact sequence

\begin{align}
0 \to X_{\ast}(T^{\text{sc}}) \to X_{\ast}(T) \to \pi_1(G) \to 0
\end{align}

defining $\pi_1(G)$ induces a short exact sequence

\begin{align}
0 \to X_{\ast}(T^{\text{sc}})_I \to X_{\ast}(T)_I \to \pi_1(G)_I \to 0,
\end{align}

because $X_{\ast}(T^{\text{sc}})_I$ is torsion-free since $X_{\ast}(T^{\text{sc}})$ is an induced Galois module by 4.4.16 of BT2. Taking the long exact sequence in cohomology for the Frobenius action we see that the surjectivity of $X_{\ast}(T)_I^{\sigma} \to \pi_1(G)_I^{\sigma}$ is equivalent to the injectivity of

\begin{align}
X_{\ast}(T^{\text{sc}})_{\Gamma} \to X_{\ast}(T)_{\Gamma},
\end{align}

where $\Gamma$ is now the full Galois group. This injectivity follows because $X_{\ast}(T^{\text{sc}})_{\Gamma}$ is torsion free since $X_{\ast}(T^{\text{sc}})$ is an induced Galois module. Finally the map $X_{\ast}(T)_I^{\sigma} \to \pi_1(G)_I^{\sigma}$ factors through $\pi_1(M)_I^{\sigma} \to \pi_1(G)_I^{\sigma}$ and since $\pi_1(M) \simeq \pi_1(J_b)$, we are done.

§.§ Main results for abelian type Shimura varieties

Let $(G,X)$ be a Shimura datum of abelian type and let $p>2$ be a prime such that $G_{\mathbb{Q}_p}$ is quasi-split and splits over a tamely ramified extension. Let $U_p \subset G(\qp)$ be a parahoric subgroup and consider the tower of Shimura varieties $\{\mathbf{Sh}_{G, U^p U_p}\}_{U^p}$ over E with its action of $G(\afp)$, where $U^p$ varies over compact open subgroups of $G(\afp)$. Then by Theorem 0.1 of <cit.>, this tower of Shimura varieties has a $G(\afp)$-equivariant extension to a tower of flat normal schemes $\{\mathscr{S}_{G, U^p U_p}\}_{U^p}$ over $\mathcal{O}_{E_{(v)}}$, where $v \mid p$ is a prime of the reflex field $E$.
Let $\mu'$ be the dominant representative of the conjugacy class $\{\mu_h^{-1}\}$ where $\mu_h$ is the Hodge cocharacter associated to $X$ and let $\mu=\sigma(\mu')$. Let $b \in B(G,X)$ be the unique basic $\sigma$-conjugacy class, and let $J_{b}/\mathbb{Q}_p$ be its twisted centraliser. Consider the following sets of hypotheses on $(G,X)$ and $U_p$.

* The parahoric subgroup $U_p$ is very special.

* The group $J^{\text{ad}}_{b}$ has no factors that are of compact type and either $\mathbf{Sh}_U$ is proper or Conjecture <ref> holds for an auxiliary Hodge type Shimura datum of very special level.

* The Shimura datum $(G,X)$ admits an auxiliary Hodge type Shimura datum that is of PEL type A with $G_{\mathbb{Q}_p}$ unramified.

We need one more technical assumption which has to do with being able to reduce to the case of a Hodge type Shimura variety with a connected parahoric, see Remark <ref>.

* All factors of $(G^{\text{ad}},X^{\text{ad}})$ that are of type $D^{\mathbb{H}}$ split over an unramified extension (at $p$), and for those factors the parahoric subgroup $U_p=\mG_J(\mathbb{Z}_p)$
is contained in a hyperspecial subgroup.

Suppose that $(G,X)$ and $U_p$ satisfy (T3) or that they satisfy (P1) and either (T1) or (T2). Then there is an $G(\afp) \times \langle \Phi \rangle$-equivariant bijection

\begin{align} \label{eq:BijMainThm}
\mathscr{S}_{G, U^p U_p}(\ovfp) \simeq \coprod_{\phi} S_{\tau(\phi_0)}(\phi),
\end{align}

where $\phi$ runs over conjugacy classes of admissible morphisms $\mathfrak{Q} \to \mathfrak{G}_G$.

The theorem in the (T1) case is essentially due to Rong Zhou and the theorem in the (T3) case is essentially due to Kottwitz Kottwitzpoints and Rapoport-Zink <cit.>.

As in Theorem <ref>, we do not construct an action of $Z_G(\mathbb{Q}_p)$ on the left hand side. However if $G$ splits over a metacyclic extension, then $Z_G(\mathbb{Q}_p)$ acts trivially on the right hand side of (<ref>) (see Remark <ref> and Remark 3.7.10.(2) of <cit.>), and so we get a $Z_G(\mathbb{Q}_p)$-equivariant statement for free.

Let $(G,X)$ be as above, let $U_p$ denote an arbitrary parahoric and suppose that $G^{\text{ad}}$ is $\mathbb{Q}$-simple and that (P1) and (T2) hold. Let $w \in {}^K \admu$ and let $\mathscr{Sh}_{U, \ovfp}\{w\}$ be the corresponding EKOR stratum, where $K$ is the type of $U_{p}$ (c.f. Section <ref>). Suppose that it is not contained in the basic locus, then

\begin{align}
\mathscr{Sh}_{U, \ovfp}\{w\} \to \mathscr{Sh}_{U, \ovfp}
\end{align}

induces a bijection on connected components.

Theorem <ref> is a special case of Theorem <ref>, because Conjecture <ref> holds for unramified groups by Proposition 6.20 of <cit.>, combined with Theorem 1.2 of <cit.> and because all type A Shimura varieties admit auxiliary Hodge type data of PEL type (see Appendix B of <cit.> and Proposition 1.4 of <cit.>). By the same reasoning, Theorem <ref> is a special case of Theorem <ref>, except that we have to prove irreducibility of nonbasic Ekedahl-Oort strata for unramified PEL type Shimura varieties of type A.

Let $\shg\{w\}$ denote the nonbasic EO stratum that we are trying to show is `connected'. By Proposition 4.4 of <cit.>, it suffices to prove that the prime-to-$\Sigma$ Hecke operators coming from $G^{\text{sc}}$ act transitively on the fibers of

\begin{align}
\pi_0(\shg\{w\}) \to \pi_0(\shg),
\end{align}

where $\Sigma$ is a finite set of primes including $p$. There is a Hecke-equivariant and finite étale surjective map $\shge(w) \to \shg\{w\}$, so it
suffices to show the same statement for $\pi_0(\shge(w))=\pi_0(\shge(\le \! w))$. Proposition <ref> (see Remark <ref>) tells us that each connected component $\shge(\le \! w)$ intersects $\shge(\tau)$. The closure relations then give us a surjective map $\shge(\tau) \to \pi_0(\shge(\le \! w)$, hence it is enough to show that the prime-to-$\Sigma$ Hecke operators act transitively on the fibers of

\begin{align}
\shge(\tau) \to \pi_0(\shge)=\pi_0(\shg).
\end{align}

Rapoport-Zink uniformisation (Theorem <ref>) and the discussion in Section <ref> tells us that there is a commutative diagram

\begin{equation}
\begin{tikzcd}
\shge(\tau) \arrow[r, "\sim"] \arrow{d} & I(\mathbb{Q}) \backslash G(\afp) \times \frac{J_b(\qp)}{J_{b, \emptyset}(\zp)}/U^p \arrow[d, "\beta"] \\
\pi_0(\shge) \arrow[r, "\sim"] & I(\mathbb{Q}) \backslash \frac{G(\afp)}{G^{\text{sc}}(\afp)} \times \frac{J_b(\qp)}{J_{b, \emptyset}(\zp) J^{\text{sc}}_b(\qp)}/U^p.
\end{tikzcd}
\end{equation}

Weak approximation (Theorem 7.8 of <cit.>) tells us that $I^{\text{sc}}(\mathbb{Q})$ is dense in

\begin{align}
J_b^{\text{sc}}(\qp) \times \prod_{\ell \in ( \Sigma \setminus \{p\})} G^{\text{sc}}(\mathbb{Q}_{\ell}),
\end{align}

which means that $G^{\text{sc}}(\mathbb{A}_f^{\Sigma})$ acts transitively on the fibers of $\beta$.

Theorem <ref> follows from Theorem <ref>, because it can be checked on connected components of Shimura varieties. To be precise, EKOR strata on abelian type Shimura varieties are constructed from the EKOR strata on a single connected component of an auxiliary Hodge type Shimura variety, see Section 5.4 of <cit.>.

Theorem <ref> in the (T1) and (T2) cases follows by the following chain of reasoning: As in <cit.>, it suffices to show that the $\tau$-version of Conjecture <ref> holds for an auxiliary Hodge type Shimura datum, using Corollary <ref>, Remark <ref> and Lemma 4.6.13 of <cit.>. We can take this auxiliary Hodge type datum to be a product of quasi-simple groups that all satisfy the assumptions of Theorem <ref> by Lemma 4.6.22 of <cit.>. [To see that we can choose the parahoric to be connected in type $D^{\mathbb{H}}$-cases, use the argument in the proof of part (5) of Theorem 4.6.23 of op. cit. which relies on the assumption (P1)].

Theorem <ref> tells us that the $\tau$-version of Conjecture <ref> holds for each of these groups, and it is not hard to see that this implies that it holds for their product.

In the (T3) case, we first reduce from the abelian type to the Hodge type case as above. In our situation, these Hodge type Shimura varieties can be chosen to be of PEL type by Appendix B of <cit.> in combination with Proposition 1.4 of <cit.>, and we consider the Rapoport-Zink integral models. These are flat and normal by the main theorem of <cit.>, and they come with tautological closed embeddings (for sufficiently small level away from $p$) into Siegel modular varieties of parahoric level. It follows that they are isomorphic to the (normalisation) of the Zariski closure of their generic fibre in the Siegel modular variety, and one can argue as in Section 7 of <cit.> that these models are isomorphic to the Kisin-Pappas integral models.

It follows from Proposition 4.4 of <cit.> that the $\ovfp$ points of Rapoport-Zink spaces of parahoric level agree with the $\ovfp$-points of the corresponding affine Deligne-Lusztig variety. It follows from the moduli description (c.f. Section 6 of <cit.>) that we can produce maps from the set of $\ovfp$ points of our Rapoport-Zink space into the set of $\ovfp$-points of our Shimura variety. To be precise, assumption 6.18 of <cit.> is satisfied and then Proposition 9.1.(i) gives us uniformisation of isogeny classes.

§ ISOGENY CLASSES IN SHIMURA VARIETIES WITH VERY SPECIAL LEVEL STRUCTURE, BY RONG ZHOU

The aim of this Appendix is to prove some results concerning the structure of isogeny classes on integral models of Shimura varieties. In <cit.>, we showed these results would follow from the existence of a natural map from a certain union of affine Deligne–Lusztig varieties into the special fiber of $\scrS_{\rmK}(G,X)$ (cf. <cit.>). Moreover, such a map was shown to exist under some restrictive hypothesis on the group $G$, namely that the group at $p$ was residually split. In this Appendix, we show the existence of such a map for general quasi-split groups at $p$, but with the restriction that the level structure at $p$ is a very special parahoric subgroup. For this we follow the method in <cit.>.

§.§ Connected components of affine Deligne-Lusztig varieties
In this section obtain a bound on the set of connected components of certain affine Deligne–Lusztig varieties.

We follow the notation of section <ref>. Thus $G$ is a reductive group over $\bbQ_p$ and $\{\mu\}$ is a geometric conjugacy class of cocharacters of $G_{\overline{\bbQ}_p}$. We assume that $G$ is quasi-split and we let $\calI$ be the Iwahori group scheme corresponding to a $\sigma$-stable alcove $\fka$ in the building for $G$. We fix $\calG$ a very special standard parahoric group scheme for $G$. Then $\calG$ corresponds to a $\sigma$-stable special point $\fks$ lying in the closure of $\fka$ and we write $K\subset \bbS$ for the subset of simple affine reflections  which preserve $\fks$. The projection $\tilde W\rightarrow W_0$ induces an isomorphism $W_K\cong W_0$.

As explained in <cit.>, we have an identification
$$W_K\backslash\tilde W/W_K\cong X_*(T)^+_I.$$

By <cit.>, there exists a reduced root system $\Sigma$ (the échelonnage root system) such that
$$W_a\cong W(\Sigma)\ltimes Q^\vee(\Sigma),$$
where $W(\Sigma)$ (resp. $Q^\vee(\Sigma)$) is the Weyl group (resp. coroot lattice) of $\Sigma$. We define a partial order $\preccurlyeq$ on $X_*(T)^+_I$ by setting $\lambda\preccurlyeq \lambda'$ if $\lambda-\lambda'$  can be written as a sum of positive coroots in $Q^\vee(\Sigma)$ with positive integral coefficients. Then by <cit.>, the Bruhat order on $W_K\backslash\tilde W/W_K$ agrees with the partial order $\preccurlyeq$. It follows that for $\mu\in X_*(T)_I^+$, we have
$$\Adm(\mu)_K=\{\lambda\in X_*(T)^+_I| \lambda\preccurlyeq \mu\}.$$

Let $b\in G(\brQ)$ such that $[b]\in B(G,\mu)$. We have the affine Deligne–Lusztig variety $X(\mu,b)_K$ defined in 2.1.4. We also set
$$\Adm(\mu)^K:=W_K\Adm(\mu)W_K\subset \tilde W$$
and define
$$X(\mu,b)^K:=\bigcup_{w\in \Adm(\mu)^K}X_w(b)$$

which is a locally closed subscheme of the Witt vector affine flag variety $\mathrm{Gr}_{\calI}$. Then there is a natural map
\begin{equation}\label{eqn: fibration ADLV}X(\mu,b)^K\rightarrow X(\mu,b)_K\end{equation}
which is equivariant for the action of the $\sigma$-centralizer group $J_b(\brQ)$. In fact, (<ref>) is a fibration with connected fibers and hence induces a $J_b(\brQ)$-equivariant  bijection
\begin{equation}\label{eqn: fibration ADLV 2}\pi_0(X(\mu,b)^K)\xrightarrow{\sim} \pi_0(X(\mu,b)_K).\end{equation}

Let $M$ denote the centraliser of $\overline\nu_{[b]}$ and we fix a representative $b$ of $[b]$ such that $\nu_{[b]}=\overline\nu_{[b]}$. The existence of such a representative follows from the same argument as in <cit.> which also shows that $b\in M(\brQ)$. Then $b$ is a basic element of $M$, in other words $\overline{\nu}_{[b]}$ is central in $M$.

We use a subscript $M$ to denote the corresponding objects for $M$. Thus $\tilde W_M$ (resp. $W_{a,M}$) denotes the Iwahori Weyl group (resp. affine Weyl group) for $M$. The intersection $M(\brQ)\cap\calG(\brZ)$ arises as the $\brZ$-points of a very special parahoric group scheme $\calM$ for $M$, which is standard for the alcove $\fka_M$ for $M$ determined by $\fka$. We write $\calI_M$ for the Iwahori group scheme of $M$ determined by $\fka_M$ and we let  $K_M\subset \bbS_M$ denote the subset of simple affine reflections for $M$ corresponding to $\calM$.
We let $\Sigma_M$ denote the échelonnage root system for $M$ so that
$$W_{a,M}\cong W(\Sigma_M)\ltimes Q^\vee(\Sigma_M).$$

For $x\in \pi_1(M)_I$, we write $\tau_x\in \Omega_M$ for the corresponding length 0 element and we let $\tau_x=t^{\mu_x}w_x$ for $w_x\in W_K$. Then the map $x\mapsto \mu_x$ induces a bijection
\begin{equation}\label{eqn: I set}\pi_1(M)_I\cong \{\lambda\in X_*(T)_I|\text{$\lambda$ is $M$-dominant and $M$-minuscule}\},\end{equation}
here $M$-minuscule means minuscule with respect to the root system $\Sigma_M$. We define the set
$$I_{\mu,b,M}:=\{x\in \pi_1(M)_I| \kappa_M(b)=x, \mu_x\preccurlyeq\mu\}.$$

Via the bijection (<ref>), we also consider $I_{\mu,b,M}$ as a subset of the set of $M$-minuscule and $M$-dominant elements in $X_*(T)_I$. For each $\lambda\in I_{\mu,b,M}$, we have the affine Deligne–Lustig variety
$$X^M(\lambda,b)_{K_M}:=\{m\in M(\brQ)/\calM(\brZ)| m^{-1}b\sigma(m)\in \calM(\brZ)\dot t^{\lambda'}\calM(\brZ), \lambda'\preccurlyeq_M\lambda\}$$

for the group $M$. It is equipped with a natural map
\begin{equation}\label{eqn: ADLV Levi}X^M(\lambda,b)_{K_M} \rightarrow X(\mu,b)_K\end{equation}

which is equivariant for the action of the $\sigma$-centralizer group $J_b(\bbQ_p)$.

Our main theorem on the connected components of affine Deligne–Lusztig varieties is the following.

$J_b(\bbQ_p)$ acts transitively on $\pi_0(X(\mu,b)_K)$. In particular, for any $\lambda\in I_{\mu,b,M}$ the map (<ref>) induces a surjection
$$\pi_0(X^M(\lambda,b)_{K_M})\rightarrow \pi_0(X(\mu,b)_K).$$

The theorem is stated for  $G$ a quasi-split reductive group over $\bbQ_p$. However the result makes sense  for general quasi-split groups over any local field $F$ and can be proved in exactly the same way.

We follow the strategy of <cit.> and <cit.> where this result was proved for $G$-unramified. The result follows from the following two Propositions.

The natural map
$$\coprod_{\lambda\in I_{\mu,b,M}} X^M(\lambda,b)_{K_M}\rightarrow X(\mu,b)_K$$
induces a surjection
$$\coprod_{\lambda\in I_{\mu,b,M}} \pi_0(X^M(\lambda,b)_{K_M})\rightarrow \pi_0(X(\mu,b)_K).$$

Let $\lambda\in I_{\mu,b,M}$. The image of the natural  map
$$\pi_0(X^M(\lambda,b)_{K_M})\rightarrow \pi_0(X(\mu,b)_K)$$
does not depend on the choice of $\lambda\in I_{\mu,b,M}$.

Fix $\lambda \in I_{\mu,b,M}$. By Proposition <ref> and Proposition <ref>, the map
$$\pi_0(X^M(\lambda,b)_{K_M})\rightarrow \pi_0(X(\mu,b)_K)$$
is surjective.	By <cit.>, $J_b(\bbQ_p)$ acts transitively on $\pi_0(X^M(\lambda,b)_{K_M})$, and hence on $\pi_0(X(\mu,b)_K)$.

We now proceed to prove the two Propositions. Note that by a standard reduction (see <cit.>), it suffices to prove the Proposition when $G$ is adjoint and $\bbQ_p$-simple. We may and do assume this from now on.

§.§ Proof of Proposition <ref>

In the case of unramified groups, the Proposition is <cit.>. Here we prove the general case using a different method based on the Deligne–Lusztig reduction method for affine Deligne–Lusztig varieties in the affine flag variety.

We begin with some preliminaries regarding $\sigma$-conjugacy classes in Iwahori Weyl groups. For any element $w\in \tilde W$ we let $n$ be a sufficiently divisible integer such that $\sigma^n$ acts trivially on $\tilde W $ and $w\sigma(w)\dotsc\sigma^{n-1}(w)=t^\lambda$ for some $\lambda\in X_*(T)_I$. We set $\nu_w:=\frac{t^\lambda}{n}\in X_*(T)_{I,\bbQ}$ and $\overline{\nu}_w\in X_*(T)_{I,\bbQ}^+$ for the dominant representative of $\nu_w$. We let $\kappa(w)\in \pi_1(G)_\Gamma$ denote the image of $w$ under the projection $\tilde{W}\rightarrow \pi_1(G)_I\rightarrow\pi_1(G)_\Gamma$. We write $B(\tilde W,\sigma)$ for the set of $\sigma$-conjugacy classes in $\tilde W$. Then $w\mapsto [\dot{w}]$ induces a well-defined map $\Psi:B(\tilde W,\sigma)\rightarrow B(G)$ and we have a commutative diagram:

\[\xymatrix{B(\tilde W,\s)\ar[rr]^\Psi \ar[rd]_{(\overline\nu,\kappa)}&& B(G)\ar[ld]^{(\overline\nu,\kappa)}\\
& (X_*(T)_{I,\mathbb{Q}}^+)^\sigma\times \pi_1(G)_{\Gamma}&.}\]

We will need the following Lemma. We write $J\subset K\cong\bbS_0$ for the subset corresponding to $M$. Recall that an element $w\in \tilde W$ is said to be $\sigma$-straight if
for all $n$.
Let $w\in \tilde W$ be $\sigma$-straight such that $\dot w\in[b]$; in particular $\overline\nu_b=\overline \nu_w$. Let $u\in {^JW_0}$ such that $u(\nu_w)=\overline{\nu}_b$ and set $w_\sharp:= uw\sigma(u)^{-1}$. Then $w_\sharp\in \Omega_M$.

It suffices to show that $w_\sharp\in W_M$ and

The first statement follows since $w_\sharp(\overline\nu_w)=\overline\nu_w$.
By <cit.>, the element $w$ is $(\nu_w,\sigma)$-fundamental. Thus
where $M_{\nu_w}$ is the centralizer of $\nu_w$ and  $\calI_{M_{\nu_w}}(\brZ):=\calI(\brZ)\cap M_{\nu_w}(\brQ)$. Since $u\in {^JW_0}$, we have $u\calI_{M_{\nu_w}}(\brZ)u^{-1}=\calI_M(\brZ)$. It follows that
\begin{align*}w_\sharp\calI_M(\brZ)w_\sharp^{-1} &= w_\sharp\sigma(\calI_M(\brZ))w_\sharp^{-1}\\
&=uw \sigma(\calI_{M_{\nu_w}}(\brZ))w^{-1}u^{-1}\\
\end{align*}

as desired.

By (<ref>), it suffices to show the natural map
$$\coprod_{\lambda\in I_{\mu,b,M}} X^M(\lambda,b)^{K_M}\rightarrow X(\mu,b)^K$$
induces a surjection
$$\coprod_{\lambda\in I_{\mu,b,M}} \pi_0(X^M(\lambda,b)^{K_M})\rightarrow \pi_0(X(\mu,b)^K).$$

Let $Y$ be a connected component of $X(\mu,b)^K$. Then by <cit.>, there exists a $\sigma$-straight element $w\in\Adm(\mu)^K$, such that $Y\cap X_w(b)\neq \emptyset$. Let $w_\sharp$ denote the element constructed in Lemma <ref> and $u\in {^JW_0}$ such that $uw\sigma(u)=w_\sharp$. Then we claim that $[b]_M= [\dot w_\sharp]_M\in B(M)$. Indeed we have $\nu_{ w_\sharp}=\overline\nu_w=\overline \nu_b$. Therefore the image of $[b]_M$ and $[\dot w_\sharp]_M$ in $\pi_1(M)_I$ coincide up to torsion. On the other hand, the image of $[b]_M$ and $[\dot w_\sharp]_M$ in $\pi_1(G)_I$ coincide and we have $\ker(\pi_1(M)_I\rightarrow \pi_1(G)_I)$ is torsion free. It follows that $\kappa_M([b]_M)=\kappa_M([\dot w_\sharp]_M)$ and hence $[b]_M=[\dot w_\sharp]_M$. Thus we may replace $b$ by $\dot w_\sharp$.

We will show that $Y\cap X_{w_\sharp}^M(\dot w_\sharp)\neq \emptyset$. Since $w_\sharp\in\Adm^M(\lambda)^{K_M}$, where $\lambda\in I_{\mu,b,M}$ corresponds to the image of $w_\sharp$ in $\pi_1(M)_I$, it follows that $X_{w_\sharp}(\dot w_\sharp)\subset X^M(\lambda,\dot w_\sharp)^{K_M}$;  this implies the Proposition.

For any affine root $\alpha$, we let $\calU_{-\alpha_i}$ denote the affine root subgroup corresponding to $-\alpha_i$ over $\brZ$. By <cit.>, $\calU_{-\alpha_i}$ is the group scheme associated to a finite free $\brZ$-module. For any $\epsilon>0$, we let $\calU_{-\alpha_i+\epsilon}(\brZ)$ be the subgroup of $\calU_{-\alpha_i}(\brZ)$ corresponding to the affine function $-\alpha+\epsilon$. Similarly we write $\calU_{-\alpha_i+}(\brZ)$  for the union of $\calU_{-\alpha_i+\epsilon}(\brZ)$ over all $\epsilon>0$. As the notation suggests, these  arise as the $\brZ$-points of group schemes $\calU_{-\alpha_i+\epsilon}$ and $\calU_{-\alpha_i+\epsilon}$ over $\brZ$, and the quotient $\calU_{-\alpha_i}(\brZ)/\calU_{-\alpha_i+}(\brZ)$ is a 1-dimensional vector space over $ k$. We choose a group scheme homomorphism
$$f:\bbG_a\rightarrow \calU_{-\alpha}$$
which lifts the map $k\cong \calU_{-\alpha_i}(\brZ)/\calU_{-\alpha_i+\epsilon}(\brZ)$.

For $R$ a perfect $k$-algebra and $a\in R$, the map
$$h_\alpha:R\rightarrow \calU_{-\alpha}(W(R))$$

$$a\mapsto f([a])$$
where $[a]\in W(R)$ is the Teichmüller lift of $a$, induces a $ k$-scheme morphism
$$h_\alpha:\bbA^{1,p^{-\infty}}\rightarrow L^+\calU_{-\alpha},$$

where $\bbA^{1,p^{-\infty}}$ denotes the perfection of $\bbA^1$ over $k$. The induced morphism $\bbA^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calI}$ extends to a morphism $\bbP^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calI}$ also denoted $h_\alpha$.

Let $g_0\calI\in Y\cap X_w(\dot w_\sharp)$ with $g_0\in G(\brQ)$. By <cit.>, we may choose $g_0$ such that $g_0^{-1}\dot w_\sharp g_0= \dot{w}$. Let $s_n\dotsc s_1$ be a reduced word decomposition for $u$ (note that $s_i\in K$) and we write $u_i$ for the element $s_1\dotsc s_i\in \tilde W$ and $u_0=e$. We write $g_i\in G(\brQ)$for the element $g_0\dot u_i$. We will prove by induction that $g_i\calI\in Y$ for $i=0,1,\dotsc,n$; clearly this is true for $i=0$.

Assume $g_i\in Y$ and we let $\alpha_{i+1}$ denote the positive affine root corresponding to $s_{i+1}$. We consider the map
$$g:=g_ih_{\alpha_{i+1}}:\bbP^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calI}$$

A simple calculation shows that
$$g(s)^{-1}\dot{w}^{\sharp} \sigma(g(s))\in \bigcup_{v\in\Adm(\mu)^K} \calI\dot{v}\calI$$

where $s\in \bbP^{1,p^{-\infty}}(k)$. Moreover we have $g(0)=g_i\calI$ and $g(\infty)=g_{i+1}\calI$. Thus the image of $g$ is a curve which connects $g_i\calI$ and $g_{i+1}\calI$ and hence $g_{i+1}\calI\in Y$ as desired. A simple calculation shows that $g_n\calI$ lies in the image  of $X^M_{w_\sharp}({\dot{w}_\sharp})$ as desired.

§.§ Proof of Proposition <ref>

When $G$ is unramified, this Proposition follows from the proof of <cit.> when $\mu$ is minuscule; the general case is proved in <cit.>. The main input is the construction of explicit curves in $X(\mu,b)_K$ which connect points in $X^M(\lambda,b)_{K_M}$ and $X^M(\lambda',b)_{K_M}$ for $\lambda\neq\lambda'\in I_{\mu,b,M}$. The construction of these curves relied on certain combinatorial results concerning the root system for $G$. The exact same method of proof works in our setting; however there are a few subtleties which we now explain.

First, the explicit curves were constructed in <cit.> and <cit.> using root subgroups of $G_{\brQ}$ which are all isomorphic to $\bbG_a$ when the group is unramified. In general, the root subgroups are more complicated thus one needs to be more careful. However we are still able to give a uniform construction of the curves that we need.

Secondly, we need to generalize the combinatorial results to general quasi-split groups $G$. It turns out there is a systematic way to deduce these combinatorial results for quasi-split $G$ from the case of unramified groups which we now explain.

Recall we have assumed $G$ is adjoint and simple. As in <cit.>, there is an unramified adjoint group $H$ over $F$ such that the pair $(W',\sigma')$ consisting of the Iwahori Weyl group for $G'$ and the action of Frobenius is identified with the pair $(W,\sigma)$. Moreover the échelonnage root system $\Sigma$ is identified with the absolute root system $\Sigma'$ for $G'$, and we have an isomorphism

$$X_*(T)_I\cong X_*(T'),$$
where $T'$ is a suitable maximal $\brQ$-split torus of $G'$.

We use a superscript $'$ to denote the corresponding objects for $G'$. Then $\calG$ determines a hyperspecial subgroup $\calG'$ for $G'$ and we write $K'\subset \bbS' $ for the corresponding subset of simple reflections. Then $M$ determines a Levi subgroup $M'$ of $G'$ and hence a subset $J'\subset K'$. It follows that the combinatorial data
\begin{equation}\label{eqn: comb data}(\Sigma,X_*(T)_I,\sigma, J,\mu,\kappa_M(b))\end{equation}
is identified with the corresponding data for $G'$. Thus any result which only depends on the data (<ref>), can be reduced to the case of unramified groups. The combinatorial results that we need are already proved in the case of unramified groups in <cit.> and <cit.>. We therefore take the convention that whenever we need certain results which depend on the data (<ref>), we will refer to the relevant result in <cit.> or <cit.>.

We now proceed with the proof of Proposition <ref>. Let $x,x'\in \pi_1(M)_I$. We write $x \overset{(\alpha,r)}\rightarrow x'$ for some $\alpha\in \Sigma$ and $r\in\bbN$ if $x-x'=\alpha^\vee-\sigma^r(\alpha^\vee)$ and
$$\mu_x, \mu_{x+\alpha^\vee}, \mu_{x-\sigma^r(\alpha^\vee)},\mu_{x'}\preccurlyeq \mu.$$
We write $x\overset{(\alpha,r)}\rightarrowtail x'$ if $x \overset{(\alpha,r)}\rightarrow x'$ and neither

$$x \overset{(\alpha,i)}\rightarrow x+\alpha^\vee-\sigma^i(\alpha^\vee) \overset{(\sigma^{i}(\alpha),r-i)}\rightarrow x'$$

$$x \overset{(\sigma^i(\alpha),r-i)}\rightarrow x+\sigma^i(\alpha^\vee)-\sigma^r(\alpha^\vee) \overset{(\alpha,i)}\rightarrow x'$$
for any $i\in[1,r-1]$.

We let
$$\langle\ ,\ \rangle: Q(\Sigma)\otimes_{\bbZ}\bbR\times X_*(T)_I\otimes_{\bbZ}\bbR\rightarrow \bbR$$
be the natural pairing, where $Q(\Sigma)$ is the root lattice of $\Sigma$. For any element $\alpha\in \Sigma$, we write $\calO_\alpha$ for the $\sigma$-orbit of $\alpha$. We let $h$ denote the number of connected components of the Dynkin diagram of $G$ over $\brQ$; then we have  $\#\calO_\alpha\in\{h,2h,3h\}$.

[<cit.>]Let $x\neq x'\in I_{\mu,b,M}$. Then there exists $x_j\in \pi_1(M)_I$, $\alpha_j\in \Sigma-\Sigma_M$ and $r_j\in\bbN$ for $j\in[0,m-1]$ such that

* $\alpha_j^\vee$ is $M$-dominant and $M$-minuscule.

* $r_j\in [1,h]$ if $\#\calO_{\alpha_j}\in\{h,2h\}$ and $r_j\in [1,2h-1]$ if $\#\calO_{\alpha_j}=3h$.

* $x_0=x, x_m=x'$ and we have $x_j \overset{(\alpha_j,r_j)}\rightarrow x_{j+1}$ for $j\in[0,m-1]$.

This follows from <cit.> by the above discussion. Note that in loc. cit., the result is stated for  $M$ a Levi subgroup such that $b$ is superbasic in $M$. However, one checks that the same proof works for any $M$ as long as $I_{\mu,b,M}$ contains a weakly dominant element. Here $\lambda\in X_*(T)_I$ is weakly dominant if $\langle \alpha, \lambda\rangle\geq-1$ for any positive root $\alpha\in \Sigma$. But as in <cit.>, any element $\lambda\in I_{\mu,b,M}$ is weakly dominant, so the result applies to our $M$.

We now construct certain curves inside $LG$ which we will use to connect points in $X(\mu,b)_K$.
Let $\alpha\in \Sigma$ be a  root. Then $\alpha$ determines a relative root $\widetilde{\alpha}$ of $G$ over $\brQ$ which we always take to be the short root. We let $G_\alpha$ denote the simply connected cover of the (semi-simple) group generated by $U_{\widetilde\alpha}$ and $U_{-\widetilde\alpha}$ and we write
$$i_\alpha:G_\alpha\rightarrow G$$
for the natural map. We let $\calG_\alpha$ denote the very special parahoric of $G_\alpha$  such that $\calG(\brZ):=G_\alpha(\brQ)\cap  i_\alpha^{-1}(\calG(\brZ))$.

If $\widetilde\alpha$ is not divisible then we have an isomorphism

$$G_\alpha\cong \mathrm{Res}_{\breve K/\brQ}SL_2,$$
where $\breve K/\brQ$ is a finite extension. Then up to conjugacy $\calG_\alpha$ is identified with the very special parahoric $SL_2(\calO_{\breve K})\subset G_\alpha(\brQ)$ and there is an isomorphism
$$f_{\widetilde\alpha}:\mathrm{Res}_{\breve K/\brQ}\bbG_a\xrightarrow{\sim}U_{\widetilde\alpha}.$$

If $\widetilde\alpha$ is divisible, then there is an isomorphism

\begin{equation}G_\alpha\cong \mathrm{Res}_{\breve K/\brQ}SU_3,\end{equation}
where $SU_3$ is the special unitary group over $\breve K$ associated to a quadratic extension $\breve K'/\breve K$.

We recall the presentation of the $\breve K$-group $\SU_3$ in <cit.>.

We let $\tau\in \mathrm{Gal}(\breve K'/\breve K)$ denote the nontrivial element and we consider the Hermitian form on $\breve K'^3$ given by

The group $\SU_3$  is the special unitary group attached to this form. For $i=-1,1$ and $c,d\in \breve K'$ such that $\tau(c)c+d+\tau(d)=0$, we define
where $I_3$ is the identity matrix and $(g_{rs})$ is the matrix with entries $g_{-i,0}=-\tau(c)$, $g_{0,i}=c$, $g_{-i,i}=d$ and $g_{rs}=0$ otherwise. The root subgroups are then given by
$$U_{\pm\widetilde{\alpha}}(\breve  K)=\{u_{\pm 1}(c,d)|c,d\in \breve K', \tau(c)c+\tau(d)+d=0\}$$

$$U_{\pm2\widetilde{\alpha}}(\breve K)=\{u_{\pm 1}(0,d)|c,d\in \breve K', \tau(d)+d=0 \}.$$

We consider the very special parahoric $G_{\alpha}(\breve F)\cap GL_3(\calO_{\breve K'})$ of $G_\alpha(\breve F)$; we call this the standard parahoric. Let $\pi\in\breve K'$ be a uniformizer such that $\tau(\pi)=-\pi$ and let  $s\in GL_3(\breve K')$ denote the element $\mathrm{diag}(\pi,1,1)$. Then the subgroup of $G_\alpha(\brQ)$ defined by
$$G_\alpha(\brQ)\cap s\breve GL_3(\calO_{\breve K'}) s^{-1}$$

is a very special parahoric subgroup of $G_\alpha(\brQ)$ which we shall call the non-standard parahoric. Up to conjugacy, these are the only very special parahorics of $SU_3$.

For $\alpha\in \Sigma$, we define a map $u_\alpha:\bbA^{1,p^{-\infty}}\rightarrow LU_{\widetilde{\alpha}}\subset LG$ as follows. Let $R$ be a perfect ring of characteristic $p$ and $a\in R$ will denote an arbitrary element. We consider the following three separate cases.

* $G_\alpha\cong \mathrm{Res}_{\breve K/\brQ}SL_2$ and $\calG_\alpha(\brZ)=SL_2(\calO_{\breve K})$.

We define $u_\alpha$ to be the map induced by
$$a\mapsto i_\alpha(f_{\tilde\alpha}(\pi^{-1}[a])).$$

* $G_\alpha\cong \mathrm{Res}_{\breve K/\brQ}SU_3$ and $\calG_\alpha$ is the standard parahoric subgroup.

We define $u_\alpha$ to be the map
$$a\mapsto i_\alpha(u_{1}(0,\pi^{-1}[a])).$$

* $G_\alpha\cong \mathrm{Res}_{\breve K/\brQ}SU_3$ and $\calG_\alpha$ is the  non-standard parahoric subgroup.

We define $u_\alpha$ to be the map
$$a\mapsto i_\alpha(u_1([a],\frac{[a]^2}{2})).$$

A  calculation  using the presentations of $SL_2$ or $SU_3$ above gives the following lemma (cf. <cit.>).

* Let $\lambda,\delta\in X_*(T)_I$ and $\alpha,\beta\in\Sigma$ such that $Q(\Sigma)\cap(\bbZ\alpha+\bbZ\beta)$ is of type $A_2$, $A_1\times A_1$ or $A_1$ and such that
$$\delta,\delta+\alpha^\vee,\delta-\beta^\vee,\delta+\alpha^\vee-\beta^\vee\preccurlyeq \lambda.$$

Then for all $y,z\in k$, we have
$$u_\alpha(z) \dot{t}^\delta u_\beta(y)\in \bigcup_{\lambda'\preccurlyeq\lambda}L^+\calG\dot t^{\lambda'} L^+\calG.$$

* Let $\alpha,\beta\in \Sigma$ and $\lambda\in X_*(T)_I$ such that $\langle\alpha,\beta^\vee\rangle=\langle\beta,\alpha^\vee\rangle=-1$ and $\langle\alpha,\lambda\rangle\geq 2$. Then for any $y,z\in k$ we have
$$u_\beta(z)(\dot t^\lambda u_\alpha(y)\dot t^{-\lambda})u_{\beta}(-z)\in L^+\calG$$

The following Lemma  is the analogue of <cit.>.

Let $x,x'\in \pi_1(M)_I$, $\alpha\in\Sigma-\Sigma_M$ and $r\in \bbN$ such that

* $\alpha^\vee$ is $M$-dominant and $M$-minuscule.

* $r\in [1,h]$ if $\#\calO_\alpha\in\{h,2h\}$ and $r\in [1,2h-1]$ if $\#\calO_\alpha=3h$.

* $x \overset{(\alpha,r)}\rightarrow x'$.

Then for any $P\in X^M(\mu_x,b)_{K_M}$, there exists $P'\in X^M(\mu_{x'},b)_{K_M}$ such that $P$ and $P'$ lie in the same connected component of $X(\mu,b)_K$ and we have

As in <cit.>, we may assume that $x\overset{(\alpha,r)}\rightarrowtail x'$. Moreover, arguing as in <cit.>, it suffices to show that there exists $P\in X^M(\mu_x,b)_{K_M}$ and $P'\in X^M(\mu_{x'},b)_{K_M}$ such that $P$ and $P'$ lie in the same connected component of $X(\mu,b)_K$ and we have $\kappa_M(P)-\kappa_M(P')=\sum_{i=0}^{r-1}\sigma^i(\alpha^\vee)\in\pi_1(M)_I.$

Let $b_x=\dot t^{\mu_x}\dot w_x$; then $b_x$ is basic in $M$ and since $\kappa_M(b)=\kappa_M(b_x)\in \pi_1(M)_\Gamma$, there exists $g_x\in M(\breve F)$ such that $g_x^{-1}b\sigma(g_x)=b_x$. We define $P:=g_xL^+\calM$ so that $P\in X^M(\mu_x,b)_{K_M}$.

We first consider the case $r\in [1,h]$. For an element $g\in LG$, we write ${^{b_x\sigma}g}$ for the element $b_x\sigma(g)b_x^{-1}$. We define a map $u:\bbA^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$  given by
$$u(z)= g_xu_{\alpha}(z){^{b_x\sigma}u_{\alpha}(z)}\dotsc {^{(b_x\sigma)^{r-1}}u_{\alpha}(z)}L^+\calG$$

Then by ind-projectivity of $\mathrm{Gr}_{\calG}$, $u$ extends to a map $g:\bbP^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$. As in <cit.>, for any $z\in k$ we have

\begin{align*}
g(z)^{-1}b\sigma(g(z))&\in L^+\calG u_\alpha(-z)b_x\sigma\ {^{(b_x\sigma)^{r-1}}u_{\alpha}(z)}L^+\calG\\
&=L^+\calG u_\alpha(-z)\dot{t}^{\mu_x}u_{w_x\sigma^r(\alpha)}(c\sigma^r(z)) L^+\calG
\end{align*}

for some $c\in k^\times$.	Here we use  <cit.>  which shows that $w_x\sigma^{i}(\alpha)=\sigma^{i}(\alpha)$ and $\langle\sigma^i(\alpha),\mu_x\rangle=0$ for $i\in [1,r-1]$. By <cit.>, we have
Thus by Lemma <ref> (1), we have
$$g(z)^{-1}b\sigma(g(z))\in \bigcup_{\mu'\preccurlyeq \mu}L^+\calG\dot t^{\mu'} L^+\calG$$
and hence $g$ factors through $X(\mu,b)_K$. Moreover one computes that
$$P':=g(\infty)=g_x\dot t^{-\sum_{i=0}^{r-1}\sigma^i(\alpha^\vee)}L^+\calG,$$
which lies in the image of $X^M(\mu_{x'},b)_{K_M}$.

We now consider the case $r\in [h+1,2h-1]$. In this case $\#\calO_\alpha=3h$ and each connected component of the Dynkin diagram of $G$ over $\brQ$ is of type $D_4$. Then either $J=\emptyset$ or $J=\calO_\beta$ where $\beta$ is the unique root in $\Sigma$ with $\sigma^h(\beta)$ and such that $\beta,\alpha$ lie in the same connected component of $\Sigma$. We consider the following two cases.

Case (i): Either $\langle\beta,\mu_x\rangle=0$ or $\langle\beta,\alpha^\vee\rangle=0$. Then as in <cit.>, we have $\langle\sigma^j(\alpha),\mu_x\rangle=0$ and $w_x(\sigma^j(\alpha))=\sigma^j(\alpha)$ for $j\in[1,r-1]$. Then we may define $u:\bbA^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$ by
$$u(z)= g_xu_{\alpha}(z){^{b_x\sigma}u_{\alpha}(z)}\dotsc {^{(b_x\sigma)^{r-1}}u_{\alpha}(z)}L^+\calG$$
as above. Then $u$ extends to $g:\bbP^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$ and the same computation shows that $g$ is a curve connecting $P=g(0)\in X^M(\mu_x,b)$ and $P':=g(\infty)=g_x\dot t^{-\sum_{i=0}^{r-1}\sigma^i(\alpha^\vee)}L^+\calG\in X^M(\mu_{x'},b)$.

Case (ii): $\langle\beta,\alpha^\vee\rangle=-1$ and $\langle\beta,\mu_x\rangle=1$. Then by <cit.>, upon switching the roles of $x$ and $x'$ if necessary, we may assume that
$$\langle \sigma^r(\beta),\mu_x\rangle=\langle\sigma^{r-h}(\alpha),\mu_x\rangle=\langle\sigma^h(\alpha),\mu_x\rangle=0.$$

We define $u:\bbA^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$ by
$$u(z)= g_x{^{(b_x\sigma)^{r-1}}u_{\alpha}(z)}{^{(b_x\sigma)^{r-2}}u_{\alpha}(z)}\dotsc u_{\alpha}(z)L^+\calG$$

Then $u$ extends to $g:\bbP^{1,p^{-\infty}}\rightarrow \mathrm{Gr}_{\calG}$ and we have

\begin{align*}
g(z)^{-1}b\sigma(g(z))&\in L^+\calG u_{\sigma^{r-h}(\alpha)}(-c_2z)(\dot t^{\lambda}u_{\sigma^r(\alpha)+\sigma^r(\beta)}(c_1z)\dot t^{-\lambda} )u_{\sigma^{r-h}(\alpha)}(c_2z) u_\alpha(-z)b_xL^+\calG
\end{align*}
where $\lambda\in X_*(T)_I$ satisfies $\langle \sigma^r(\alpha)+\sigma^r(\beta),\lambda\rangle\geq2$.
By Lemma <ref> (1) we have
$$u_\alpha(-z)b_x\in \bigcup_{\mu'\preccurlyeq\mu} L^+\calG \dot{t}^{\mu'} L^+\calG$$
and by Lemma <ref> (2) we have
$$u_{\sigma^{r-h}(\alpha)}(-c_2z)(\dot t^{\lambda}u_{\sigma^r(\alpha)+\sigma^r(\beta)}(c_1z)\dot t^{-\lambda} )u_{\sigma^{r-h}(\alpha)}(c_2z).$$

It follows that

$$g(z)^{-1}b\sigma(g(z))\in \bigcup_{\mu'\preccurlyeq\mu} L^+\calG \dot{t}^{\mu'} L^+\calG$$
and hence $g$ factors through $X(\mu,b)_K$. A similar calculation to the above shows that $g$ is a curve connecting $P=g(0)\in X^M(\mu_x,b)$ and $P':=g(\infty)\in X^M(\mu_{x'},b)$ with $\kappa_M(P)-\kappa_M(P')=\sum_{i=0}^{r-1}\sigma^i(\alpha^\vee)$.

This follows by combining Lemma <ref> and Lemma <ref>.

§.§ Structure of isogeny classes

We will use Theorem <ref> to give a description of the isogeny classes in $\scrS_{\rmK}(G,X)$. We assume that $p>2$, $p\nmid |\pi_1(G_{\mathrm{der}})|$ and that $G_{\bbQ_p}$ is quasi-split and splits over a tamely ramified extension of $\bbQ_p$. We now follow the notation in section 3, so $(G,X)$ is a Shimura datum of Hodge type. We let $\rmK=\rmK_p\rmK^p$ where $\rmK^p\subset G(\bbA_f^p)$ is a compact open subgroup and $\rmK_p$ is a very special connected parahoric subgroup of $G(\bbQ_p)$; we write $\calG$ for the corresponding parahoric group scheme.

By construction, the integral model $\scrS_{\rmK}(G,X)$ is equipped with a map
$$\scrS_{\rmK}(G,X)\rightarrow \scrS_{\rmK'}(\mathrm{GSp}(V),S^\pm)$$
where $\scrS_{\rmK'}(\mathrm{GSp}(V),S^\pm)$ is an integral model for a suitable Siegel Shimura variety. Here $\rmK'=\rmK'_p\rmK'^p$ where $\rmK'_p\subset \mathrm{GSp}(V_{\bbZ_p})$ is the stabilizer of a lattice $V_{\bbZ_p}$ and $\rmK'^p\subset \mathrm{GSp}(V_{\bbA_f^p})$ is a sufficiently small compact open subgroup containing $\rmK^p$. Let $V_{\bbZ_{(p)}}:=V_{\bbZ_p}\cap V_{\bbQ}$ and we fix a set of tensors $s_\alpha \in V_{\bbZ_{(p)}}^\otimes$ whose stabilizer  $\calG_{\bbZ_{(p)}}$ satisfies $\calG_{\bbZ_{(p)}}\otimes_{\bbZ_{(p)}} \bbZ_p\cong\calG$.

Let $x\in \scrS_{\rmK}(G,X)(k)$. Then attached to $x$ is an abelian variety $\calA_x$ over $k$. We write $\bbD_x$ for the contravariant Dieudonné module associated to the $p$-divisible group $\scrG_x$ of $\calA_x$; then $\bbD_x$ is equipped with a corresponding set of tensors $s_{\alpha,0,x}$. Similarly, for $\ell \neq p$, the $\ell$-adic Tate modules $T_\ell\calA_x$ are equipped with tensors $s_{\alpha,\ell,x}\in T_{\ell}\calA_x^\otimes$.

By <cit.>, there is an isomorphism

$$V_{\bbZ{(p)}}\otimes_{\bbZ_{(p)}}\brZ\cong \bbD_x$$
taking $s_\alpha$ to $s_{\alpha,0,x}$.
Under this identification, the Frobenius on $\bbD_x$ is of the form $\varphi=b\sigma$ for some $b \in G(\brQ)$; then $b$ is well-defined up to $\sigma$-conjugation by $\G(\brZ)$.

We write $\{\mu_h^{-1}\}$ for the conjugacy class of the inverse of the Hodge cocharacter. We let $\mu'\in X_*(T)^+_I$ denote the image of a dominant representative of the conjugacy $\{\mu_h^{-1}\}$ and we define $\mu=\sigma(\mu')$. Then by the argument in <cit.>, we have
$$b\in \calG(\brZ)\dot{w}\calG(\brZ)$$
for some $w\in\Adm(\mu)_K$; it follows that $1\in X(\mu,b)$. As in <cit.>, there is a natural map
$$i_x':X(\mu,b)_K(k)\rightarrow \scrS_{\rmK'}(\mathrm{GSp}(V),S^\pm)(k)$$
defined using Dieudonné theory which sends $1$ to the image of $x$.

Let $r$ be the residue degree of the extension $E_v/\bbQ_p$. Then $X(\mu,b)_K$ is equipped with an action $\Phi$ given by $\Phi(g)=(b\sigma)^r(g)$.

Suppose $\rmK_p$ is a very special connected parahoric subgroup of $G(\bbQ_p)$. Then there exists a unique map
$$i_x:  X(\mu,b)_K(k)\rightarrow \scrS_{\rmK}(G,X)(k)$$
lifting $i'_x$ such that $s_{\alpha,0,i_x(g)}=s_{\alpha,0,x}$ and $\Phi\circ i_x=i_x\circ \Phi$, where $\Phi$ acts on $\scrS_{\rmK}(G,X)(k)$ via the geometric $r$-Frobenius.

For notational  simplicity we write $X(\mu,b)_K$ for $X(\mu,b)_K(k)$. The uniqueness and compatibility with $\Phi$  is proved in the same way as <cit.>. We may thus define $X(\mu,b)_K^\circ\subset X(\mu,b)_K$ as the maximal subset which admits such a lifting. We therefore need to show that  $X(\mu,b)_K^\circ=X(\mu,b)_K$. To do this we follow the strategy of <cit.>.

Arguing as in <cit.>, we have that $X(\mu,b)^\circ_K$ is a union of connected components. Note that the key input <cit.> needed for this can be proved verbatim in our setting.

It therefore suffices to show that the map
$$X(\mu,b)_K^\circ\rightarrow \pi_0(X(\mu,b)_K)$$
is a surjection.

Let $M\subset G_{\bbQ_p}$ be the standard Levi subgroup given by the centralizer of the Newton cocharacter $\overline\nu_b$. By Theorem <ref>, there exists $\lambda\in I_{\mu,b,M}$ and an element
$$g\in X(\mu,b)^\circ_K\cap X^M(\lambda,b)_M.$$

Upon replacing $x$ by $i_x(g)$ and using the diagram <cit.>, we may assume $b\in M(\brQ)$. Since $b$ is basic in $M$ and using <cit.>, we may further  assume that  $b=\dot\tau_\lambda$ where $\tau_\lambda\in\Omega_M$ corresponds to $\kappa_M(b)\in \pi_1(M)_I$.

Arguing as in <cit.>, we find that <cit.> is satisfied, in other words, the filtration on $\bbD_x\otimes _{\brZ}k$ lifts to a filtration on $\bbD_x\otimes_{\brZ}\brK$ for some $\brK/\brQ$ finite which is induced by an $M$-valued cocharacter $\mu_y$. We may therefore let $\tilde{\scrG}/\calO_{\brK'}$ be an $(M,\mu_y)$-adapted lifting of $\scrG$ (cf. <cit.>) which corresponds to a point $\tilde{x}\in\scrS_{\rmK}(G,X)(\calO_{\brK'})$. The construction in <cit.> gives us a map
$$\iota:M(\bbQ_p)/\calM(\bbZ_p)\rightarrow X^M(\lambda,b)_{K_M},\ \ g\mapsto g_0$$

which induces a surjection
$$M(\bbQ_p)/\calM(\bbZ_p)\rightarrow \pi_0(X^M(\lambda,b)_{K_M})$$
by <cit.>.
Moreover, the image of $\iota$ lands in $X(\mu,b)_{K}^\circ$. Therefore, by Theorem <ref>, $X(\mu,b)_{K}^\circ$ intersects every connected component of $X^M(\lambda,b)_{K_M}$, and hence  $X(\mu,b)_{K}^\circ\rightarrow \pi_0(X(\mu,b)_{K})$ is a surjection as desired.

The Proposition implies that <cit.> is satisfied. Hence we obtain the following Theorem (cf. <cit.>).

We let $\scrS_{\rmK_p}(G,X):=\varprojlim_{\rmK^p}\scrS_{\rmK_p\rmK^p}(G,X)$. Recall that $x,x'\in\scrS_{\rmK_p}(G,X)(k)$ are said to lie in the same isogeny class if there exists a quasi-isogeny $\calA_x\rightarrow\calA_{x'}$ such that induced maps $\bbD_{x'}\rightarrow\bbD_x$ and $T_{\ell}\calA_x\otimes\bbQ_\ell\rightarrow T_{\ell}\calA_{x'}\otimes\bbQ_\ell$ sends $s_{\alpha,0,x'}$ to $s_{\alpha,0,x}$ and $s_{\alpha,\ell,x}$ to $s_{\alpha,\ell,x'}$ for all $\ell\neq p$. For $x\in \scrS_{\rmK_p}(G,X)(k)$ we let $I_x$ denote the reductive $\bbQ$-group associated to $x$ as in <cit.>.

Let $p>2$ and $(G,X)$ a Shimura datum of Hodge type with $G_{\bbQ_p}$ tamely ramified and quasi split. We assume that $p\nmid|\pi_1(G_{\mathrm{der}})|$ and that $\rmK_p$ is a very special connected parahoric subgroup of $G(\bbQ_p)$.

* Let $x\in \scrS_{\rmK_p}(G,X)(k)$ and $b\in G(\brQ)$ the associated element. The  isogeny class of $x$ has the form

$$I_x(\bbQ) \backslash X(\mu,b)\times G(\bbA_f^p).$$

* Each isogeny class of $\scrS_{\rmK_p}(G,X)(k)$ contains a point $x$ which is the reduction of a special point on $\mathrm{Sh}_{\rmK_p}(G,X)$.

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