2.1.1

Let $\mathbb {S}:=\mathrm {Res}_{{\mathbb {C}}/{\mathbb {R}}}\mathbb {G}_m$ be the Deligne torus, and let $h : \mathbb {S} \rightarrow G_{{\mathbb {R}}}\simeq \prod _{i=1}^g \operatorname {GL}_{2, {\mathbb {R}}}$ be the morphism which on ${\mathbb {R}}$-points is given by

\[ z=a+ib \in \mathbb{S}({\mathbb{R}}) \mapsto \bigg(\begin{array}{@{}cc@{}} a & b\\ -b & a \end{array}\bigg)^g \in G({\mathbb{R}}). \]

Let $K^\circ _\infty : =\prod _{i=1}^g\mathrm {SO}_2({\mathbb {R}})\subset G({\mathbb {R}})$ (a maximal compact connected subgroup). The $G({\mathbb {R}})$-conjugacy class of $h$, denoted by $X$, is identified with $G({\mathbb {R}})/Z({\mathbb {R}})K^\circ _\infty \simeq ({\mathbb {C}}\smallsetminus {\mathbb {R}})^g$, and the couple $(G, X)$ is a Shimura datum with reflex field ${\mathbb {Q}}$. For every compact open subgroup $K \subset G({\mathbb {A}}_f)$ the Shimura variety $Sh_K(G)$ is a quasi-projective variety with a canonical model over the reflex field ${\mathbb {Q}}$, with complex analytic uniformisation

\[ Sh_K(G)({\mathbb{C}})=G({\mathbb{Q}})\backslash X\times G({\mathbb{A}}_f)/K. \]

The Shimura varieties $Sh_K(G)$ are called Hilbert modular varieties. If $K\subset G({\mathbb {A}}_f)$ is neat (which we will always assume in what follows), then $Sh_K(G)$ is smooth; moreover, if $K'\subset K$ is a normal compact open subgroup then the map $Sh_{K'}(G)\rightarrow Sh_{K}(G)$ is a finite étale Galois cover.

We have a map

(2.1.1.1)\begin{equation} Sh_K(G)({\mathbb{C}}) \rightarrow F^\times \backslash \{\pm 1\}^g \times {\mathbb{A}}_{F, f}^\times / \det(K)\simeq F^{\times, +} \backslash {\mathbb{A}}_{F, f}^\times / \det(K) \end{equation}

induced by the map sending $(x_\infty, x_f) \in G({\mathbb {R}}) \times G({\mathbb {A}}_f)$ to $(\mathrm {sgn} \det x_\infty, \det x_f)$. An element in $F^\times$ acts on $\{\pm 1\}^g$ by multiplying by the sign of the image via each real embedding. Fibres of the map in (2.1.1.1) are connected components of the source.

2.1.2 Hecke action

We have the Hecke algebra $\mathbb {T}_K(G) :={\mathbb {Z}}[K\backslash G({\mathbb {A}}_f)/K]$ of compactly supported, $K$-bi-invariant functions on $G({\mathbb {A}}_f)$, with multiplication given by convolution. Every element of $\mathbb {T}_K(G)$ gives rise to a correspondence on $Sh_K(G)$ as follows: given $g \in G({\mathbb {A}}_f)$, let $K_g:=K \cap g K g^{-1}$; we have a correspondence $[KgK]$ on $Sh_K(G)$ given by the following diagram.

Here the vertical maps are the canonical projections and the upper horizontal map on complex points is induced by right multiplication by $g$ on $G({\mathbb {A}}_f)$. Therefore, we obtain an action of $\mathbb {T}_K(G)$ on the cohomology groups $H^i(Sh_K(G), {\mathbb {F}}_\ell )$ as well as on the cohomology groups with compact support $H^i_{\rm c}(Sh_K(G), {\mathbb {F}}_\ell )$, where $\ell$ is a prime number. In the rest of the paper, we will rather work with a smaller Hecke algebra, defined as follows: fix a finite set $S$ of places of $F$ containing all the infinite places, all the places $v\mid \ell$, and all the finite places $v$ such that $K_v$ is not conjugate to $\operatorname {GL}_2({\mathcal {O}}_v)$, where ${\mathcal {O}}_v$ is the ring of integers in the completion $F_v$ of $F$ at $v$. Let

\[ \mathbb{T}:= \bigotimes_{v\not \in S} {' \; {\mathbb{Z}}[\operatorname{GL}_2({\mathcal{O}}_v)\backslash \operatorname{GL}_2(F_v)/\operatorname{GL}_2({\mathcal{O}}_v)]} \]

denote the abstract spherical Hecke algebra away from $S$. For every $v\not \in S$, choose a uniformiser $\varpi _v$ of ${\mathcal {O}}_v$. We denote by $T_v$ the double coset $\operatorname {GL}_2({\mathcal {O}}_v)(\begin{smallmatrix} \varpi _v & 0\\ 0 & 1 \end{smallmatrix})\operatorname {GL}_2({\mathcal {O}}_v)$ and by $S_v$ the double coset $\operatorname {GL}_2({\mathcal {O}}_v)(\begin{smallmatrix} \varpi _v & 0\\ 0 & \varpi _v \end{smallmatrix})\operatorname {GL}_2({\mathcal {O}}_v)$, seen as elements of $\mathbb {T}$. The algebra $\mathbb {T}$ is commutative and is generated by the operators $T_v$, $S_v^{\pm 1}$ for $v \not \in S$. The inversion map on $G$ induces a map $\iota : \mathbb {T}\rightarrow \mathbb {T}$; for a maximal ideal $\mathfrak {m} \subset \mathbb {T}$ we denote by $\mathfrak {m}^\vee$ its image via $\iota$.

2.1.3 Locally symmetric spaces

We have the locally symmetric space $X_K(G)$ attached to $G$ and $K$, defined as follows:

\[ X_K(G):=G({\mathbb{Q}})\backslash G({\mathbb{R}})\times G({\mathbb{A}}_f)/({\mathbb{R}}_{>0} K_\infty^\circ) K. \]

Letting $Z_K:=Z({\mathbb {Q}})\cap K$, the quotient $T_K:=Z({\mathbb {R}})/\{\pm 1\}^g{\mathbb {R}}_{>0}Z_K$ is a torus of dimension $g-1$ by Dirichlet's unit theorem, and the projection $X_K(G) \rightarrow Sh_K(G)$ is a $T_K$-bundle (see [Reference GrafGra16, Lemma 3.1.2]). Let us define $K_\infty :=\prod _{i=1}^g\mathrm {O}_2({\mathbb {R}})$, and let us consider the space

\[ \bar{X}_K(G):=G({\mathbb{Q}})\backslash G({\mathbb{R}})\times G({\mathbb{A}}_f)/({\mathbb{R}}_{>0}K_\infty)K. \]

In other words, we are quotienting by a maximal compact subgroup at infinity instead of its connected component of the identity. This is the space used in [Reference ScholzeSch15] (see the introduction there). The matrix $(\begin{smallmatrix}-1 & 0\\ 0 & 1\end{smallmatrix})$ normalises $\mathrm {SO}_2({\mathbb {R}})$; hence, we get a right action of $C:=({\mathbb {Z}}/2{\mathbb {Z}})^g$ on $X_K(G)$, and we have $\bar {X}_K(G)=X_K(G)/C$. The recipe given above endows the (Betti) cohomology of $X_K(G)$ and $\bar {X}_K(G)$ with an action of $\mathbb {T}$, which commutes with the action of $C$. Furthermore, if we make $C$ act on $F^\times \backslash \{\pm 1\}^g \times {\mathbb {A}}_{F, f}^\times / \det (K)$ by switching signs at archimedean places, then the map in (2.1.1.1) is $C$-equivariant.

Let $\hat {C}$ be the set of characters of $C$ with values in ${\mathbb {F}}_\ell ^\times$. The following lemma relates the cohomology of the spaces $Sh_K(G), X_K(G)$ and $\bar {X}_K(G)$.

3.1.1 Quaternionic Shimura data

As in the previous section we use the notation $G:=\mathrm {Res}_{F/{\mathbb {Q}}}\mathrm {GL}_2$; we also let $T_F:=\mathrm {Res}_{F/{\mathbb {Q}}}\mathbb {G}_m$. Let $K\subset G({\mathbb {A}}_f)$ be a neat compact open subgroup. Fix a prime $p>2$ which is totally split in $F$ and such that $K=K^{p}K_p$ with $K_p=\operatorname {GL}_2({\mathcal {O}}_F\otimes {\mathbb {Z}}_p)$.

Recall that $\Sigma _\infty$ denotes the set of real places of $F$. We will denote by $\Sigma _p$ the set of embeddings of $F$ into $\bar {{\mathbb {Q}}}_p$, which we identify with the set of prime ideals $\mathfrak {p} \subset {\mathcal {O}}_F$ lying above $p$. We fix an isomorphism $\iota _p : \bar {{\mathbb {Q}}}_p \buildrel \sim \over \to {\mathbb {C}}$, inducing a bijection $\iota _{p, \infty }: \Sigma _p \buildrel \sim \over \to \Sigma _\infty$. For every subset $T \subset \Sigma _p$ we set $T_\infty :=\iota _{p, \infty }(T)$ and we denote by $B_T$ the quaternion algebra over $F$ ramified precisely at $T \coprod T_\infty$. We let $G_T:=\mathrm {Res}_{F/{\mathbb {Q}}}B_T^\times$, and we fix an isomorphism $G_T({\mathbb {A}}_f^{(p)})\simeq G({\mathbb {A}}_f^{(p)})$. The group $G_{T, {\mathbb {R}}}$ is isomorphic to $\prod _{\tau \in T_\infty } \mathbb {H}^\times \times \prod _{\tau \in \Sigma _\infty \smallsetminus T_\infty } \operatorname {GL}_{2, {\mathbb {R}}}$ where $\mathbb {H}$ is the algebra of Hamilton quaternions. Let $X_T$ be the $G_T({\mathbb {R}})$-conjugacy class of the morphism $h_T: \mathbb {S} \rightarrow G_{T, {\mathbb {R}}}$ sending an ${\mathbb {R}}$-point $z=a+ib \in \mathbb {S}({\mathbb {R}})$ to $(z^\tau )_{\tau \in \Sigma _\infty }$, where $z^\tau =1$ for $\tau \in T_\infty$ and $z^\tau =(\begin{smallmatrix} a & b\\ -b & a \end{smallmatrix})$ if $\tau \in \Sigma _\infty \smallsetminus T_\infty$. The couple $(G_T, X_T)$ is a weak Shimura datum (in the sense of [Reference Tian and XiaoTX16, § 2.2]) whose reflex field $F_T$ can be described as follows: the group $\mathrm {Aut}({\mathbb {C}}/{\mathbb {Q}})$ acts on $\Sigma _\infty$ by post-composition. Let $\Gamma _T \subset \mathrm {Aut}({\mathbb {C}}/{\mathbb {Q}})$ be the subgroup preserving $T_\infty$; then $F_T={\mathbb {C}}^{\Gamma _T}\subset {\mathbb {C}}$.

Let $K_T=K_{T, p}K^{p} \subset G_T({\mathbb {A}}_f)$ be the compact open subgroup such that $K^p\subset G_T({\mathbb {A}}_f^{(p)})\simeq G({\mathbb {A}}_f^{(p)})$ is the subgroup chosen above and $K_{T, p}=\prod _{\mathfrak {p} \mid p}K_{T, \mathfrak {p}}$ is of the following type: if $\mathfrak {p} \in \Sigma _p \smallsetminus T$, then we fix an isomorphism $\rho _\mathfrak {p}: B_T(F_\mathfrak {p})^\times \rightarrow \operatorname {GL}_2(F_\mathfrak {p})$ and we take $K_{T, \mathfrak {p}}:=\rho _\mathfrak {p}^{-1}(\operatorname {GL}_2(\mathcal {O}_\mathfrak {p}))$. If $\mathfrak {p} \in T$, then we take $K_{T, \mathfrak {p}}$ to be the group of units in the unique maximal order in the division quaternion algebra $B_T \otimes _F F_\mathfrak {p}$.

3.1.2 The auxiliary $CM$ extension

Choose a $CM$ extension $E/F$ such that every place $\mathfrak {p} \in \Sigma _p$ is inert in $E$; in particular, $B_T \otimes _F E$ is isomorphic to the matrix algebra $M_2(E)$. Let $c \in \mathrm {Gal}(E/F)$ be the non trivial element and let $\Sigma _{E, \infty }$ be the set of complex embeddings of $E$: it comes with a restriction map $\Sigma _{E, \infty }\rightarrow \Sigma _\infty$ whose fibres are the orbits for the action of $\mathrm {Gal}(E/F)$ sending an embedding $\tilde {\tau } \in \Sigma _{E, \infty }$ to $\tilde {\tau }^c:=\tilde {\tau }\circ c$. Choose a set $\tilde {T} \subset \Sigma _{E, \infty }$ containing exactly one lift of each $\tau \in T_\infty$. For every $\tilde {\tau } \in \Sigma _{E, \infty }$, define an integer $s_{\tilde {\tau }} \in \{0, 1, 2\}$ as follows:

• • if $\tilde {\tau }_{|F} \not \in T_\infty$ then $s_{\tilde {\tau }}=1$;

• • if $\tilde {\tau } \in \tilde {T}$ then $s_{\tilde {\tau }}=0$;

• • if $\tilde {\tau }^c \in \tilde {T}$ then $s_{\tilde {\tau }}=2$.

For each $\tau \in T_\infty$ (respectively, $\tau \in \Sigma _\infty \smallsetminus T_\infty$) we choose the isomorphism $E\otimes _{F, \tau }{\mathbb {R}}\simeq {\mathbb {C}}$ induced by the embedding $\tilde {\tau } \in \Sigma _{E, \infty }\smallsetminus \tilde {T}$ lifting $\tau$ (respectively, an arbitrary lift of $\tau$). Let $T_E:=\mathrm {Res}_{E/{\mathbb {Q}}}\mathbb {G}_m$; via the previous choices we obtain an isomorphism $T_{E}({\mathbb {R}})= \prod _{\tau \in \Sigma _\infty }(E\otimes _{F, \tau }{\mathbb {R}})^\times \simeq \prod _{\tau \in \Sigma _\infty }{\mathbb {C}}^\times$. Let $h_{T, E}: \mathbb {S} \rightarrow T_{E, {\mathbb {R}}}$ be the morphism which on ${\mathbb {R}}$-points sends $z=a+ib \in \mathbb {S}({\mathbb {R}})$ to $(z^\tau )_{\tau \in \Sigma _\infty }$, where $z^\tau =z$ (respectively, $z^\tau =1$) if $\tau \in T_\infty$ (respectively, $\tau \in \Sigma _\infty \smallsetminus T_\infty$). The reflex field $E_T\supset F_T$ of the $T_E({\mathbb {R}})-$conjugacy class of $h_{T, E}$ is the subfield of ${\mathbb {C}}$ fixed by the stabiliser in $\mathrm {Aut}({\mathbb {C}}/{\mathbb {Q}})$ of $\tilde {T}$.

3.1.3 Unitary Shimura data

We denote by $H_T$ the algebraic group fitting in the exact sequence

\[ 0 \rightarrow T_F \rightarrow G_T \times T_E \rightarrow H_T \rightarrow 0, \]

where the map $T_F \rightarrow G_T \times T_E$ is given by $a \mapsto (a, a^{-1})$. Note that, because of Hilbert's theorem 90, the map $G_T \times T_E \rightarrow H_T$ induces surjections on ${\mathbb {Q}}$-points and ${\mathbb {A}}_f$-points. The $(G_T \times T_E)({\mathbb {R}})-$conjugacy class of the map $h_T\times h_{T, E}: \mathbb {S}\rightarrow (G_T \times T_E)_{\mathbb {R}}$ and the $H_T({\mathbb {R}})$-conjugacy class of the induced map $h_{H_T}: \mathbb {S}\rightarrow H_{T, {\mathbb {R}}}$ can both be identified with $X_T \simeq \prod _{\tau \in \Sigma _\infty \smallsetminus T_\infty }({\mathbb {C}} \smallsetminus {\mathbb {R}})$, and they give rise to weak Shimura data $(G_T \times T_E, X_T)$ and $(H_T, X_T)$ with reflex field $E_T$. We let $X_{T}^+:=\prod _{\tau \in \Sigma _\infty \smallsetminus T_\infty }({\mathbb {C}} \smallsetminus {\mathbb {R}})^+$, where $({\mathbb {C}} \smallsetminus {\mathbb {R}})^+ \subset ({\mathbb {C}} \smallsetminus {\mathbb {R}})$ is the upper half-plane.

Let us denote by $D_T$ the tensor product $B_T\otimes _F E$ and by $\overline {(\cdot )}: D_T \rightarrow D_T$ the tensor product of the main involution on $B_T$ and complex conjugation on $E$. For every ${\mathbb {Q}}-$algebra $R$ we have a canonical isomorphism

\[ (B_T\otimes_{\mathbb{Q}} R)\otimes_{F \otimes_{\mathbb{Q}} R}(E\otimes_{\mathbb{Q}} R)\simeq (B_T\otimes_F E)\otimes_{{\mathbb{Q}}}R, \]

hence we get a map $G_T(R) \times T_E(R) \rightarrow (D_T \otimes _{\mathbb {Q}} R)^\times$. This yields an identification

\[ H_T(R)=\{g \in D_T\otimes_{\mathbb{Q}} R \mid g \bar{g} \in (F \otimes_{\mathbb{Q}} R)^\times\}. \]

The latter description allows to see $H_T$ as a unitary group as follows: given an element $\sigma \in D_T$ such that $\bar {\sigma }=-\sigma$ the map sending $g \in D_T$ to $g^*=\sigma ^{-1}\bar {g}\sigma$ is an involution of $D_T$. Let us denote by $V_T$ the ${\mathbb {Q}}$-vector space underlying $D_T$, together with its natural structure of left $D_T$-module, so that $\mathrm {End}_{D_T}(V_T)=D_T^{op}$. Consider the skew-Hermitian pairing

\begin{align*} \psi_T: V_T \times V_T & \rightarrow E\\ (v, w) & \mapsto \mathrm{Tr}_{D_T/E}(v\sigma w^*); \end{align*}

for every $g \in D_T$ and $v, w \in V_T$ we have

\[ \psi_T(vg, wg)=\mathrm{Tr}_{D_T/E}(vg \sigma (wg)^*)=\mathrm{Tr}_{D_T/E}(vg \bar{g}\sigma w^*)=g\bar{g}\psi_T(v, w), \]

hence, for every ${\mathbb {Q}}$-algebra $R$,

\[ H_T(R)=\{(g, c(g)) \in \mathrm{End}_{D_T \otimes_{\mathbb{Q}} R}(V_T \otimes_{\mathbb{Q}} R)\times (F \otimes_{\mathbb{Q}} R)^\times \mid \psi_T(vg, wg)=c(g)\psi(v, w)\}. \]

3.2.1

Fix a subset $T\subset \Sigma _p$; the reduced norm on $B_T$ gives rise to a map $\mathrm {Nm}: G_T \rightarrow T_F$. Let $T_{H_T}:=(T_F \times T_E)/T_F$, where the embedding of $T_F$ in $T_F \times T_E$ is given by $a \mapsto (a^2, a^{-1})$. Letting $N_{H_T}: H_T \rightarrow T_{H_T}$ be the map induced by $\mathrm {Nm} \times \mathrm {Id}: G_T \times T_E \rightarrow T_F \times T_E$, we obtain a commutative diagram

where the top arrow is compatible with the Deligne homomorphisms. Recall that we have fixed a compact open subgroup $K_T \subset G_T({\mathbb {A}}_f)$. Take a compact open subgroup $K_E=K_{E, p} K_E^{p} \subset T_E({\mathbb {A}}_f)$ and a compact open subgroup $U_T \subset H_T({\mathbb {A}}_f)$ containing the image of $K_T \times K_E$. Let

\begin{align*} C_{K_T}&:=F^{\times, +}\backslash {\mathbb{A}}_{F, f}^\times/\mathrm{Nm}(K_T), \quad C_{K_E}:=E^{\times}\backslash {\mathbb{A}}_{E, f}^\times/K_E\\ C_{K_T \times K_E}&:=C_{K_T}\times C_{K_E}, \quad C_{U_T}:=T_{H_T}({\mathbb{Q}})^+\backslash T_{H_T}({\mathbb{A}}_f)/N_{H_T}(U_T), \end{align*}

where $T_{H_T}({\mathbb {Q}})^+:=(F^{\times, +}\times E^\times )/F^\times$. We have maps $C_{K_T}\rightarrow C_{K_T}\times C_{K_E} \rightarrow C_{U_T}$, where the first map sends $C_{K_T}$ to (the equivalence class of) 1 on the second component. Letting $G_T({\mathbb {Q}})^+\subset G_T({\mathbb {Q}})$ be the subgroup of elements with totally positive norm, and $H_T({\mathbb {Q}})^+:=(G_T({\mathbb {Q}})^+\times T_E({\mathbb {Q}}))/T_F({\mathbb {Q}})$, we obtain the following commutative diagram.

(3.2.1.1)

If $T \neq \Sigma _p$ (so that the spaces in the top row are positive dimensional), the fibres of the vertical maps are connected components of the complex analytic spaces in the first row. The upper left (respectively, upper right) space is identified with the complex points of $Sh_{K_T}(G_T)$ (respectively, $Sh_{U_T}(H_T)$). If $T=\Sigma _p$, then we take this as a definition of (the complex points of) $Sh_{K_T}(G_T)$ and $Sh_{U_T}(H_T)$.

Following [Reference Diamond, Kassei and SasakiDKS23], we will study the relation between the top left and the top right space of the above diagram. To do so, we introduce the following notion, borrowed from [Reference Diamond, Kassei and SasakiDKS23, § 2.3.1].

3.2.3

Given $K_T$, it is always possible to choose $K_E$ sufficiently small with respect to $K_T$, cf. [Reference Diamond, Kassei and SasakiDKS23, § 2.3.1]. The main point is that the norm map ${\mathcal {O}}_E^\times \rightarrow {\mathcal {O}}_F^\times$ has finite kernel, hence we can choose $K_E$ satisfying the first condition. Note that this can be done independently of $K_T$; given $K_T$, we can then shrink $K_E$ so that it satisfies the second and third conditions. Furthermore, we may choose $K_E$ of the form $(\mathcal {O}_E\otimes {\mathbb {Z}}_p)^\times K_E^{p}$ (if $K_T$ is as in § 3.1.1).

3.2.4 Hecke algebras

Let $u: G_T \rightarrow H_T$ be the composite of the inclusion $G_T \rightarrow G_T \times T_E$ (whose second component is the composition of the structure map and the identity section) and the projection $G_T \times T_E \rightarrow H_T$. If $U_T \subset H_T({\mathbb {A}}_f)$ contains the image of $K_T \times K_E$, then $u$ induces a map $K_T\backslash G_T({\mathbb {A}}_f)/K_T \rightarrow U_T\backslash H_T({\mathbb {A}}_f)/U_T$, which, in turn, induces a (set theoretic) pullback map $r: \mathbb {T}_{U_T}(H_T)\rightarrow \mathbb {T}_{K_T}(G_T)$. On the other hand, the inclusion $G_T \rightarrow G_T \times T_E$ gives rise to a morphism of Hecke algebras $i: \mathbb {T}_{K_T}(G_T) \rightarrow \mathbb {T}_{U_T}(H_T)$ sending the characteristic function of a double coset $K_TgK_T$ to that of $U_T(g, 1)U_T$.

Proof. Let us see $G_T$ as a subgroup of $G_T \times T_E$ and denote by $q:(G_T \times T_E)(\mathbb {A}_f)\rightarrow H_T(\mathbb {A}_f)$ the quotient map. Given $g \in G_T(\mathbb {A}_f)$, we need to prove that

\[ q^{-1}(U_T(g, 1)U_T)\cap G_T(\mathbb{A}_f)=K_TgK_T. \]

Take $x$ belonging to the set on the left-hand side above. Then there exist $(k_1, e_1), (k_2, e_2) \in K_T \times K_E$ and $a \in T_F(\mathbb {A}_f)$ such that

\[ x=(a, a^{-1})(k_1, e_1)(g, 1)(k_2, e_2). \]

As $x \in G_T(\mathbb {A}_f)$, we must have $a^{-1}e_1e_2=1$, hence $e_1e_2=a \in T_F(\mathbb {A}_f)\cap K_E\subset K_T$. It follows that

\[ x=(ak_1gk_2, 1) \in K_TgK_T. \]

3.2.6

The above lemma implies, in particular, that the map $i: \mathbb {T}_{K_T}(G_T) \rightarrow \mathbb {T}_{U_T}(H_T)$ is injective. In what follows, we will use this map to identify $\mathbb {T}_{K_T}(G_T)$ with a sub-algebra of $\mathbb {T}_{U_T}(H_T)$. We will work with the Hecke algebra $\mathbb {T}=\bigotimes _{v}'{\mathbb {Z}}[\operatorname {GL}_2({\mathcal {O}}_v)\backslash \operatorname {GL}_2(F_v)/\operatorname {GL}_2({\mathcal {O}}_v)]$ as in $\S ~$2.1.2, where the product runs over the set of places of $F$ lying above a rational prime different from $p$ and such that the component of $K_T$ at $v$ is hyperspecial.

The statements in the next lemma are established, in a slightly different setting, in the proof of [Reference Diamond, Kassei and SasakiDKS23, Lemma 2.3.1].

Proof. For the reader's convenience, let us briefly explain how the main points of the argument in the proof of [Reference Diamond, Kassei and SasakiDKS23, Lemma 2.3.1] adapt to our situation.

(i) This follows from the fact that the inclusion $T_F\rightarrow T_{H_T}$ sending $a$ to $(a, 1)$ is split by the map $(a, b) \mapsto aN_{E/F}(b)$, which induces splittings

\begin{align*} T_{H_T}({\mathbb{Q}})^+=(F^{\times, +}\times E^\times)/F^\times & \rightarrow F^{\times, +} \; & \mathrm{N}_{H_T}(U_T)& \rightarrow \mathrm{Nm}(K_T)\\ (a, b) & \mapsto aN_{E/F}(b) & (\mathrm{Nm}(k), l) & \mapsto \mathrm{Nm}(k)N_{E/F}(l) \end{align*}

(note that the last formula gives a well-defined map $\mathrm {N}_{H_T}(U_T) \rightarrow \mathrm {Nm}(K_T)$ because of the assumptions that $U_T$ is the image of $K_T \times K_E$ and $N_{E/F}(K_E)\subset \mathrm {Nm}(K_T)$) of the inclusions

\begin{align*} F^{\times, +} & \rightarrow T_{H_T}({\mathbb{Q}})^+=(F^{\times, +}\times E^\times)/F^\times \; & \mathrm{Nm}(K_T)& \rightarrow \mathrm{N}_{H_T}(U_T)\\ a & \mapsto (a, 1) & \mathrm{Nm}(k) & \mapsto (\mathrm{Nm}(k), 1). \end{align*}

(ii) If $T=\Sigma _p$, there is nothing to prove. Assume that $T \neq \Sigma _p$. Then the claim follows from the equality, valid for every $g \in G_T({\mathbb {A}}_f)$:

\[ G_T({\mathbb{Q}})^+\cap g K_T g^{-1}=H_T({\mathbb{Q}})^+\cap g U_T g^{-1}. \]

Clearly the group on the left-hand side is contained in the group on the right-hand side. Conversely, let $h \in H_T({\mathbb {Q}})^+\cap g U_T g^{-1}$. On the one hand, we can write (the equivalence class of) $h$ as $h=a \cdot e$ for some $a \in G_T({\mathbb {Q}})^+, e \in E^\times$; on the other hand, $h=g(k \cdot y)g^{-1}$ for some $k \in K_T, y \in K_E$. It follows that $ae=g(ky)g^{-1}$, which implies that $e\cdot c(e)^{-1}=y\cdot c(y)^{-1} \in E^\times \cap \{ {y}/{y^c} \mid y \in K_E\}=\{1\}$. Therefore, $y \in K_E \cap T_F({\mathbb {A}}_f)\subset K_T$. This yields $h \in G_T({\mathbb {Q}})^+\cap g K_T g^{-1}$.

3.2.8

Keeping the notation and assumptions of the previous lemma, we set $I:=C_{U_T}/C_{K_T}$. For each $\alpha \in I$, let $Sh_{U_T}^{\alpha }(H_T)({\mathbb {C}}) \subset Sh_{U_T}(H_T)({\mathbb {C}})$ be the subspace consisting of connected components mapping to the $C_{K_T}$-coset inside $C_{U_T}$ given by $\alpha$. We obtain a decomposition

\[ Sh_{U_T}(H_T)({\mathbb{C}})=\coprod_{\alpha \in I} Sh_{U_T}^{\alpha}(H_T)({\mathbb{C}}) \]

into open and closed subspaces; the subspace corresponding to the coset containing the identity is identified with $Sh_{K_{T}}(G_T)({\mathbb {C}})$ (if $\Sigma _p=T$, cf. the argument in the second part of the proof of [Reference Diamond, Kassei and SasakiDKS23, Lemma 2.3.1]).

Proof. We will work over the complex numbers throughout this proof, and omit this from the notation for simplicity. The decomposition $Sh_{U_T}(H_T)=\coprod _{\alpha \in I} Sh_{U_T}^{\alpha }(H_T)$ introduced above yields a direct sum decomposition

(3.2.9.1)\begin{equation} H^i(Sh_{U_T}(H_{T}), {\mathbb{F}}_{\ell})=\bigoplus_{\alpha \in I} H^i(Sh_{U_T}^{\alpha}(H_T), {\mathbb{F}}_{\ell}) \end{equation}

and each summand on the right-hand side is preserved by the action of $\mathbb {T}$. Indeed, the Hecke algebra $\mathbb {T}$ is spanned by characteristic functions of double cosets $K_TgK_T$, whose component at a place $v$ is of the form $\operatorname {GL}_2(\mathcal {O}_v)\big (\begin{smallmatrix} \varpi _v^{a_v} & 0\\ 0 & \varpi _v^{b_v} \end{smallmatrix}\big )\operatorname {GL}_2(\mathcal {O}_v)$, for some uniformiser $\varpi _v$ of $\mathcal {O}_v$ and $a_v, b_v \in {\mathbb {Z}}$. The group $K_E$ is sufficiently small with respect to $K_{T, g}:=K_T \cap gK_Tg^{-1}$; letting $U_{T, g}\subset H_T(\mathbb {A}_f)$ be the image of $K_{T, g} \times K_E$ (which coincides with the group $U_{T} \cap gU_Tg^{-1}$), the Hecke correspondence

restricts to a correspondence on each $Sh_{U_T}^{\alpha }(H_T)$. Indeed, the map $Sh_{U_{T, g}}(H_T)\rightarrow Sh_{U_{T, g^{-1}}}(H_T)$ induced by right multiplication by $(g, 1) \in H_T({\mathbb {A}}_f)$ preserves each $Sh_{U_{T, g}}^\alpha (H_T)$, as $N_{H_T}((g, 1)) \in {\mathbb {A}}_{F, f}^\times \subset T_{H_T}({\mathbb {A}}_f)$. Hence, each summand in (3.2.9.1) is preserved by the action of $\mathbb {T}$.

On the other hand, we have an action of $T_E(\mathbb {A}_f)$ on $Sh_{U_T}(H_{T})$: the action of $e \in T_E(\mathbb {A}_f)$ sends each $Sh_{U_T}^{\alpha }(H_T)$ to $Sh_{U_T}^{e\alpha }(H_T)$. In particular, $T_E(\mathbb {A}_f)$ acts transitively on the set of subvarieties $Sh_{U_T}^{\alpha }(H_T)$, inducing isomorphisms, for every $\alpha \in I$:

\[ H^i(Sh_{U_T}^{\alpha}(H_T), {\mathbb{F}}_{\ell})\simeq H^i(Sh_{U_T}^{1}(H_T), {\mathbb{F}}_{\ell})=H^i(Sh_{K_{T}}(G_T), {\mathbb{F}}_{\ell}). \]

Finally, the action on cohomology induced by the $T_E(\mathbb {A}_f)$-action on $Sh_{U_T}(H_{T})$ commutes with the action of $\mathbb {T}$. It follows that the above isomorphism induces an isomorphism

(3.2.9.2)\begin{equation} H^i(Sh_{U_T}(H_T), {\mathbb{F}}_{\ell})_\mathfrak{m}\simeq \bigoplus_{\alpha \in I} H^i(Sh_{K_{T}}(G_T), {\mathbb{F}}_{\ell})_\mathfrak{m}. \end{equation}

The isomorphism (3.2.9.2) (which also holds with ${\mathbb {Q}}_\ell$-coefficients) implies part (i). Furthermore, the previous argument also applies to compactly supported cohomology, yielding a direct sum decomposition of $H^i_c(Sh_{U_T}(H_T), {\mathbb {F}}_{\ell })_\mathfrak {m}$ as in (3.2.9.2). Hence we obtain a similar decomposition for the kernel and cokernel of the map $H^i_c(Sh_{U_T}(H_T), {\mathbb {F}}_{\ell })_\mathfrak {m}\rightarrow H^i(Sh_{U_T}(H_T), {\mathbb {F}}_{\ell })_\mathfrak {m}$, from which part (ii) follows.

3.3.1

Fix a totally negative element $\mathfrak {d} \in \mathcal {O}_F$ coprime to $p$; choose isomorphisms $\theta _T: M_2(E)=D_{\emptyset } \buildrel \sim \over \to D_{T}$ for every non-empty subset $T \subset \Sigma _p$. For every such $T$, choose an element $\delta _T \in D_T$ as in [Reference Tian and XiaoTX16, Lemma 3.8], and let $\sigma _T=\sqrt {\mathfrak {d}}\delta _T$. Via the construction in § 3.1.3, such a choice gives an involution $*_T$ on each $D_T$. By [Reference Tian and XiaoTX16, Lemma 5.4] we may, and will, choose the elements $\delta _T$ in such a way that these involutions are respected by the isomorphisms $\theta _T$. Let $\mathcal {O}_{D_\emptyset }=M_2(\mathcal {O}_E)\subset D_{\emptyset }$ and $\mathcal {O}_{D_T}=\theta _T(\mathcal {O}_{D_\emptyset })$ for every non-empty subset $T \subset \Sigma _p$.

Take $K_T\subset G_T(\mathbb {A}_f)$ as in § 3.1.1, $K_E=(\mathcal {O}_E\otimes {\mathbb {Z}}_p)^\times K_E^{p} \subset T_E({\mathbb {A}}_f)$ sufficiently small with respect to $K_T$ and let $U_T\subset H_T(\mathbb {A}_f)$ be the image of $K_T \times K_E$. The inverse of the chosen isomorphism $\iota _p: \bar {{\mathbb {Q}}}_p \buildrel \sim \over \to {\mathbb {C}}$ determines a distinguished $p$-adic place $\wp$ of the reflex field $E_T \subset {\mathbb {C}}$. Let $E_\wp \subset \bar {{\mathbb {Q}}}_p$ be the completion of $E_T$ at $\wp$, and ${\mathcal {O}}_\wp$ its ring of integers. Following [Reference Diamond, Kassei and SasakiDKS23, § 2] and [Reference Tian and XiaoTX16, § 3], an integral model of $Sh_{U_T}(H_T)$ over ${\mathcal {O}}_\wp$ can be constructed as follows. Consider the functor sending an ${\mathcal {O}}_\wp$-scheme $S$ to the set of isomorphism classes of tuples $(A, \iota, \lambda, \eta )$ where:

1. (i) $A/S$ is an abelian scheme of dimension $4g$;

2. (ii) $\iota : \mathcal {O}_{D_T}\rightarrow \mathrm {End}_S(A)$ is an embedding;

3. (iii) $\lambda : A \rightarrow A^\vee$ is a ${\mathbb {Z}}_{(p)}^\times$-polarisation whose attached Rosati involution coincides with $*_T$ on $\mathcal {O}_{D_T}$;

4. (iv) $\eta$ is a $U_T$-level structure, in the sense of [Reference Diamond, Kassei and SasakiDKS23, § 2.2.2].

Furthermore, the above data are required to satisfy the following conditions.

1. (a) For every $b \in \mathcal {O}_E$, the characteristic polynomial of $\iota (b)$ acting on $\mathrm {Lie}(A/S)$ equals

   \[ \prod_{\tilde{\tau} \in \Sigma_{E, \infty}}(X-\tilde{\tau}(b))^{2s_{\tilde{\tau}}}, \]
   where the integers $s_{\tilde {\tau }}$ were defined in § 3.1.2.

2. (b) The kernel $\ker (\lambda [p^\infty ]): A[p^\infty ]\rightarrow A^\vee [p^\infty ]$ is a finite flat subgroup scheme contained in $\prod _{\mathfrak {p} \in T}A[\mathfrak {p}]$ and such that for each $\mathfrak {p} \in T$ the rank of $\ker (\lambda [p^\infty ])\cap A[\mathfrak {p}]$ is $p^4$.

3. (c) The cokernel of $\lambda _*: H_1^{dR}(A/S)\rightarrow H_1^{dR}(A^\vee /S)$ is locally free of rank two over $\bigoplus _{\mathfrak {p} \in T} \mathcal {O}_S\otimes _{{\mathbb {Z}}_p} (\mathcal {O}_E\otimes _{\mathcal {O}_F}\mathcal {O}_F/\mathfrak {p})$ (here $H_1^{dR}(A/S)$ denotes the de Rham homology sheaf of $A/S$).

This functor is represented by a scheme $Y_{U_T}(H_T)$ which is an infinite disjoint union of smooth, quasi-projective (respectively, projective if $T$ is non-empty) ${\mathcal {O}}_\wp$-schemes. The group $\mathcal {O}_{F, (p)}^{\times, +}$ acts on this scheme as follows: an element $u \in \mathcal {O}_{F, (p)}^{\times, +}$ sends $(A, \iota, \lambda, \eta )$ to $(A, \iota, u\lambda, \eta )$. This action factors through the group $\mathcal {O}_{F, (p)}^{\times, +}/N_{E/F}(U_T \cap E^\times )$, and the resulting quotient is a smooth quasi-projective scheme giving the desired integral model of $Sh_{U_T}(H_T)$.

4.2.1

Let $\overline {\mathrm {Ig}}_T^b$ be the scheme representing the functor that sends a ring $R$ of characteristic $p$ to the set of isomorphism classes of data $(r, A, \iota, \lambda, \eta, \phi )$ defined as follows:

1. (i) $r: \bar {\mathbb {F}}_p\rightarrow R$ is a ring morphism;

2. (ii) $(A, \iota, \lambda, \eta )/\mathrm {Spec} \, R$ is a datum as in the definition of the integral model of $Sh_{U_T}(H_T)$, satisfying conditions (a), (b) and (c) in § 3.3;

3. (iii) $\phi =(\phi _\mathfrak {p})_{\mathfrak {p} \in \Sigma _p}: A[p^\infty ] \buildrel \sim \over \to \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p, r}R$ is an isomorphism commuting with the $\mathcal {O}_{D_T}$-action and such that each $\phi _\mathfrak {p}$ respects the polarisation up to a ${\mathbb {Z}}_p^\times$-factor.

Two tuples $(r, A, \iota, \lambda, \eta, \phi ), (r', A', \iota ', \lambda ', \eta ', \phi ')$ are said to be isomorphic if $r=r'$ and there is an isomorphism of abelian schemes over $\mathrm {Spec} \, R$ between $A$ and $A'$ commuting with all the additional data.

Forgetting everything but the structure map $r$ we see that $\overline {\mathrm {Ig}}_T^b$ is fibred over $\bar {\mathbb {F}}_p$; in what follows, if there is no danger of confusion, we will abusively denote its points with values in an $\bar {\mathbb {F}}_p$-algebra $(R, r)$ just by $(A, \iota, \lambda, \eta, \phi ) \in \overline {\mathrm {Ig}}_T^b(R)$. The functor defining $\overline {\mathrm {Ig}}_T^b$ is representable: indeed, $\overline {\mathrm {Ig}}_T^b\rightarrow Y_{U_T}(H_T)_{\bar {\mathbb {F}}_p}$ relatively represents the moduli problem parametrising isomorphisms $\phi : A[p^\infty ]\buildrel \sim \over \to \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p, r}R$ as above. By definition, such an isomorphisms is a compatible sequence of isomorphisms $\phi _k: A[p^k]\buildrel \sim \over \to \mathbb {X}^b_T[p^k]\times _{\bar {\mathbb {F}}_p, r}R$ for $k \geq 0$. Hence $\overline {\mathrm {Ig}}_T^b$ is the inverse limit of the schemes $\overline {\mathrm {Ig}}_{T, k}^b\rightarrow Y_{U_T}(H_T)_{\bar {\mathbb {F}}_p}$ parametrising isomorphisms $\phi _k$ as above; each of these moduli problems is relatively representable (cf. [Reference Caraiani and ScholzeCS17, Proposition 4.3.3]) and the transition maps are finite.

Recall that we have an action of ${\mathcal {O}}_{F, (p)}^{\times, +}$ on $Y_{U_T}(H_T)_{\bar {\mathbb {F}}_p}$. For each $k \geq 0$, consider the quotient $\Delta _k:=\mathcal {O}_{F, (p)}^\times /\{N_{E/F}(u), u \in U_T \cap \mathcal {O}_E^\times, u \equiv 1 \pmod {p^k}\}$; set $\Delta :=\varprojlim _k \Delta _k$. The action of $\mathcal {O}_{F, (p)}^{\times, +}$ on $\overline {\mathrm {Ig}}_{T, k}^b$ lifting the action on $Y_{U_T}(H_T)_{\bar {\mathbb {F}}_p}$ factors through $\Delta _k$, hence we get an action of $\Delta$ on $\overline {\mathrm {Ig}}_T^b$. We define $\mathrm {Ig}_T^b:=\overline {\mathrm {Ig}}_T^b/\Delta$.

For each $\mathfrak {p} \in \Sigma _p$ fix an element $x_\mathfrak {p} \in {\mathcal {O}}_F^+$ with $\mathfrak {p}$-adic valuation one, and with $\mathfrak {p}'$-adic valuation zero for every $\mathfrak {p}' \in \Sigma _p \smallsetminus \{\mathfrak {p}\}$. The following lemma gives a description of the functor of points of $\overline {\mathrm {Ig}}_T^b$ in terms of abelian schemes up to $p$-quasi-isogeny; see also [Reference Caraiani and ScholzeCS17, Lemma 4.3.4].

Proof. Let $R$ be an $\bar {\mathbb {F}}_p$-algebra. An isomorphism class of data $(A, \iota, \lambda, \eta, \phi )$ corresponding to an $R$-point of $\overline {\mathrm {Ig}}_T^b$ gives rise to a tuple as in the statement of the lemma. Conversely, let $(A, \iota, \lambda, \eta, \rho )$ be a datum as in the statement of the lemma; multiplying $\rho$ by suitable powers of the elements $x_\mathfrak {p}$ we obtain an isogeny, abusively still denoted by the same symbol, $\rho : A[p^\infty ]\rightarrow \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R$, commuting with $\mathcal {O}_{D_T}$-action and polarisation (the latter up to a ${\mathbb {Q}}_p^\times$-factor on each component). Letting $B=A/\ker (\rho )$, the map $\rho$ induces an isomorphism $\bar {\rho }: B[p^\infty ]\xrightarrow {\sim } \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R$, and this property characterises uniquely $B$ among abelian schemes in the $p$-power isogeny class of $A$. In order to complete the proof we need to endow $B$ with extra structures $\iota _B, \lambda _B, \eta _B, \phi$ in such a way that $(B, \iota _B, \lambda _B, \eta _B, \phi )$ is a point of $\overline {\mathrm {Ig}}^b_T$, and the quotient map $q: A\rightarrow B$ respects the extra structures as in the statement of the lemma.

Let us start by defining the $\mathcal {O}_{D_T}$-action on $B$. Given $o \in \mathcal {O}_{D_T}$ we consider the self-quasi-isogeny $\iota _B(o):=q \circ \iota (o) \circ q^{-1}$ of $B$. Let us look at the following diagram.

By assumption, the largest square and the left square in the diagram commute; it follows that the right square also commutes. We deduce that $\iota _B(o): B \rightarrow B$ is a morphism (and not just a quasi-isogeny) as the same is true for $\iota _T^b(o): \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R \rightarrow \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R$. Hence, the map sending $o$ to $\iota _B(o)$ endows $B$ with an $\mathcal {O}_{D_T}$-action, which commutes with the quotient map $A \rightarrow B$ by construction.

Let us now define the polarisation $\lambda _B$. We consider the quasi-isogeny $\lambda '_B:=(q^\vee )^{-1}\circ \lambda \circ q^{-1}: B \rightarrow B^\vee$. It fits in the following diagram.

By assumption, the map $\rho =\bar {\rho }\circ q$ can be written as $\rho =(\rho _\mathfrak {p})_{\mathfrak {p} \in \Sigma _p}$ where each $\rho _\mathfrak {p}$ respects polarisations on the $\mathfrak {p}$-component of the relevant $p$-divisible groups up to a factor $c_\mathfrak {p} \in {\mathbb {Q}}_p^\times$, i.e. we have $(\rho _\mathfrak {p}^\vee )^{-1} \circ \lambda _\mathfrak {p} \circ (\rho _\mathfrak {p})^{-1}=c_\mathfrak {p}\lambda _{\mathbb {X}^{b_\mathfrak {p}}_T}$. For each $\mathfrak {p}$ let $v_\mathfrak {p}$ be the valuation of $c_\mathfrak {p}$, and set $\lambda _B:=(\prod _{\mathfrak {p} \in \Sigma _p} x_\mathfrak {p}^{-v_\mathfrak {p}})\lambda '_B$; then the isomorphism $\bar {\rho }$ commutes with the polarisations $\lambda _B$ and $\lambda _{\mathbb {X}^b_T}$ up to a ${\mathbb {Z}}_p^\times$-factor on each component.

Let $\eta _B$ be the (prime to $p$) level structure on $B$ induced by $\eta$ and $q$. Let the isomorphism $\phi$ be given by $\bar {\rho }$. Note that changing the polarisation on $A$ by a product of powers of the elements $x_\mathfrak {p}$ does not affect the polarisation $\lambda _B$, hence the isomorphism class of $(B, \iota _B, \lambda _B, \eta _B, \phi )$ only depends on the isomorphism class of $(A, \iota, \lambda, \eta, \rho )$ (in the sense of the lemma).

Furthermore, by construction, the map $q$ commutes with polarisations up to a product of powers of the elements $x_\mathfrak {p}$, and respects all the other additional structures; therefore, the data $(A, \iota, \lambda, \eta, \rho )$ and $(B, \iota _B, \lambda _B, \eta _B, \phi )$ are isomorphic in the sense of the lemma.

Finally, the datum $(B, \iota _B, \lambda _B, \eta _B)$ satisfies conditions (a), (b) and (c) in § 3.3, hence is a point of $\overline {\mathrm {Ig}}_T^b$. This follows from Remark 3.3.3 and from the fact that the $p$-divisible group $\mathbb {X}^b_T$ comes from an abelian variety with extra structure satisfying the same conditions.

Theorem 4.2.4 Let $b \in B(H_{T, {\mathbb {Q}}_p}, \mu _T)$ and let $B\subset \Sigma _p \smallsetminus T$ be the subset of places $\mathfrak {p}$ such that $b_\mathfrak {p}$ is basic. Let $T'=T \coprod B$ and let $b' \in B(H_{T', {\mathbb {Q}}_p}, \mu _{T'})$ be the element associated with $b$. Then there is an isomorphism

\[ \mathrm{Ig}^b_T \simeq \mathrm{Ig}^{b'}_{T'} \]

whose induced map in cohomology is equivariant with respect to the action of the Hecke operators outside $p$.

Proof. It suffices to produce an isomorphism $\overline {\mathrm {Ig}}^b_T \simeq \overline {\mathrm {Ig}}^{b'}_{T'}$ which satisfies the requirements in the theorem and is in addition ${\mathcal {O}}_{F, (p)}^{\times, +}$-equivariant. Recall that in § 3.3 we have chosen isomorphisms

\[ (D_\emptyset, *_\emptyset, \mathcal{O}_{D_\emptyset}) \simeq (D_T, *_T, \mathcal{O}_{D_T}), \]

i.e. isomorphisms $\theta _T: D_\emptyset \buildrel \sim \over \to D_T$ respecting orders and involutions; we use these isomorphisms to identify the above data, and we denote them by $(D, *, \mathcal {O}_D)$ in this proof. With this notation, Lemma 4.2.2 tells us that, for every $T \subset \Sigma _p$ and every $\bar {\mathbb {F}}_p$-algebra $R$, the set $\overline {\mathrm {Ig}}_T^b(R)$ is the set of isomorphism classes of data $(A, \iota, \lambda, \eta, \rho )$ where:

1. (i) $A/\mathrm {Spec} \, R$ is an abelian scheme of dimension $4g$;

2. (ii) $\iota : \mathcal {O}_{D}\rightarrow \mathrm {End}_R(A)$ is an embedding;

3. (iii) $\lambda : A \rightarrow A^\vee$ is a ${\mathbb {Z}}_{(p)}^\times$ polarisation whose attached Rosati involution coincides with $*$ on $\mathcal {O}_{D}$;

4. (iv) $\eta$ is a level structure, defined as in § 3.3;

5. (v) $\rho : A[p^\infty ]\rightarrow \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R$ is a quasi-isogeny commuting with $\mathcal {O}_{D}$-action and polarisation (the latter up to a constant).

In particular, the only datum in the description of the functor of points of $\overline {\mathrm {Ig}}_T^b$ which depends on $T$ is the quasi-isogeny $\rho : A[p^\infty ]\rightarrow \mathbb {X}^b_T\times _{\bar {\mathbb {F}}_p}R$. Hence, in order to complete the proof it suffices to show that, if $b$ is associated with $b'$, then there is a quasi-isogeny $\rho _{T, T'}: \mathbb {X}^b_T\rightarrow \mathbb {X}^{b'}_{T'}$ commuting with the extra structure. The functor sending $(A, \iota, \lambda, \eta, \rho )$ to $(A, \iota, \lambda, \eta, \rho _{T, T'}\circ \rho )$ will then give the desired isomorphism $\overline {\mathrm {Ig}}^b_T \simeq \overline {\mathrm {Ig}}^{b'}_{T'}$; as ${\mathcal {O}}_{F, (p)}^{\times, +}$ only acts modifying polarisations by prime to $p$ quasi-isogenies, this isomorphism is ${\mathcal {O}}_{F, (p)}^{\times, +}$-equivariant. The existence of a quasi-isogeny $\rho _{T, T'}$ as above follows from the construction in the proof of [Reference Tian and XiaoTX16, Lemma 5.18]. Indeed, letting ${\mathcal {D}}_T$ be the Dieudonné module of $\mathbb {X}^b_T$, the argument in [Reference Tian and XiaoTX16, proof of Lemma 5.18, p. 2185] produces a Dieudonné module $p{\mathcal {D}}_T \subset N \subset {\mathcal {D}}_T$, giving rise to a closed finite subgroup scheme $Z\subset \mathbb {X}^b_T[p]$; moreover $\mathbb {X}^b_T/Z$ is isomorphic to $\mathbb {X}^{b'}_{T'}$, and the construction in [Reference Tian and XiaoTX16, proof of Lemma 5.18, p. 2185] endows the quotient $\mathbb {X}^b_T/Z$ with extra structures respected by the isogeny $\mathbb {X}^b_T/Z\rightarrow \mathbb {X}^b_T$ whose composite with the quotient map is multiplication by $p$.

5.1.1

Let $\breve {{\mathcal {O}}}_\wp$ be the ring of integers of the completion of the maximal unramified extension of $E_\wp$, and let $\mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp }$ be the category of $\breve {{\mathcal {O}}}_\wp$-algebras in which $p$ is nilpotent. Following [Reference Caraiani and ScholzeCS17, Definition 4.3.11], we consider the functor $\mathfrak {X}^b: \mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp } \rightarrow \mathrm {Sets}$ sending $R$ to the set of isomorphism classes of data $(A, \iota, \lambda, \eta, \varphi )$, where:

1. (i) $(A, \iota, \lambda, \eta )/\mathrm {Spec} \, R$ is a datum as in § 3.3; and

2. (ii) $\varphi =\bigoplus _{\mathfrak {p} \in \Sigma _p}\varphi _\mathfrak {p}: A[p^\infty ]\times _{R} R/p=\bigoplus _{\mathfrak {p} \in \Sigma _p}A[\mathfrak {p}^\infty ]\times _{R} R/p\rightarrow \bigoplus _{\mathfrak {p} \in \Sigma _p} \mathbb {X}^{b_\mathfrak {p}}\times _{\bar {\mathbb {F}}_p}R/p$ is a quasi-isogeny commuting with the ${\mathcal {O}}_D$-action, and such that each $\varphi _\mathfrak {p}$ respects the polarisation up to a ${\mathbb {Q}}_p^\times$-factor.

We will now decompose $\mathfrak {X}^b$ as a product of Rapoport–Zink spaces and a formal lift of an Igusa variety we introduced before, as in [Reference Caraiani and ScholzeCS17, Lemma 4.3.12]. The existence of such lifts is a consequence of the next lemma, which is analogous to [Reference Caraiani and ScholzeCS17, Corollary 4.3.5].

Proof. We have to show that, for every ring $R$, the absolute Frobenius $\mathrm {Fr}: \overline {\mathrm {Ig}}^b \rightarrow \overline {\mathrm {Ig}}^b$ induces a bijection on $R$-points. We will use the description of the functor of points of $\overline {\mathrm {Ig}}^b$ given in Lemma 4.2.2. As the map $\mathrm {Fr}$ is not a morphism of $\bar {\mathbb {F}}_p$-schemes, in this proof we need to take the $\bar {\mathbb {F}}_p$-algebra structure on $R$ into account. Let $(r, A, \iota, \lambda, \eta, \rho )$ be an $R$-point of $\overline {\mathrm {Ig}}^b$; its image via Frobenius is the $R$-point $(r^{(p)}, A^{(p)}, \iota ^{(p)}, \lambda ^{(p)}, \eta ^{(p)}, \rho ^{(p)})$ where:

1. (i) $r^{(p)}$ is the composite of $r$ and the Frobenius on $\bar {\mathbb {F}}_p$;

2. (ii) $A^{(p)}:=A \times _{R, \mathrm {Fr}_R}R$;

3. (iii) ($\iota ^{(p)}, \lambda ^{(p)}, \eta ^{(p)}$) are induced from $(\iota, \lambda, \eta )$ by functoriality;

4. (iv) $\rho ^{(p)}:=\rho \times \mathrm {Id}: A[p^\infty ] \times _{R, \mathrm {Fr}_R}R \rightarrow (\mathbb {X}^b\times _{\overline {\mathbb {F}}_p, r} R) \times _{R, \mathrm {Fr}_R} R = \mathbb {X}^b\times _{\overline {\mathbb {F}}_p, r^{(p)}} R$.

Let $F_A: A \rightarrow A^{(p)}$ be the relative Frobenius morphism. Then $F_A$ commutes with $\iota, \iota ^{(p)}$ (as well as with the level structures) by functoriality. Furthermore, we have

\[ (F_A)^\vee \circ \lambda^{(p)} \circ F_A= (F_A)^\vee \circ F_{A^\vee} \circ \lambda=p \lambda \]

where the first equality holds true by functoriality of relative Frobenius and the second follows from the fact that $(F_A)^\vee$ is the Verschiebung on $A^\vee$, and the composite of Verschiebung and Frobenius is multiplication by $p$. Finally, we have

\[ \rho^{(p)}\circ F_A=F_{\mathbb{X}^b}\circ \rho, \]

where $F_{\mathbb {X}^b}: \mathbb {X}^b \times _{\overline {\mathbb {F}}_p, r} R \rightarrow \mathbb {X}^b \times _{\overline {\mathbb {F}}_p, r^{(p)}} R$ is the relative Frobenius. Writing $\prod _{\mathfrak {p} \in \Sigma _p}x_\mathfrak {p}=pt$ with $t \in {\mathcal {O}}_{F, (p)}^{\times, +}$, we deduce that the $R$-point $(r^{(p)}, A^{(p)}, \iota ^{(p)}, \lambda ^{(p)}, \eta ^{(p)}, \rho ^{(p)})$ coincides with the $R$-point

\[ \bigg(r^{(p)}, A, \iota, \frac{1}{t}\lambda, \eta, F_{\mathbb{X}^b}\circ \rho\bigg). \]

Let us denote by $r^{(-p)}$ the composition of $r$ and the inverse of Frobenius on $\overline {\mathbb {F}}_p$; we have the Verschiebung map $V_{\mathbb {X}^b}: \mathbb {X}^b \times _{\bar {\mathbb {F}}_p, r} R \rightarrow \mathbb {X}^b \times _{\bar {\mathbb {F}}_p, r^{(-p)}} R$, and the map $\overline {\mathrm {Ig}}^b \rightarrow \overline {\mathrm {Ig}}^b$ which on $R$-points is given by

\[ (r, A, \iota, \lambda, \eta, \rho) \mapsto \bigg(r^{(-p)}, A, \iota, t\lambda, \eta, \frac{1}{p}V_{\mathbb{X}^b} \circ \rho\bigg) \]

is the inverse of the map induced by $\mathrm {Fr}: \overline {\mathrm {Ig}}^b \rightarrow \overline {\mathrm {Ig}}^b$.

5.1.3 The formal Igusa variety

Since the Igusa variety $\overline {\mathrm {Ig}}^b$ is perfect, it lifts canonically to a flat $p$-adic formal scheme $\overline {\mathfrak {Ig}}^b: \mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp } \rightarrow \mathrm {Sets}$ described as follows (see [Reference Caraiani and ScholzeCS17, p. 719]): fix a lift (up to quasi-isogeny) $\hat {\mathbb {X}}^{b}/\breve {{\mathcal {O}}}_\wp$ of $\mathbb {X}^b$ with extra structure. For $R \in \mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp }$, the $R$-points of $\overline {\mathfrak {Ig}}^b$ are isomorphism classes of data $(A, \iota, \lambda, \eta, \phi )$ as in § 4.2.1, except that the target of $\phi$ is $\hat {\mathbb {X}}^{b}_R:=\hat {\mathbb {X}}^{b}\times _{\breve {{\mathcal {O}}}_\wp }R$.

The group $\Delta$ defined in § 4.2 acts on $\overline {\mathfrak {Ig}}^b$, and we set $\mathfrak {Ig}^b:=\overline {\mathfrak {Ig}}^b/\Delta$.

5.1.4 Rapoport–Zink spaces

For every $\mathfrak {p} \in \Sigma _p$ we have the Rapoport–Zink space $\mathfrak {M}^{b_\mathfrak {p}}$ which is the formal scheme $\mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp } \rightarrow \mathrm {Sets}$ sending $R$ to the set of isomorphism classes of data $({\mathcal {G}}, \iota, \lambda, \rho )$, where:

1. (i) ${\mathcal {G}}$ is a $p$-divisible group over $R$;

2. (ii) $\iota : {\mathcal {O}}_{D, \mathfrak {p}}\rightarrow \mathrm {End}_R({\mathcal {G}})$ is an action of ${\mathcal {O}}_{D, \mathfrak {p}}:={\mathcal {O}}_D\otimes _{{\mathcal {O}}_F}{\mathcal {O}}_{F_\mathfrak {p}}$;

3. (iii) $\lambda$ is a polarisation compatible with the involution on ${\mathcal {O}}_{D, \mathfrak {p}}$ induced by $*$ and such that the Kottwitz condition is satisfied;

4. (iv) $\rho : {\mathcal {G}} \times _{R}R/p \rightarrow \mathbb {X}^{b_\mathfrak {p}}\times _{\bar {\mathbb {F}}_p}R/p$ is a quasi-isogeny commuting with the ${\mathcal {O}}_{D, \mathfrak {p}}$-action, and respecting polarisations up to a ${\mathbb {Q}}_p^\times$-factor.

Furthermore, we require $\lambda$ to be principal if $\mathfrak {p} \not \in T$. If $\mathfrak {p} \in T$, we ask the cokernel of the map induced by $\lambda$ on Lie algebras of the universal vector extensions of the relevant $p$-divisible groups to be locally free of rank two over $R \otimes _{{\mathbb {Z}}_p}\mathcal {O}_E/\mathfrak {p}\mathcal {O}_E$.

Two tuples $({\mathcal {G}}, \iota, \lambda, \rho )$ and $({\mathcal {G}}', \iota ', \lambda ', \rho ')$ are regarded as isomorphic if there exists an isomorphism ${\mathcal {G}} \rightarrow {\mathcal {G}}'$ commuting with $\iota$, $\iota '$, $\rho$ and $\rho '$, and respecting polarisations up to a ${\mathbb {Z}}_p^\times$-factor. Let $\mathfrak {M}^b:=\prod _{\mathfrak {p} \in \Sigma _p} \mathfrak {M}^{b_\mathfrak {p}}$.

5.1.5

We will now define a map $\alpha : \overline {\mathfrak {Ig}}^b \times _{\breve {{\mathcal {O}}}_\wp } \mathfrak {M}^b \rightarrow \mathfrak {X}^b$. We will denote the object to which some extra structure is attached by a subscript.

Fix $R \in \mathrm {Nilp}_{\breve {{\mathcal {O}}}_\wp }$ and take

\[ (A, \iota_A, \lambda_A, \eta_A, \phi_A) \in \overline{\mathfrak{Ig}}^b(R) \quad \text{and} \quad ({\mathcal{G}}_\mathfrak{p}, \iota_{{\mathcal{G}}_\mathfrak{p}}, \lambda_{{\mathcal{G}}_\mathfrak{p}}, \rho_{{\mathcal{G}}_\mathfrak{p}})_{\mathfrak{p} \in \Sigma_p} \in \mathfrak{M}^b(R). \]

Let ${\mathcal {G}}=\bigoplus _{\mathfrak {p} \in \Sigma _p}{\mathcal {G}}_\mathfrak {p}$ and let $\rho _ {\mathcal {G}}: {\mathcal {G}} \rightarrow \hat {\mathbb {X}}^b_R$ be the quasi-isogeny lifting $\bigoplus _{\mathfrak {p} \in \Sigma _p}\rho _{{\mathcal {G}}_\mathfrak {p}}$; we get a quasi-isogeny $\rho _{\mathcal {G}}^{-1}\circ \phi _A: A[p^\infty ]\rightarrow {\mathcal {G}}$ which commutes with the action of ${\mathcal {O}}_D$ and such that each component respects polarisations up to a ${\mathbb {Q}}_p^\times$-factor. We need to construct a point $(B, \iota _{B}, \lambda _{B}, \eta _{B}, \varphi _{B}) \in \mathfrak {X}^b(R)$.

Let $B$ be the unique abelian scheme in the $p$-isogeny class of $A$ such that $\rho _{\mathcal {G}}^{-1}\circ \phi _A$ induces an isomorphism $\gamma =(\gamma _\mathfrak {p})_{\mathfrak {p} \in \Sigma _p}: B[p^\infty ] \buildrel \sim \over \to {\mathcal {G}}$. We endow $B$ with an action $\iota _B:{\mathcal {O}}_D \rightarrow \mathrm {End}_R(B)$ and with a level structure $\eta _{B}$ defined as in the proof of Lemma 4.2.2. To define the polarisation $\lambda _{B}$, look at the following diagram.

As in the proof of Lemma 4.2.2 we may rescale the quasi-isogeny $(q^\vee )^{-1}\circ \lambda _A\circ q^{-1}$ and get a polarisation $\lambda _B$ on $B$ such that $\gamma$ interchanges it with $\lambda _{\mathcal {G}}$ up to a ${\mathbb {Z}}_p^\times$-factor on each component. Finally, let $\bar {\gamma }$ be the reduction of $\gamma$ modulo $p$ and

\[ \varphi_B:=(\bigoplus_{\mathfrak{p} \in \Sigma_p}\rho_{{\mathcal{G}}_\mathfrak{p}})\circ \bar{\gamma}: B[p^\infty]\times_R R/p\rightarrow \mathbb{X}^b\times_{\bar{\mathbb{F}}_p}R/p. \]

Note that, by construction, the polarisation $\lambda _B$ is unchanged if ${\mathcal {G}}$ is replaced by a $p$-divisible group ${\mathcal {G}}'$ endowed with an isomorphism to ${\mathcal {G}}$ respecting the extra structures, the polarisation being respected up to a ${\mathbb {Z}}_p^\times$-factor on each component. Therefore, the resulting map $\alpha : \overline {\mathfrak {Ig}}^b \times _{\breve {{\mathcal {O}}}_\wp } \mathfrak {M}^b \rightarrow \mathfrak {X}^b$ is well-defined.

Proof. We will construct an inverse $\beta : \mathfrak {X}^b \rightarrow \overline {\mathfrak {Ig}}^b \times _{\breve {{\mathcal {O}}}_\wp } \mathfrak {M}^b$ of the map $\alpha$. Take $(B, \iota _{B}, \lambda _{B}, \eta _{B}, \varphi _{B}) \in \mathfrak {X}^b(R)$ and let ${\mathcal {G}}:=B[p^\infty ]$, endowed with the additional structures coming from $B$, and with the trivialisation $\rho _{\mathcal {G}}:=\varphi _B$. This gives an $R$-point of $\mathfrak {M}^b$. Finally, define $(A, \iota _A, \lambda _A, \eta _A, \phi _A)$ as follows: the abelian scheme $A$ is the unique one in the $p$-isogeny class of $B$ such that the lift of $\varphi _{B}$ induces an isomorphism $\phi _A: A[p^\infty ] \rightarrow \hat {\mathbb {X}}^b_R$, and the additional structures are obtained from those on $B$ as in the discussion before the statement of the lemma.

With the notation introduced in the construction of the map $\alpha$, the composite $\beta \circ \alpha$ sends

\[ ({\mathcal{G}}, \iota_{{\mathcal{G}}}, \lambda_{{\mathcal{G}}}, \bigoplus_{\mathfrak{p} \in \Sigma_p}\rho_{{\mathcal{G}}_\mathfrak{p}}), (A, \iota_A, \lambda_A, \eta_A, \phi_A) \]

to

\[ (B[p^\infty], \iota_B[p^\infty], \lambda_B[p^\infty], \bigoplus_{\mathfrak{p} \in \Sigma_p}\rho_{{\mathcal{G}}_\mathfrak{p}} \circ \bar{\gamma}), (A, \iota_A, \lambda_A, \eta_A, \phi_A). \]

The map $\gamma$ gives an isomorphism, respecting polarisations up to ${\mathbb {Z}}_p^\times$ on each factor, between $(B[p^\infty ], \iota _B[p^\infty ], \lambda _B[p^\infty ], \bigoplus _{\mathfrak {p} \in \Sigma _p}\rho _{{\mathcal {G}}_\mathfrak {p}} \circ \bar {\gamma })$ and $({\mathcal {G}}, \iota _{{\mathcal {G}}}, \lambda _{{\mathcal {G}}}, \bigoplus _{\mathfrak {p} \in \Sigma _p}\rho _{{\mathcal {G}}_\mathfrak {p}})$, hence $\beta \circ \alpha$ is the identity map. Similarly one checks that $\alpha \circ \beta$ is the identity.

5.2.1 Generic fibres of formal schemes

We will need to work with the adic generic fibres of the formal schemes introduced in the previous section. Recall that to a formal scheme $\mathfrak {M}$ over $\breve {{\mathcal {O}}}_\wp$ (locally admitting a finitely generated ideal of definition) one can attach an adic space $\mathfrak {M}^{ad}$ as in [Reference Scholze and WeinsteinSW13, Proposition 2.2.1] over $\mathrm {Spa}(\breve {{\mathcal {O}}}_\wp, \breve {{\mathcal {O}}}_\wp )$; one can then take the adic generic fibre

\[ \mathcal{M}:=\mathfrak{M}^{ad}\times_{\mathrm{Spa}(\breve{{\mathcal{O}}}_\wp, \breve{{\mathcal{O}}}_\wp)}\mathrm{Spa}(\breve{E}_\wp, \breve{{\mathcal{O}}}_\wp), \]

where $\breve {E}_\wp$ is the fraction field of $\breve {{\mathcal {O}}}_\wp$. With this notation, the product formula in Lemma 5.1.6 becomes, on the generic fibre:

(5.2.1.1)\begin{equation} \overline{\mathcal{I}g}^b \times_{\mathrm{Spa}(\breve{E}_\wp, \breve{{\mathcal{O}}}_\wp)} \mathcal{M}^b \xrightarrow{\sim} \mathcal{X}^b. \end{equation}

The functors of points on complete affinoid $(\breve {E}_\wp, \breve {{\mathcal {O}}}_\wp )$-algebras of the adic spaces in the previous formula can be described using [Reference Scholze and WeinsteinSW13, Proposition 2.2.2]. It is easier, and sufficient for our purposes, to give such a description restricted to the category $\mathrm {Perf}_{\breve {E}_\wp }$ of perfectoid Huber pairs $(R, R^+)$ over $(\breve {E}_\wp, \breve {{\mathcal {O}}}_\wp )$. For example, the functor of points of $\mathcal {X}^b$ is the sheafification in the analytic topology of the functor

\begin{align*} \mathrm{Perf}_{\breve{E}_\wp} & \rightarrow \mathrm{Sets}\\ (R, R^+) & \mapsto \varprojlim_n \mathfrak{X}^b(R^+/p^n). \end{align*}

In other words, up to sheafification, an $(R, R^+)$-point of $\mathcal {X}^b$ is a compatible collection of data $(A_n, \iota _n, \lambda _n, \eta _n, \varphi _n)$ over $R^+/p^n$ as defined in § 5.1. Note that giving a compatible sequence of quasi-isogenies $\varphi _n$ just amounts to giving $\varphi _1$; on the other hand, we may see the compatible sequence $(A_n, \iota _n, \lambda _n, \eta _n)$ as a formal scheme with extra structure over $\operatorname {Spf} R^+$, which as above can be regarded as an adic space; we will denote it by $(\mathcal {A}, \iota, \lambda, \eta )$. Hence, the compatible sequence of data $(A_n, \iota _n, \lambda _n, \eta _n, \varphi _n)$ uniquely corresponds to a datum of the form $(\mathcal {A}, \iota, \lambda, \eta, \varphi )$.

A similar description can be given of the functor of points of the other objects appearing in (5.2.1.1), as well as of the good reduction locus $\mathcal {Y}_{U}(H)^\circ$ inside the analytification $\mathcal {Y}_{U}(H)$ of the space $Y_{U}(H)_{E_\wp }$; the space $\mathcal {Y}_{U}(H)^\circ$ is defined as the generic fibre of the completion along the special fibre of $Y_{U, {\mathcal {O}}_\wp }(H)$. In particular, [Reference Scholze and WeinsteinSW13, Lemma 2.2.2] applies and yields a description of the functor of points of $\mathcal {Y}_{U}(H)^\circ$ similar to that discussed above for $\mathcal {X}^b$. Note that the inclusion $\mathcal {Y}_{U}(H)^\circ \subset \mathcal {Y}_{U}(H)$ is an equality if $T$ is non-empty, but it is strict if $T$ is empty.

We will now introduce versions with infinite level at $p$ of the spaces considered so far.

5.2.2 The space $\mathcal {M}^b_{\infty }$

We will first define the Rapoport–Zink space at infinite level $\mathcal {M}^b_{\infty }:=\prod _{\mathfrak {p} \in \Sigma _p}\mathcal {M}^{b_\mathfrak {p}}_{\infty }$. Each $\mathcal {M}^{b_\mathfrak {p}}_{\infty }$ is a pro-étale cover of the generic fibre $\mathcal {M}^{b_\mathfrak {p}}$ of the adic space attached to the formal scheme $\mathfrak {M}^{b_\mathfrak {p}}$.

Fix $\mathfrak {p} \in \Sigma _p$; as recalled above, the functor of points $\mathrm {Perf}_{\breve {E}_\wp } \rightarrow \mathrm {Sets}$ of $\mathcal {M}^{b_\mathfrak {p}}$ is the sheafification of the functor sending $(R, R^+)$ to $\varprojlim _n \mathfrak {M}^{b_\mathfrak {p}}(R^+/p^n)$. Giving such a compatible system of $(R^+/p^n)$-points amounts to giving a compatible collection of $p$-divisible groups with extra structure $(\mathcal {G}_n, \iota _n, \lambda _n)$ as in § 5.1.4 and a quasi-isogeny (respecting extra structures) $\rho : \mathcal {G}_1 \rightarrow \mathbb {X}^{b_\mathfrak {p}} \times _{\bar {\mathbb {F}}_p} R^+/p$. Now, by [Reference MessingMes72, Lemma (4.16)], giving a compatible sequence of $p$-divisible groups with extra-structure $(\mathcal {G}_n, \iota _n, \lambda _n)$ is equivalent to giving a $p$-divisible group with extra structure $(\mathcal {G}, \iota, \lambda )$ over $R^+$. For every $n \geq 1$, the Tate module functor $T(\mathcal {G}_n)$ from $R^+/p^n$-algebras to sets sending $S$ to $\varprojlim _{k}\mathcal {G}_n[p^k](S)$ is represented by an affine scheme, flat over $\operatorname {Spec}(R^+/p^n)$, by [Reference Scholze and WeinsteinSW13, Proposition 3.3.1]. Taking the limit over $n$ and passing to the adic generic fibre we obtain an adic space which we denote by $\mathcal {T}(\mathcal {G})$.