l-Adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Abstract

Let $(\mathsf {G},\mathsf {X})$ be a Shimura datum of Hodge type. Let p be an odd prime such that $\mathsf {G}_{\mathbb {Q}_p}$ splits after a tamely ramified extension and $p\not \mid |\pi _1(\mathsf {G}^\mathrm{der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf {G}_{\mathbb {Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan’s formula for the l-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo p of the Kisin–Pappas integral model.

Appendices

Appendix A: Homomorphisms of constant F-crystals

In order to show that the Igusa variety associated to a Barsotti–Tate group $(\mathbb {X},(t_\alpha ))$ is a flat closed subvariety of the Igusa variety associated to $\mathbb {X}$, we will need the proposition below. To improve readability, we use the following notation. For any profinite set S and topological ring R, we denote

$$\begin{aligned} R^S :={{\,\mathrm{Map}\,}}_\mathrm{cont}(S,R). \end{aligned}$$

Unless stated explicitly otherwise, we will assume that R is equipped with the discrete topology.

Proposition A.1

Let $M_1, M_2$ be F-crystals over an algebraically closed field k. Denote $H = {{\,\mathrm{Hom}\,}}(M_1,M_2)$ with the p-adic topology. We denote by the same symbol the associated k-ring scheme ${{\,\mathrm{Spec}\,}}C^\infty (H,k)$, which may also be described via the functor of points $H(S) = H^{\pi _0(S)}.$ Then H represents the functor

$$\begin{aligned} (\mathrm{PerfSch}_k)&\rightarrow (\mathrm{Set}) \\ S&\mapsto {{\,\mathrm{Hom}\,}}(M_{1,S},M_{2,S}) \end{aligned}$$

Proof

The proof is identical to the proof of [54, Lemma 3.9], except for the last sentence. Here Rapoport and Richartz use $W(R)_\mathbb {Q}^\sigma = \mathbb {Q}_p$, i.e. require ${{\,\mathrm{Spec}\,}}R$ to be connected. The same argument still works for us if we replace it by $W(R)_\mathbb {Q}^\sigma = \mathbb {Q}_p^{\pi _0({{\,\mathrm{Spec}\,}}R)}$ (w.r.t. the p-adic topology), which is proven in Lemma A.2 and Lemma A.3 below.$\square$

For any $\mathbb {F}_q$-algebra R we have a canonical embedding $\mathbb {F}_q^{\pi _0({{\,\mathrm{Spec}\,}}R)} \hookrightarrow R$ given as follows. For any $f \in \mathbb {F}_q^{\pi _0({{\,\mathrm{Spec}\,}}R)}$ consider the clopen sets $U_{f,x} \subset {{\,\mathrm{Spec}\,}}R$ given by the preimage of $x \in \mathbb {F}_q$ w.r.t. the composition ${{\,\mathrm{Spec}\,}}R \rightarrow \pi _0({{\,\mathrm{Spec}\,}}R) {\mathop {\rightarrow }\limits ^{f}} \mathbb {F}_q$. Then there exists a unique idempotent $e_{f,x} \in R$ such that $U_{f,x} = D(e_{f,x})$. We define the image of f in R to be $\sum _{x \in \mathbb {F}_q} x \cdot e_{f,x}$.

Lemma A.2

For any $\mathbb {F}_q$-algebra R one has $\{r \in R \mid r^q = r\} = \mathbb {F}_q^{\pi _0({{\,\mathrm{Spec}\,}}R)}$.

Proof

The inclusion “$\supseteq$” is obvious. Thus let $r \in R$ with $r^q - r = 0$. Then ${{\,\mathrm{Spec}\,}}R = \coprod _{x \in \mathbb {F}_q} V(r-x)$, thus r defines an locally constant function

$$\begin{aligned} \varphi : {{\,\mathrm{Spec}\,}}R&\rightarrow \mathbb {F}_q \\ \mathfrak {p}&\mapsto r \mod \mathfrak {p}. \end{aligned}$$

Let $f \in \mathbb {F}_q^{\pi _0({{\,\mathrm{Spec}\,}}R)}$ be the corresponding element. Then $r-f$ is nilpotent, say $(r-f)^{q^n} = 0$. Now

$$\begin{aligned} r-f = r^{q^n} - f^{q^n} = (r-f)^{q^n} = 0. \end{aligned}$$

$\square$

Lemma A.3

Let R be a perfect ring of characteristic p and S be a profinite set. We have

$$\begin{aligned} W(R^S) = W(R)^S \end{aligned}$$

where we consider the p-adic topology on W(R).

Proof

Recall that W(R) is uniquely characterised by the property that it is torsion-free, p-adically complete such that $W(R)/p = R$. As a direct consequence we obtain that for any directed system $(R_\lambda )$ of perfect rings $W(\varinjlim R_\lambda )$ is the p-adic completion of $\varinjlim W(R_\lambda )$. Now let $S = \varprojlim S_\lambda$ with $S_\lambda$ finite. We obtain

$$\begin{aligned} W(R^S)&= W(\varinjlim \, R^{S_\lambda }) \\&= (\varinjlim \, W(R^{S_\lambda }))^{\wedge \, p-ad} \\&= (\varinjlim \, W(R)^{S_\lambda })^{\wedge \, p-ad} \\&= {{\,\mathrm{Map}\,}}_\mathrm{loc.~const.}(S,W(R))^{\wedge \,p-ad} \\&= W(R)^S. \end{aligned}$$

$\square$

Appendix B: The Künneth formula for proétale torsors

The cohomology of compact support is normally only defined for schemes of finite type. We define it for a slightly larger class of schemes, as we also want to calculate it for pro-étale torsors over varieties.

Definition/Lemma B.1

Let $f:P \rightarrow S$ be a morphism of schemes where S is noetherian and f factorises as $P \xrightarrow {g} X \xrightarrow {h} S$ where h is locally of finite type and g is integral. We define the derived pushforward with compact support for étale sheaves on P by

$$\begin{aligned} \mathrm {R}f_! :=\mathrm {R}h_! \circ \mathrm {R}g_*\end{aligned}$$

This definition does not depend on the choice of factorisation above.

Proof

Let $P \xrightarrow {g} X \xrightarrow {h} S$ and $P \xrightarrow {g'} X' \xrightarrow {h'} S$ be two factorisations as above. If $g'$ factorises as $P \xrightarrow {g} X \xrightarrow {g''} X'$, then $g''$ is finite and thus

$$\begin{aligned} \mathrm {R}h'_! \circ \mathrm {R}g'_*= \mathrm {R}h'_! \circ \mathrm {R}g''_*\circ \mathrm {R}g_*= \mathrm {R}h'_! \circ \mathrm {R}g''_! \circ \mathrm {R}g_*= \mathrm {R}h_! \circ R g_*\end{aligned}$$

In general, since

$$\begin{aligned} Rh_! = \varinjlim _{U \subset X \text { qc open}}\, Rh_{U !} \end{aligned}$$

where $h_U$ is the composition of the open embedding $U \hookrightarrow X$ with h, we may assume that h and $h'$ are of finite type. Writing $P = \varprojlim X_\lambda$, where $X_\lambda$ is finite over X, we see that $g'$ factors over some $X_\lambda$ since $X'$ is of finite presentation over S. Replacing X by $X_\lambda$, we are in the situation discussed above. $\square$

In the case where $S = {{\,\mathrm{Spec}\,}}k$, we may simply write $\mathrm {R}\Gamma _c(X,-)$ instead of $\mathrm {R}f_!$ and $H_c^i(X,-)$ instead of $\mathrm {R}^i f_!$.

Remark B.2

For special cases of f, the definition of $\mathrm {R}f_!$ simplifies.

• (1) If $f:X\rightarrow S={{\,\mathrm{Spec}\,}}k$ is locally of finite type, then this definition of $\mathrm {R}\Gamma _c(X,-)$ coincides with Definition 2.2.

• (2) If X is proper, $\mathrm {R}f_! = \mathrm {R}f_*$.

• (3) Writing $P = \varprojlim X_\lambda$ with $X_\lambda \xrightarrow {g_\lambda } X$ finite, assume that the sheaf $\mathscr {L}$ over the étale site of P is defined as the pullback of a sheaf $\mathscr {L}_\lambda$ over the étale site of $X_\lambda$. By [61, 09YQ] we have $\mathrm {R}f_! \mathscr {L}= \varinjlim \mathrm {R}f_{\lambda , !} \mathscr {L}_\lambda$ where $f_\lambda = h \circ g_\lambda$. Thus the usual ad hoc definition of cohomology with compact support for infinite level Shimura varieties coincides with the definition above.

We fix a locally profinite group J with a compact open pro-p-subgroup $K_0$ and denote by dh the Haar measure such that $dh(K_0) = 1$. From now on we assume that $f: P \rightarrow X$ is a proétale J-torsor and that $\mathscr {L}$ is a smooth J-equivariant abelian sheaf on P such that p is invertible in $\mathscr {L}$. The J-action defines an effective descent datum on $\mathscr {L}$; let $\mathscr {L}_X$ denote the corresponding sheaf on X. One obtains the following explicit description of the stalks of $\mathrm {R}^0 f_!\mathscr {L}$.

Lemma B.3

Let $\bar{x}$ be a geometric point of X.

• (1) At any geometric point $\bar{x}$ of X,  the stalk of $\mathrm {R}^0 f_! \mathscr {L}$ at $\bar{x}$ is canonically isomorphic to $C_c^\infty (f^{-1}(\bar{x}),\mathscr {L}_{X,\bar{x}})$. Moreover,  we have a canonical J-action on $f_!\mathscr {L}$ which is given on stalks by the natural J-action on $C_c^\infty (f^{-1}(\bar{x}),\mathscr {L}_{X,\bar{x}})$.

• (2) There exists a (noncanonical) morphism $\int _f: \mathrm {R}^0 f_!\mathscr {L}\rightarrow \mathscr {L}_X,$ which is given on stalks by

  $$\begin{aligned} C_c^\infty (f^{-1}(\bar{x}),\mathscr {L}_{X,\bar{x}}) \rightarrow \mathscr {L}_{X,\bar{x}}, \varphi \mapsto \int _{f^{-1}(\bar{x})} \varphi dh. \end{aligned}$$

  In particular,  $\int _f$ is J-equivariant and the definition only depends on the choice of dh.

Proof

For a compact open subgroup K denote $X_K :=P/K$ and let $P \xrightarrow {g_K} X_K \xrightarrow {f_K} X$ be the canonical factorisation of f and $\mathscr {L}_K :=f_K^*\mathscr {L}_X$ the descent of $\mathscr {L}$ to $X_K$.

For $K \subset K_0$ let $g_{K,K_0}: X_{K} \rightarrow X_{K_0}$ be the canonical projection. Then

$$\begin{aligned} g_{K_0\,*} \mathscr {L}= \varinjlim _{K}\, g_{K,K_0\, *} \mathscr {L}_{K} = \varinjlim _{K}\, g_{K,K_0\,*}g_{K,K_0}^*\mathscr {L}_{K_0}. \end{aligned}$$

Since $K_0/K$ is finite, the morphism $g_{K,K_0}$ finite étale. Thus $g_{K,K_0\, *} = g_{K,K_0\,!}$ is the left adjoint of $g_{K,K_0}^*$. Hence the stalk at a geometric point $\overline{y} \in X_{K_0}$ equals (see e.g. [61, Tag 0710])

$$\begin{aligned} (g_{K_0\,*} \mathscr {L})_{\bar{y}} = \varinjlim _{K} (g_{K,K_0\,*}g_{K,K_0}^*\mathscr {L}_{K_0})_{\bar{y}} = \varinjlim _K \bigoplus _{g_{K,K_0}^{-1}(\bar{y})} \mathscr {L}_{K_0,\bar{y}} = C_c^\infty (g_{K_0}^{-1}(\bar{y}),\mathscr {L}_{K_0,\bar{y}}). \end{aligned}$$

Since $J/K_0$ is discrete, $f_{K_0}$ is étale; thus by the same argument as above

$$\begin{aligned} (f_{K_0\, !} g_{K_0\,*}\mathscr {L})_{\bar{x}} = \bigoplus _{\bar{y}\in f_{K_0}^{-1}(\bar{x})} (g_{K_0\,*}\mathscr {L})_{\bar{y}} = C_c^\infty (f^{-1}(\bar{x}),\mathscr {L}_{X,\bar{x}}). \end{aligned}$$

Since $f_!$ is functorial, the J-action on $\mathscr {L}$ induces an J-action on $f_!\mathscr {L}$. Tracing through the definitions, one easily checks that the action of J on $(f_{!} \mathscr {L})_{\bar{x}} = C_c^\infty (f^{-1}(\bar{x}),\mathscr {L}_{X,\bar{x}})$ is indeed the canonical action induced by the J-action of $f^{-1}(\bar{x})$.

The morphism $\int _f$ is defined as follows. Let $\tau _{K/K_0} :=\frac{1}{[K_0:K]} \cdot {{\,\mathrm{Tr}\,}}:g_{K,K_0\,*} \mathscr {L}_K \rightarrow \mathscr {L}_{K_0}$. On stalks, this morphism is given by

$$\begin{aligned} \bigoplus _{\bar{z}\in g_{K,K_0}^{-1}(\bar{y})} \mathscr {L}_{K_0,\bar{y}} \rightarrow \mathscr {L}_{K_0,\bar{y}}, (f_{\bar{z}}) \mapsto \frac{1}{[K_0:K]} \cdot \sum _{\bar{z}\in g_{K,K_0}^{-1}(\bar{y})} f_{\bar{z}}, \end{aligned}$$

i.e. taking the average. Hence $\tau :=\varinjlim \, \tau _{K,K_0}:g_{K_0\,*} \mathscr {L}\rightarrow \mathscr {L}_{K_0}$ is given on stalks by taking the integral $\int _{K_0} dh$. We define $\int _f$ as the concatenation

$$\begin{aligned} \mathrm {R}^0f_!\mathscr {L}\xrightarrow {f_{K_0\, !}(\tau )} f_{K_0\, !} \mathscr {L}_{K_0} \xrightarrow {{{\,\mathrm{Tr}\,}}} \mathscr {L}_X. \end{aligned}$$

By the above calculation it indeed coincides with the integral over $f^{-1}(\bar{x})$ on stalks.    $\square$

Corollary B.4

$\mathrm {R}^0 f_!\mathscr {L}$ satisfies property $(\mathcal {P})$ of [43, Def. 5.3], i.e. $(\mathrm {R}^0 f_!\mathscr {L})_{\bar{x}} \cong {{\,\mathrm{c-Ind}\,}}_1^J \mathscr {L}_{X,\bar{x}}$.

In this article, we will consider sheaves which have alongside the $J = J_b(\mathbb {Q}_p)$-action an action of the Weil group $W = W_E$ of the p-adic completion of the Shimura field. We remark that the above construction still works if we consider W-equivariant sheaves for some profinite group W acting on the underlying schemes. For simplicity we assume that $\mathscr {L}$ is a (W-equivariant) sheaf of $\mathbb {Z}/l^r\mathbb {Z}$-modules for a prime $l \not = p$. Let $\mathcal {H}_r = C_c^\infty (J_b(\mathbb {Q}_p),\mathbb {Z}/l^r\mathbb {Z})$ and denote by $\Lambda$ the module $\mathbb {Z}/l^r\mathbb {Z}$ as trivial $\mathcal {H}_r$-representation.

The following statements are a formal consequence of Corollary B.4, and properties of $R\Gamma _c$ (cf. [49, § 7.2][43, § 5.3])

Proposition B.5

(Cf. [49, Prop. 7.2.2], [43, Thm. 5.11]) $\mathrm {R}\Gamma _c(X,f_!\mathscr {L}_X) = \Lambda \otimes ^L_{\mathcal {H}_r} \mathrm {R}\Gamma _c(P,\mathscr {L})$

Theorem B.6

(Cf. [49, Thm. 7.2.3], [43, Thm. 5.13]) Assume that we may write $P = P_1 \times P_2$ where $P_1$ and $P_2$ are J-equivariant and locally of finite presentation up to Galois cover and assume that

$$\begin{aligned} \mathscr {L}= \mathscr {E}_1 \boxtimes \mathcal {E}_2 \end{aligned}$$

where $\mathscr {E}_i$ is a smooth J-equivariant $\mathbb {Z}/l^r\mathbb {Z}$-sheaf on $P_i$ with continuous W-action. Then

$$\begin{aligned} \mathrm {R}\Gamma _c(X, \mathcal {L}_X) = \mathrm {R}\Gamma _c(P_1,\mathscr {E}_1) \otimes ^L_{\mathcal {H}_r} \mathrm {R}\Gamma _c(P_2,\mathscr {E}_2) \end{aligned}$$

In particular, in the case where all the sheaves are constant étale sheaves associated to $\mathbb {Z}/l^r\mathbb {Z}$, we obtain the following result.

Corollary B.7

(cf. [49, Cor. 7.2.5], [43, Thm. 5.14]) We have a spectral sequence

$$\begin{aligned} E_2^{p,q} = \bigoplus _{s+t = q} {{\,\mathrm{Tor}\,}}_{\mathcal {H}_r}^p(H^s(P_1,\mathbb {Z}/l^r\mathbb {Z}), H^t(P_2,\mathbb {Z}/l^r\mathbb {Z})) \Rightarrow H^{p+q}(X,\mathbb {Z}/l^r\mathbb {Z}). \end{aligned}$$