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.

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Notes

  1. 1.

    Variants of such a formula were also studied in [10, 53]. Readers are referred to the introduction of [21] for the works prior to loc. cit.

  2. 2.

    This is where we use the assumption that \(\mathsf {G}\) is unramified at p.

  3. 3.

    Instead of parahoric level structure, we will work with the “Bruhat–Tits level structure” for technical reasons; in other words, the level at p is given by the full stabiliser in \(\mathsf {G}(\mathbb {Q}_p)\) of some facet in the Bruhat–Tits building, not necessarily by the connected stabiliser (which is a parahoric group). Note that integral models with the “Bruhat–Tits level structure” is obtained as the normalisation of the closure of the generic fibre in some Siegel modular variety.

  4. 4.

    “Displays” over (not necessarily affine) p-adic formal schemes are described right above the statement of Proposition 4.6.

  5. 5.

    Assume that the étale \(\mathbb {Z}_l\)-sheaf \(\mathscr {F}\) is given by an inverse system \(\{\mathscr {F}_n\}\) for \(\mathscr {F}_n\in (\mathbb {Z}/l^n)\text {-}{\widetilde{X}}_{\acute{\mathrm{e}}\text {t}}\). Then a section in \(\Gamma (X,\mathscr {F})\) has proper support if its image in \(\Gamma (X,\mathscr {F}_n)\) is supported in a common quasi-compact closed subset independent of n. Therefore, \(\Gamma _c(X,\mathscr {F})\) may not coincide with the inverse limit of \(\Gamma _c(X,\mathscr {F}_n)\) in general, as it was pointed out in [24, Introduction].

  6. 6.

    In [23], \(\mathfrak {X}^\mathrm {ad}\) is denoted by \(t(\mathfrak {X})\).

  7. 7.

    We will recall the definition of compactly supported distributions in Sect. 2.4.

  8. 8.

    In [48, §3.3], the compactly supported J-equivariant cohomology is denoted by \(\mathrm {R}\Gamma _c(\mathcal {X}/J,\bullet )\), but we find this notation could lead to confusion with the compactly supported cohomology of the “quotient” \(\mathcal {X}/J\) in some suitable sense.

  9. 9.

    We would like to view \(M/\mathbb {I}_R M\) as the “first de Rham homology” of some Barsotti–Tate group, and \(M/M_1\) as the Lie algebra of the Barsotti–Tate group, which is the \((-1)\)th grading of the first de Rham homology. We follow the standard notation to let \(M_1\subset M\) denote the \(\mathbb {W}(R)\)-submodule defining the Hodge filtration (although \(M^0\) may be more natural notation as it defines the 0th filtration).

  10. 10.

    Although Faltings [9] works with the divided power completion of \(A_\mathrm{cris}(O_C)\) in place of \(A_\mathrm{cris}(O_C)\), the essentially same proof work over \(A_\mathrm{cris}(O_C)\).

  11. 11.

    Note that the two conditions Zhou imposed at the beginning of section 3 in [66], i.e. that G is quasi-split and \(\mathsf {K}_p = \mathsf {K}_p^\circ \), are not needed for this proposition. Since the construction of local models commutes with base change, we may exchange \(\mathbb {Q}_p\) by an unramified extension F such that \(G_F\) is quasi-split. Moreover \(\mathsf {K}_p = \mathsf {K}_p^\circ \) is not needed in the proof; while [66, Prop. 3.4] then proves the claim for \(\mathsf {K}_p^\circ \), we know that the canonical morphism \(\bigcup _{\tilde{w}\in {{\,\mathrm{Adm}\,}}(\mu )} \mathsf {K}_p^\circ \tilde{w}\mathsf {K}_p^\circ /\mathsf {K}_p^\circ \rightarrow \bigcup _{\tilde{w}\in {{\,\mathrm{Adm}\,}}(\mu )} \mathsf {K}_p \tilde{w}\mathsf {K}_p\) is an isomorphism by the argument of Remark 5.4.

  12. 12.

    Although our integral models are the normalisations of the integral models in [44], we can pull back the statement of [44, §7, Proposition 21] via the normalisation maps. Indeed, the formation of the formal nearby cycle sheaves commute with proper base change of formal models; cf. [3, Corollary 2.3(ii)].

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Acknowledgements

We thank Rong Zhou and Stephan Neupert for helpful conversations and in particular for sharing a preliminary version of their respective preprints with us. We are grateful to Kai-Wen Lan for his explanations about compactifications of Shimura varieties. We would also like to thank the anonymous referee, whose comments have been extremely helpful. The first named author was partially supported by the ERC starting Grant 277889 “Moduli of local G-shtukas”. The second named author was supported by the EPSRC (Engineering and Physical Sciences Research Council) in the form of EP/L025302/1.

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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. (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. (2)

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

  3. (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. (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. (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

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

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Hamacher, P., Kim, W. l-Adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients. Math. Ann. 375, 973–1044 (2019). https://doi.org/10.1007/s00208-019-01815-6

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