Local models of Shimura varieties and a conjecture of Kottwitz

Abstract

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

Appendix: Homogeneous spaces

In this section, we study the representability of certain quotients of group schemes over a two-dimensional base. The results are used in Chap. 6 to show the ind-representability of the global affine Grassmannian ${\rm Gr}_{\mathcal {G}, X}$.

11.1

We assume that A is an excellent Noetherian regular ring of Krull dimension 2. If G, H are smooth group schemes over A and H↪G is a closed group scheme immersion then by Artin’s theorem [3], the fppf quotient G/H is represented by an algebraic space which is separated of finite presentation and in fact smooth over A.

Lemma 11.1

Suppose that G ₁↪G ₂↪G ₃ are closed group scheme immersions and G ₁, G ₂, G ₃ are smooth over A. The natural morphism

$$G_3/G_1\to G_3/G_2 $$

is an fppf fibration with fibers isomorphic to G ₂/G ₁. Suppose that G ₂/G ₁ is quasi-affine (affine). If G ₃/G ₂ is a scheme, then so is G ₃/G ₁. If in addition G ₃/G ₂ is quasi-affine (resp. affine), then so is G ₃/G ₁.

Proof

The first statement follows from the fact that fppf descent is effective for quasi-affine schemes. In fact, by our assumption, G ₃/G ₁→G ₃/G ₂ is a quasi-affine morphism. If G ₃/G ₂ is quasi-affine, then G ₃/G ₁ is also quasi-affine (affine) since its structure morphism to A is a composition of quasi-affine (resp. affine) morphisms and as such is also quasi-affine (resp. affine). □

Proposition 11.2

Let $\mathcal {G}, \mathcal {H}\to S=\mathrm {Spec}(A)$ be two smooth affine group schemes with connected fibers. Assume that $\mathcal {H}$ is a closed subgroup scheme of $\mathcal {G}$.

Set $\mathcal {G}=\mathrm {Spec}(B)$, so that B is an A-Hopf algebra. Then there is a free finitely generated A-module M=A ⁿ with $\mathcal {G}$-action (i.e. a B-comodule) and a projective A-submodule W⊂M which is a locally a direct summand, such that:

• (i) There is a $\mathcal {G}$-equivariant surjection ${\rm Sym}^{\bullet}_{A}(M) \to B$ and the $\mathcal {G}$-action on M gives a group scheme homomorphism $\rho: \mathcal {G}\hookrightarrow {\rm GL}(M)$ which is a closed immersion.

• (ii) The representation ρ identifies $\mathcal {H}$ with the subgroup scheme of $\mathcal {G}$ that stabilizes W.

Proof

Write $\mathcal {G}=\mathrm {Spec}(B)$, $\mathcal {H}=\mathrm {Spec}(B')$ and let p:B→B′ be the ring homomorphism that corresponds to $\mathcal {H}\subset \mathcal {G}$. Observe that B, B′ are Hopf algebras over A. We will often refer to the B-comodules for the Hopf algebra B as “modules with $\mathcal {G}$-action”. Since $\mathcal {G}$, $\mathcal {H}$ are smooth with connected geometrical fibers both B and B′ are projective A-modules by Raynaud-Gruson [57, Proposition 3.3.1]. We start with two lemmas. □

Lemma 11.3

Let P be a projective A-module and N⊂P be a finitely generated A-submodule. Then the A-torsion submodule of P/N is finitely generated.

Proof

As P is a direct summand of a free A-module, we can assume that P=A ^I itself is free, with a basis {e _i ;i∈I}. Let n ₁,…,n _t be a set of generators of N, and write n _i =∑a _ij e _j . Then J={j∈I∣∃i such that a _ij ≠0} is a finite set and A ^I/N=A ^J/N⊕A ^I−J. The conclusion follows. □

Lemma 11.4

Let P be a projective A-module and suppose that N⊂P is a finitely generated A-submodule. If P/N is torsion free, then N is a projective A-module.

Proof

Observe that N is A-torsion free. Consider the double dual N ^∨∨. We have N⊂N ^∨∨⊂P, and N ^∨∨/N⊂P/N is torsion. Therefore, N=N ^∨∨, which is projective by our assumption that A is regular and two-dimensional. □

Now let us prove the proposition. Observe first [71, Corollary 3.2] and its proof imply (i) of the proposition. (In other words, [71] implies that such a $\mathcal {G}$ is linear, i.e. a closed subgroup scheme of ${\rm GL}_{n}$.) To obtain the proof of the whole proposition we have to refine this construction from [71] to account for the subgroup scheme $\mathcal {H}$.

Let V⊂B be a finitely generated A-submodule with $\mathcal {G}$ action that contains both a set of generators of I=ker(B→B′) and a set of generators of B as an A-algebra. Let p(V) be the image of V under p:B→B′. The following diagram is a commutative diagram of modules with $\mathcal {G}$-action

where $\mathcal {G}$ acts on V⊗ _A B, p(V)⊗ _A B, B⊗ _A B, B′⊗ _A B via the actions on the second factors.

The image of $N:=(p\otimes1)\cdot{\rm coact}(V)$ in p(V)⊗ _A B is a finite A-submodule with $\mathcal {G}$-action. Let ϵ:B→A be the unit map which splits the natural A⊂B. Let M be the image of N under $B'\otimes B\stackrel{1\otimes\epsilon}{\to} B'$. Observe that M is a finite A-module, but is not necessarily $\mathcal {G}$-stable. By [62, Proposition 2], we can choose a finite $\mathcal {G}$-stable A-module $\tilde{M}$ in B′ containing M. By Lemma 11.3, we can enlarge $\tilde{M}$ if necessary to assume that $B'/\tilde{M}$ is torsion free (so $\tilde{M}$ is projective over A by Lemma 11.4). We regard $\tilde{M}$ as a $\mathcal {G}$-stable submodule of B′⊗ _A B via $\tilde{M}\subset B'=B'\otimes_{A}A\subset B'\otimes_{A}B$ (this is indeed a $\mathcal {G}$-stable submodule since the inclusion A⊂B is $\mathcal {G}$-equivariant). Let $\tilde{N}=\tilde{M}+N$. Then $\tilde{N}$ is a finite $\mathcal {G}$-stable submodule of B′⊗ _A B, and under the map 1⊗ϵ:B′⊗ _A B→B′, $(1\otimes\epsilon)(\tilde{N})=\tilde{M}$. Observe that the torsion submodule $t((B'\otimes_{A}B)/\tilde{N})\subset(B'\otimes_{A}B)/\tilde{N}$ is a module with $\mathcal {G}$-action and maps to zero under $(B'\otimes_{A}B)/\tilde{N}\stackrel{1\otimes \epsilon}{\to} B'/\tilde{M}$. Let $\tilde{N}'$ be the preimage of $t((B'\otimes_{A}B)/\tilde{N})$ under $B'\otimes_{A}B\to(B'\otimes_{A}B)/\tilde{N}$. From Lemma 11.3, $\tilde{N}'$ is finite $\mathcal {G}$-stable A-module, and $(1\otimes \epsilon)(\tilde{N}')=\tilde{M}$. In addition, $\tilde{N}'$ is locally free since $(B'\otimes_{A}B)/\tilde{N}'$ is torsion free.

Let $\tilde{V}$ be the $\mathcal {G}$-stable A-submodule of B given by the fiber product

Observe that $(p\otimes1)\cdot{\rm comult}: B' \to B'\otimes_{A}B$ is injective. Therefore, $\tilde{V}$ is an A-submodule of $\tilde{N}'$ and therefore it is finitely generated over A. In addition, $\tilde{N}'\supset N$, $V\subset\tilde{V}$. Since $B/\tilde{V}\hookrightarrow(B'\otimes_{A}B)/\tilde{N}'$ is torsion free, $\tilde{V}$ is projective. Observe that

$$B \xrightarrow{(p\otimes1)\cdot{\rm comult} } B' \otimes_AB\stackrel{1\otimes\epsilon}{\to} B' $$

is just the projection p. Therefore, $p(\tilde{V})=\tilde{M}$.

Therefore, we obtain the following commutative diagram

with the first row finitely generated projective A-modules. Notice that $\tilde{V}\supset V$ contains a set of generators of the B-ideal I and a set of A-algebra generators of B. Hence, we obtain a closed immersion $\mathcal {G}\xrightarrow{}{\rm GL}(\tilde{V})$ of group schemes and we can see that $\mathcal {H}$ can be identified with the closed subgroup scheme of $\mathcal {G}$ that preserves the direct summand $W\subset M:=\tilde{V}$. By replacing M by M⊕M′ and W by W⊕M′ where M′ is a finitely generated projective A-module with trivial $\mathcal {G}$-action, we can assume that M is A-free as desired.

Corollary 11.5

Suppose that $\mathcal {H}\subset \mathcal {G}$ are as in Proposition 11.2. Then the fppf quotient $\mathcal {G}/\mathcal {H}$ is representable by a quasi-projective scheme over A.

Proof

By Artin’s theorem the fppf quotient $\mathcal {G}/\mathcal {H}$ is represented by an algebraic space over A. The algebraic space $\mathcal {G}/\mathcal {H}$ is separated of finite type and even smooth over A, the quotient $\mathcal {G}\to \mathcal {G}/\mathcal {H}$ is also smooth. Take M and W as in Proposition 11.2 and set $P:=\bigwedge^{\operatorname {rk}W}M$ and $L=\bigwedge^{\operatorname {rk}W}W\subset\bigwedge^{\operatorname {rk}W}M=A^{r}$, where $r={{\rm rank}(M)\choose{\rm rank}(W)}$. Then, $\mathcal {H}$ is the stabilizer of [L] in ${\rm Proj}( \bigwedge^{\operatorname {rk}W}M)={\mathbb{P}}^{r-1}_{A}$. We obtain a morphism $f: \mathcal {G}\to {\mathbb{P}}^{r-1}_{A}$. This gives a monomorphism $\bar{f}: \mathcal {G}/\mathcal {H}\to {\mathbb{P}}^{r-1}_{A}$ which is a separated quasi-finite morphism of algebraic spaces. By [33, 6.15], $\mathcal {G}/\mathcal {H}$ is a scheme and we can now apply Zariski’s main theorem to $\bar{f}$. We obtain that $\bar{f}$ is a composition of an open immersion with a finite morphism and we can conclude that $\mathcal {G}/\mathcal {H}$ is quasi-projective. (See [13, proof of Theorem 2.3.1] for a similar argument.) □

Remark 11.6

General homogeneous spaces over Dedekind rings are schemes [1], but this is not always the case when the base is a Noetherian regular ring of dimension 2; see [56, X]. In loc. cit. Raynaud asks if $\mathcal {G}/\mathcal {H}$ is a scheme when both $\mathcal {G}$ and $\mathcal {H}$ are smooth and affine over a normal base and $\mathcal {H}$ has connected fibers. The above Corollary gives a partial answer to this question.

Corollary 11.7

Suppose that $\mathcal {G}$ is a smooth affine group scheme with connected fibers over A. Then there n≥1 and a closed subgroup scheme embedding $\mathcal {G}\hookrightarrow{\rm GL}_{n}$, such that the fppf quotient ${\rm GL}_{n}/\mathcal {G}$ is represented by a smooth quasi-affine scheme over A.

Proof

By Proposition 11.2 applied to $\mathcal {G}$ and $\mathcal {H}=\{e\}$, we see that there is a closed subgroup scheme embedding $\rho: \mathcal {G}\hookrightarrow{\rm GL}_{m}$ (this follows also directly from [71]). Now apply Corollary 11.5 and its proof to the pair of the group ${\rm GL}_{m}$ with its closed subgroup scheme $\mathcal {G}$. We obtain a ${\rm GL}_{m}$-representation ρ′:GL _m →GL _r =GL(M) that induces a locally closed embedding $\mathrm {GL}_{m}/\mathcal {G}\hookrightarrow{\mathbb{P}}^{r-1}$. Denote by $\chi: \mathcal {G}\to \mathbb {G}_{\mathrm {m}}={\rm Aut}_{A}(L)$ the character giving the action of $\mathcal {G}$ on the A-line L (as in the proof of Corollary 11.5) and consider $\mathcal {G}\to \mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}$ given by g↦(ρ′(g),χ ⁻¹(g)). Consider the quotient $(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}$; to prove it is quasi-affine it is enough to reduce to the case that A is local. Then L is free, L=A⋅v, and $\mathcal {G}$ is the subgroup scheme of $\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}$ (acting by (g,a)⋅m=aρ′(g)(m)) that fixes v. This gives a quasi-finite separated monomorphism $(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}\rightarrow{\mathbb{A}}^{r}$ and so by arguing as in the proof of Corollary 11.5 we see that $(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})/\mathcal {G}$ is quasi-affine. Consider now the standard diagonal block embedding $\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}}\hookrightarrow \mathrm {GL}_{m+1}$. The quotient $\mathrm {GL}_{m+1}/(\mathrm {GL}_{m}\times \mathbb {G}_{\mathrm {m}})$ is affine and we can conclude using Lemma 11.1. □