REGULAR MODELS OF CERTAIN SHIMURA VARIETIES

Michael Harris^{1}

UFR de Mathématique, Université Paris 7, 2 Pl. Jussieu 75251 Paris cedex 05, FRANCE

Richard Taylor^{2}

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

Introduction

This paper develops a technique for studying the bad reduction of Shimura varieties attached to twisted unitary groups. The Shimura varieties we consider are the precise analogues in higher dimension of the modular curves X_{1}(p). It turns out that the theory is completely analogous to the well-known theory for modular curves, due to Deligne and Rapoport [DR].

Our original purpose in investigating these special cases of bad reduction was with a view to extending the techniques of Taylor-Wiles [TW] to deformations of mod ℓ Galois representations of dimension > 2. The present article has been extracted from an unpublished manuscript, mostly written between 1996 and 1998, which largely realized this objective, though undoubtedly not in optimal form [HT1]. Subsequent developments, especially an idea of Skinner and Wiles [SW] for bypassing Ribet’s level-lowering argument, lead us to hope that more complete results may be within reach.

The Taylor-Wiles method requires information about the tame ramification at q of the Galois representations associated to modular forms of level Γ_{1}(q) for certain primes q highly congruent to 1 modulo ℓ. The extension of this method to higher dimension is based on the detailed study of the singularities of the special fiber at q for level subgroups generalizing Γ_{1}(q). In the interim, our work on the local Langlands conjecture required a much more comprehensive study of bad reduction of Shimura varieties. The results of [HT2] apply in all levels, and yield the Galoistheoretic statements of the present paper as a special case.

Nevertheless, we feel the present paper is of independent interest. In the first place, the results presented here are somewhat more precise than those of [HT2], in the situations to which they apply. In particular, the congruence formula for the U-operator, proved in §4.2, is not stated explicitly as such in [HT2]. (A congruence formula in the analogue of level Γ_{0}(q), valid in considerable generality, is due to T. Wedhorn [We2]; we treat the analogue of level Γ_{1}(q).) In the second place,

Typeset by AMS-TEX

1Institut de Mathématiques de Jussieu, U.M.R. 7586 du CNRS

2Partially supported by NSF Grant DMS-9702885

our method, based on the Tate-Oort classification of group schemes of order p, is relatively elementary, and is likely to apply to Shimura varieties whose geometry is not amenable to the sort of analysis carried out in [HT2].

Finally, the modular curves X_{1}(p) play a special role in the theory of p-adic modular forms, as developed by Hida and Coleman, as well as in the related work of Gross [Gr]. We hope the approach to bad reduction developed here will contribute to the extension of this theory to modular forms in higher dimension.

The authors would like to thank the University of Durham (Harris and Taylor), Harvard University (Harris), the Institut Henri Poincaré (Taylor), and RIMS and Kyoto University (Harris) for providing a hospitable working environment at different stages of the project. We are particularly grateful to A. J. de Jong for suggesting we use the Tate-Oort classification for passing from level Γ_{0} to level Γ_{1}, as in [DR]. We also thank Gabber, Rapoport, and Zink for help in correcting sign errors in an early version of the manuscript, and Labesse for numerous conversations regarding base change. Finally, we thank Genestier for pointing out some bothersome notational inconsistencies, and suggesting a remedy.

1. Notation for unitary groups

1.1. Let E be a totally real number field of degree d over ℚ and let K be a totally imaginary quadratic extension of E; let c ∈ Gal(K/E) denote the non-trivial automorphism. We assume K contains an imaginary quadratic field K_{0}, so that K = K_{0} · E. We denote by Σ_{E} and Σ_{K} the sets of complex embeddings of E and K. Let D be a central simple algebra of dimension n^{2} over K, endowed with an involution, denoted c̃, that induces the Galois automorphism c on K; i.e., c̃ is an involution of the second kind.

We define algebraic groups U(D) = U(D, c̃) and GU(D) = GU(D, c̃) over ℚ such that, for any ℚ-algebra R,

U(D)(R) = {g ∈ D^{opp} ⊗_{ℚ} R | g · c̃(g) = 1};
GU(D)(R) = {g ∈ D^{opp} ⊗_{ℚ} R | g · c̃(g) = ν(g) for some ν(g) ∈ R^{×}}.

Thus GU(D) admits a homomorphism ν : GU(D) → 𝔾_{m} with kernel U(D). There is an algebraic group U_{E}(D) over E such that U(D) ⟶^{∼} R_{E/ℚ}U_{E}(D), where R_{E/ℚ} denotes Weil’s restriction of scalars functor. This isomorphism identifies automorphic representations of U(D) and U_{E}(D).

The groups U(D) (resp. GU(D)) are all inner forms of the same quasi-split unitary group (resp. unitary similitude group), denoted U_{0} (resp. GU_{0}). The group U_{0} is of the form U(D_{0}, χ̃(∗)_{0}) where D_{0} is the matrix algebra and χ̃(∗)_{0} is an appropriate involution. Then U_{0,∞} ≅ U(\frac{n}{2}, \frac{n}{2})^{[E:ℚ]} if n is even, U_{0,∞} ≅ U(\frac{n+1}{2}, \frac{n−1}{2})^{[E:ℚ]} if n is odd.

Let G be a reductive algebraic group over the number field E. If v is a place of E we let G_{v} = G(ℚ_{v}); if v is archimedean we let 𝔤_{v} = Lie(G_{v})_{ℂ}. We let G_{∞} denote ∏_{v|∞} G_{v}, the product taken over all archimedean places of E, and let 𝔤_{∞} = ∏_{v|∞} 𝔤_{v}.

Let π be an irreducible automorphic representation of G; i.e., an irreducible (𝔤_{∞}, K_{∞}) × G(A^{f})-module that embeds as a submodule of the space of automorphic forms relative to the chosen maximal compact subgroup K_{∞}. We write π = π_{∞} ⊗ π_{f} as usual, and say π is cohomological if π is cuspidal and if the relative Lie algebra cohomology H^{•}(𝔤_{∞}, K_{∞}; π_{∞} ⊗ V) ≠ 0 for some finite dimensional representation V of 𝔤_{∞}. We let Coh(G) denote the set of cohomological cuspidal automorphic representations of G, Coh(G, V) ⊂ Coh(G) the subset of π for which H^{•}(𝔤_{∞}, K_{∞}; π_{∞} ⊗ V) ≠ 0, with V fixed. If K ⊂ G(A^{f}) is a compact open subgroup, let Coh(G, K) denote the set of π ∈ Coh(G) such that π^{K} ≠ {0}, Coh(G, K, V) = Coh(G, K) ∩ Coh(G, V).

Let A_{0}(G) denote the space of cusp forms on G. Let Rep(G) denote the set of equivalence classes of irreducible (𝔤_{∞}, K_{∞}) × G(A^{f})-modules. If π ∈ Rep(G), we let m(π) = dim Hom(π, A_{0}(G)), where Hom denotes the space of homomorphisms of (𝔤_{∞}, K_{∞}) × G(A^{f})-modules. More generally, let S be a finite set of places of E, containing the archimedean places (for simplicity) and let Rep(G)^{S} denote the set of equivalence classes of irreducible G(A^{f,S})-modules, where G(A^{f,S}) ⊂ G(A^{f}) is the subgroup with trivial entry at every place in S. We say π^{S} ∈ Rep(G)^{S} is automorphic if

Hom_{G(A^{f,S})}(π^{S}, A_{0}(G)) ≠ 0;

i.e., if π^{S} can be extended to a cuspidal automorphic representation of G. We say π^{S} is cohomological if it can be extended to a cohomological cuspidal automorphic representation of G.

We now fix a rational prime p that splits in K_{0} and an archimedean prime s_{0} of E. We choose a central simple algebra D^{#} over K, with involution of the second kind c̃^{#}. We write G = GU(D^{#}, c̃^{#}), G′ = U(D^{#}, c̃^{#}), and view G′ alternatively as a group over ℚ or E. We assume that

(1.1.1)  G′_{∞} ≅ U(n − 1, 1) × U(n)^{[E:ℚ]−1}.

Moreover, if v is a finite prime, we assume that

(1.1.2)  G′_{v} ≅ U_{0,v} if v does not split in K/E.

In the applications, we will always assume D^{#} to be a division algebra.

We need another involution c̃^{∗} of the second kind on D^{#}; c̃^{∗} is a positive involution, so that the unitary group U(D, c̃^{∗}) is totally definite at infinity. The existence of such involutions is established as in the introduction to [HT2].

1.2. Open compact subgroups. Fix an open compact subgroup K = ∏_{v} K_{v} ⊂ G(A^{f}), and define

(1.2.1)  𝔖_{K}(ℂ) = G(ℚ)\G(A)/Z_{G}(ℝ) · K_{∞} · K; 𝔖(ℂ) = \underleftarrow{lim} 𝔖_{K}(ℂ);

the limit taken with respect to inclusion. Then 𝔖_{K} is the locally symmetric space associated to G and its subgroup K. The structure of the projective limit 𝔖(ℂ) structure as the set of complex points of a Shimura variety will be recalled below.

We will be working with specific choices of local subgroups K_{v}. For any rational prime q that splits in K_{0}, we choose a maximal compact subgroup of G_{q} in the form

(1.2.2)  ℤ_{q}^{×} × ∏_{v | q} G_{v},

the product being taken over divisors v of q in E. Here if w is a divisor of q in K_{0} lying above v and if D_{w} is isomorphic to GL(a, B_{w}) for some factorization n = ab and some division algebra B_{w} of degree b^{2} over K_{w}, then G_{v} can be taken in the form GL(a, O_{B_{w}}), where O_{B_{w}} is the maximal order of B_{w}.

We choose a finite set Q of finite places of E, each dividing a distinct rational prime that splits in K_{0}, and an additional finite place {𝔯} of E, also split in K. Primes 𝔮 ∈ Q are assumed to have the property that, if q is the rational prime divisible by 𝔮, then q splits completely in E. We let Q(ℚ) denote the set of rational primes divisible by primes in Q. For 𝔮 ∈ Q we define

Γ_{0,𝔮} = \{k ∈ GL(n, ℤ_{q}) | k ≡ ( \begin{matrix}∗_{1} & ∗ \\ 0 & ∗_{n−1} \end{matrix} ) (mod q)\}

Γ_{1,𝔮} = \{k ∈ GL(n, ℤ_{q}) | k ≡ ( \begin{matrix}1 & ∗ \\ 0 & ∗_{n−1} \end{matrix} ) (mod q)\}

as subgroups of the 𝔮-factor of 1.2.2. Then we let

K_{0,q} = ℤ_{q}^{×} × ∏_{v≠𝔮,v|q} GL(n, O_{v}) × Γ_{0,𝔮};

K_{1,q} = ℤ_{q}^{×} × ∏_{v≠𝔮,v|q} GL(n, O_{v}) × Γ_{1,𝔮},

viewed as subgroups of G_{q}.

Let q(𝔯) denote the residue characteristic of 𝔯. Let N_{0} denote the upper triangular unipotent subgroup of GL(n) and let

I_{1}(𝔯) = {k · n ∈ GL(n, O_{𝔯}) | k ≡ 1 (mod 𝔯), n ∈ N_{0}(O_{𝔯})};

I(𝔯) = ℤ_{q(𝔯)}^{×} × ∏_{v≠𝔯,v|q(𝔯)} GL(n, O_{w}) × I_{1}(𝔯).

We let

(1.2.3)  K_{q(𝔯)} = I_{𝔯}.

Lemma 1.3. For q(𝔯) sufficiently large, the locally symmetric variety 𝔖_{K}(ℂ) is smooth for K = K_{0,Q} or K_{1,Q}. Moreover, for any s ∈ G(A^{f}) the groups s^{−1}G(ℚ)s∩ K_{0,Q} and s^{−1}G(ℚ)s ∩ K_{1,Q} are trivial.

Proof. The first assertion follows from the second. Let K = K_{0,Q} or K_{1,Q}. Let x ∈ s^{−1}G(ℚ)s ∩ K for some s ∈ G(A^{f}). The subgroup of G(A) generated by x is both discrete and compact, hence finite. The group I_{𝔯} is pro-q(𝔯) and it follows that x is a root of unity of order a power of q(𝔯), lying in some extension field K′ of K that admits an embedding in D^{#}. The degree of K′ over ℚ is bounded by n[K : ℚ], hence for q(𝔯) sufficiently large we must have x = 1.

1.4. Cohomology. Let V be a finite-dimensional irreducible representation of 𝔤_{∞}, and let π be an automorphic representation in Coh(G, K, V). Let E be a field of definition for V, finite over ℚ. Fix a rational prime ℓ and an embedding of E in \overline{ℚ}_{ℓ}, and let V (\overline{ℚ}_{ℓ}) be the \overline{ℚ}_{ℓ}-form of V defined by extension of scalars from E. Let

(1.4.1)  Ṽ_{ℓ} = G(ℚ)\G(A) × V (\overline{ℚ}_{ℓ})/Z_{G}(ℝ) · K_{∞} · K

be the ℓ-adic local system over 𝔖_{K} associated to V and our chosen embedding of E in \overline{ℚ}_{ℓ}. The definition depends on K but the local systems for K′ ⊂ K are compatible with the natural map 𝔖_{K′}→𝔖_{K}, so we omit K from the notation for Ṽ_{ℓ}. We define

H^{•}(𝔖(ℂ), Ṽ_{ℓ}) = \underrightarrow{lim}_{K} H^{•}(𝔖_{K}(ℂ), Ṽ_{ℓ}).

Since π_{f} occurs in the cohomology of the complex local system associated to V, it admits a form, also denoted π_{f} over a number field E(π). Let λ be a prime of E(π) dividing ℓ, and let π_{f,λ} = π_{f} ⊗_{E(π)} E(π)_{λ}; we apply the same notation to the local components of π_{f}. Let

(1.4.2)  M_{λ}^{•}[π_{f}] = H^{•}(𝔖(ℂ), Ṽ_{ℓ})[π_{f}] = Hom_{G(A^{f})}(π_{f,λ}, H^{•}(𝔖(ℂ), Ṽ_{ℓ})),

a finite-dimensional vector space over E(π)_{λ}.

Let π be an automorphic representation of G. The restriction of π to the unitary group G′, which can be viewed as an algebraic group over E, decomposes as a direct sum of irreducible automorphic representations. Any two summands have the same local components at any finite place w dividing a rational prime q that splits in K_{0}, since at such places the similitude map splits as a product G(ℚ_{q}) ≃ G′(ℚ_{q}) × ℚ_{q}^{×}. For such a place w, we say π is non-monodromic at w if one component (hence every component) of the restriction of π to the unitary group G′ is non-monodromic at w (if G_{w} is split) or corresponds to a supercuspidal representation of GL(n, E_{w}) by the Jacquet-Langlands correspondence (if G_{w} is the multiplicative group of a division algebra).

Proposition 1.4.3. Let π be an automorphic representation of G such that π^{K_{1,Q}} ≠ 0. Suppose π_{f} is cohomological, and occurs only in cohomology in the middle degree n − 1. Then π_{q} is generic for every q that splits in K_{0}. For every w ∈ Q, π_{w} is either (a) unramified; (b) principal series attached to an n-tuple (α, β_{1}, …, β_{n−1}) of characters of E_{w}^{×}, with α tamely ramified and each β_{i} unramified; or (c) the Langlands sum of a special representation of GL(2) and an unramified representation of GL(n − 2). In cases (a) and (c), but not in case (b), π_{w} has a Γ_{0,w}-fixed vector.

Proof. The first assertion follows from the existence and properties base change of π to G_{K_{0}} ≃ GL(n)_{K} ×GL(1)_{K_{0}}, [CL, L, cf. HT,appendix D]. Let Π denote the base change. Since π_{f} occurs only in the middle degree, it follows from the results of [C2] that Π is cuspidal, hence globally generic by Shalika’s theorem. The genericity of π_{q} is then a consequence of Theorem 4.6.2 of [L]. The remaining assertions then follow easily from the Bernstein-Zelevinsky classification of admissible irreducible representations of GL(n) and from the theory of the conductor [JPSS].

Remark 1.4.4 The hypothesis that π_{f} occurs only in the middle degree is standard, and is automatically satisfied, for instance, if π is obtained by descent [C1,CL] from a cuspidal automorphic cohomological representation of GL(n)_{K}, or if π_{f} is supercuspidal at one place that splits in K (cf. [H1]).

1.5. Notation for unramified Hecke algebras.

Let CS(ℚ) denote the set of all primes of ℚ, and let CS^{+}(ℚ) be the subset of finite primes that split in K_{0} and are unramified in K. For v ∈ CS^{+}(ℚ) of residue characteristic p we choose a place v_{1} of K_{0} above v, and let Σ = Σ(v_{1}) be the set of primes of K dividing v_{1}. Then we have

(1.5.1)  G_{v} ≅ ∏_{w∈Σ} GL(n, K_{w})×ℚ_{v}^{×}.

The Hecke algebra T_{v} of G_{v} relative to any maximal compact subgroup (conjugate to ∏_{w∈Σ} GL(n, O_{K_{w}})×ℤ_{v}^{×}) is isomorphic to a polynomial algebra over ℤ[\frac{1}{p}] in the variables

{T_{i,w}, i = 1, …, n, T_{n,w}^{−1}, w ∈ Σ; T_{0,v}, T_{0,v}^{−1}}.

Here ℤ[T_{0,v}, T_{0,v}^{−1}] is the Hecke algebra of the factor ℚ_{v}^{×} in 1.5.1; by abuse of language we refer to the T_{i,w} as the Hecke operators at w, or at the prime of E below w. The Hecke operators at w are normalized so that

(1.5.2)  P_{w}(q^{−s}) = 1 + ∑_{i=1}^{n}(−1)^{i}T_{i,w}q^{−is}

is the local Euler factor at w of the motivically normalized standard L-function of GL(n). Here q is the order of the residue field k(w) and the inverse roots of P_{w}(X) are the Satake parameters, multiplied by q^{(n−1)/2}. Up to canonical isomorphism the algebra T_{v} does not depend on the choice of v_{1} above v.

The global Hecke algebra T is the tensor product over v ∈ CS^{+}(ℚ) of the T_{v}. If S is a finite subset of CS^{+}(ℚ), we let T^{S} ⊂ T be the subalgebra generated by the T_{v} for v ∉ S.

2. Shimura varieties and Galois representations

2.1. Shimura varieties defined by twisted and untwisted unitary groups.

Let E be a totally real field of degree d, K_{0} an imaginary quadratic field, K = K_{0} · E. As in 1.1, we let G be the similitude group of a division algebra D^{#} of dimension n^{2} over K with involution c̃^{#} of the second kind (with rational similitude factor). We will always assume n > 2. The unitary group G′ ⊂ G is assumed to satisfy (1.1.1) at real places and (1.1.2) at finite places; we let s_{1} denote the real place of E at which G′ has signature (n − 1, 1). The field K_{0} is assumed to be given with a fixed complex embedding 1_{K_{0}}, and we let Φ be the set of complex embeddings of K inducing 1_{K_{0}} on K_{0}. Then Φ is a CM type of K. Let t_{1} ∈ Φ be the complex place above s_{1}.

We use Φ to identify D^{#} ⊗_{ℚ} ℝ ⟶^{∼} M(n, ℂ)^{d}. In terms of this identification, we define

h_{Φ} : ℂ^{×} ≃ R_{ℂ/ℝ}𝔾_{m,ℂ}→(D^{#,opp} ⊗_{ℚ} ℝ)^{×}

by

(2.1.1)  h_{Φ}(z) = (( z · I_{n−1} ; z̄) , zI_{n}, …, zI_{n}),

where the first matrix corresponds to t_{1}. We may view h_{Φ} as a homomorphism with values in G_{ℝ}. Let X = X_{n−1} be the G_{ℝ}-conjugacy class of homomorphisms from R_{ℂ/ℝ}𝔾_{m,ℂ} to G_{ℝ} containing h_{Φ}. Then X has a natural G_{ℝ}-invariant complex structure which is isomorphic to the unit ball in ℂ^{n−1}. The pair (G, X) is the datum defining the Shimura variety 𝔖 introduced in (1.2.1), and considered in [C2,K,H1]. As in [H1], we see that the reflex field (Shimura field) E(G, X) is just t_{1}(K). 2.1.2. We fix a maximal order O^{#} ⊂ D^{#}, stable under c̃^{∗}, and let Ô^{#} = O^{#} ⊗_{ℤ} ℤ̂. Let V = D^{#}, viewed as a 2dn^{2}-dimensional vector space over ℚ with an embedding D^{#}→End(V). The homomorphism h_{Φ} takes values in D^{#,opp} = End_{D^{#}}(V), and is defined over some extension L of K which splits D^{#}. There is then an L-rational decomposition

(2.1.2.1)  V ⊗_{ℚ} L ⟶^{∼} V_{0} ⊕ V_{1},

where V_{0} (resp. V_{1}) is the z̄- (resp. z)-eigenspace for h(z); the subspaces V_{0} and V_{1} are stable under the left action of D^{#} ⊗_{ℚ} L.

We restrict our attention to open compact subgroups K ⊂ Ô^{#,×}, though this is only for convenience. Moreover, we always assume K factors as the product of subgroups of G(ℚ_{p}), as p varies over rational primes. We consider the moduli problem A_{K}, defined on the category of schemes over Spec E(G, X). For T a scheme over Spec E(G, X) = Spec K, the functor A_{K}(T) consists of equivalence classes (see below) of quadruples (A, ι, λ, β) consisting of (2.1.2.2) An abelian scheme A over T of dimension dn^{2}; (2.1.2.3) An embedding ι : O^{#}→End_{T} (A); (2.1.2.4) A polarization λ : A ⟶^{∼} Â, such that

λ ∘ ι(b) = ι(c̃^{♯}(b)) ∨ ∘λ

for all b ∈ O^{#}; (2.1.2.5) An Ô^{#}-equivariant level K structure

β : H_{1}(A, ℤ̂) = ∏_{q} T_{q}(A) ⟶^{∼} Ô^{#} (mod K),

in the sense of Kottwitz ([K2], pp. 390-391; cf. [HT2]), compatible with λ. These data are assumed to satisfy the determinant condition of Kottwitz, namely

(2.1.2.6)  det_{O_{T}} (b; LieA) = det_{L}(b; V_{1}),  b ∈ O^{#}.

We consider two quadruples (A, ι, λ, β) and (A′, ι′, λ′, β′) equivalent if there is an isogeny α : A → A′ which takes (λ, ι, β) to (tλ′, ι′, β′), where t is a positive rational number.

For K sufficiently small the functor A_{K} is representable by a smooth projective variety, also denoted A_{K}, over E(G, X) [K2,§5]. For general K we take a normal subgroup of finite index K′ ⊂ K for which A_{K′} is representable and then let A_{K} be the scheme-theoretic quotient of A_{K′} by K/K′. Then the Shimura variety 𝔖_{K}, viewed as a scheme over Spec E(G, X), is an open and closed subscheme of A_{K}. More precisely, A_{K} is isomorphic to a finite union of Shimura varieties of the form 𝔖_{K}, the number of Shimura varieties involved being given by the deviation from the Hasse principle for G (cf. [K2,§8; RZ2, pp. 301-302]). In particular, 𝔖_{K} is projective for all K, and is smooth provided K ∩ (O^{#})^{×} = {1}. Warning 2.1.2.7 Note that the level structure β defined here goes in the direction opposite to the level structures η introduced in [HT2,IV.1], as well as in [K2]. This convention, which is justified on Hodge-theoretic grounds, leads to a dualization in the associated Galois representations. 2.1.3. The level structures corresponding to the open compact subgroups of the form K_{0,Q} and K_{1,Q}, defined as in 1.2, deserve special attention. Recall that Q denotes a finite set of primes of E dividing rational primes q that split completely in K, and such that D^{#} is split at all primes dividing q. In particular, we may take L = ℚ_{q} in (2.1.2.1); then

(2.1.3.1)  V ⊗_{ℚ} L ≅ ⨁_{w | q} M(n, K_{w}) ≅ ⨁_{w | q} M(n, ℚ_{q})^{2},

where w runs through primes of E (note: not K) dividing q and the exponent 2 corresponds to the quadratic extension K/E. We let V_{w} = M(n, K_{w}) in the above decomposition. Recall that we are identifying K with the subfield E(G, X) of ℂ via t_{1}. Then the first coordinate of h_{Φ}, in the representation (2.1.1), picks out one summand of (2.1.3.1), say the summand V_{𝔮} corresponding to the prime w = 𝔮. Write V_{𝔮} = V_{𝔮^{(1)}} ⊕ V_{𝔮^{(2)}}, where 𝔮^{(1)} and 𝔮^{(2)} are the primes of K above w. These can be numbered so that

(2.1.3.2) dim_{ℚ_{q}} V_{0} ∩ V_{𝔮^{(1)}} = n; dim_{ℚ_{q}} V_{1} ∩ V_{𝔮^{(1)}} = n(n − 1)

dim_{ℚ_{q}} V_{0} ∩ V_{𝔮^{(2)}} = n(n − 1); dim_{ℚ_{q}} V_{1} ∩ V_{𝔮^{(2)}} = n.

The corresponding maximal ideals of O_{K} are denoted 𝔪_{𝔮^{(i)}}, i = 1, 2.

We let the 𝔮 chosen in the preceding paragraph be the prime in Q dividing q, and we define K_{0,q} and K_{1,q} as in 1.2 with respect to this choice of w = 𝔮. Thus K_{?,Q} = K_{?,q} × K^{q}, where ? ∈ {0, 1} and K^{q} denotes the product of the level subgroups at primes away from q. Then the datum β breaks up into q-primary and prime-to-q components, denoted in the obvious notation β_{q} and

(2.1.3.3)  β^{q} : ∏_{p≠q} T_{p}(A) ⟶^{∼} (Ô^{#})^{q} (mod K^{q}),

respectively. First consider K_{0,Q}. We let

Γ_{q} = GU(O^{#} ⊗_{ℤ} ℤ_{q})^{×} ≃ ∏_{w|q} GL(n, ℤ_{q}) × ℤ_{q}^{×},

so that Γ_{q}/K_{0,q} is naturally in bijection with ℙ^{n−1}(k(𝔮)), where k(𝔮) ≃ 𝔽_{q} is the residue field of 𝔮. Let K^{+} = Γ_{q}×K^{q}.

Let ϖ be a uniformizer in E_{𝔮}. We write t(ϖ)_{1} for the diagonal matrix with entry ϖ^{−1} as first entry and 1 elsewhere, viewed as an element of G(ℚ_{q}) ⊂ G(A_{f}). Then

(2.1.3.4)  K_{0,Q} = K^{+} ∩ t(ϖ)_{1}^{−1}K^{+}t(ϖ)_{1}.

We thus have two maps f_{i} : 𝔖_{K_{0,Q}}→𝔖_{K^{+}}, where f_{1} is defined by the inclusion K_{0,Q} ⊂ K^{+} and f_{2} is defined by inclusion in t(ϖ)_{1}^{−1}K^{+}t(ϖ)_{1}, followed by rightmultiplication by t(ϖ)_{1}^{−1}. Let x = (A, ι, λ, β) be a geometric point of 𝔖_{K}, and let f_{2}(x) = (A′, ι′, λ′, β′). Then x is determined by the quintuple (A, A′, ι, λ, β^{q}).

By modifying our definition of f_{2}, we can arrange to have an isogeny A→A′, and then realize 𝔖_{K_{0,Q}} as (an open and closed subscheme of) the moduli space parametrizing quintuples as above, with A→A′ an isogeny of a certain type. However, we do not need an actual isogeny. Let A be an abelian scheme over T as in (2.1.2.2), and let X_{q}(A) be the associated q-divisible group. Then X_{q}(A) inherits an action of O^{#} ⊗_{ℤ} ℤ_{q}. Let 𝔮^{(2)} be the prime of K chosen above. Let e_{2} be an elementary idempotent in M(n, ℤ_{q}), which we identify with O_{𝔮^{(2)}}^{#} = Ô^{#} ∩ D_{𝔮^{(2)}}^{#}. Then e_{2} · X_{q}(A) is a q-divisible group of height n, which we denote X(A)_{2}. With our choice of 𝔮^{(2)}, it follows easily from (2.1.2.6) and (2.1.3.2) that X(A)_{2} is of dimension 1. Now the rational Tate modules V_{q}(A) = T_{q}(A) ⊗ ℚ_{q} and V_{q}(A′) can be identified. One sees easily that e_{2} · T_{q}(A′) ⊃ e_{2} · T_{q}(A); hence there is an isogeny

(2.1.3.5)  X(A)_{2}→X(A′)_{2}

with kernel O_{K} isomorphic to O_{K}/𝔮^{(2)} ≅ ℤ/qℤ. We conclude that 2.1.3.6. For K^{q} sufficiently small, the scheme 𝔖_{K_{0,Q}} is an open and closed subscheme of the moduli scheme parametrizing quintuples

(A, ι, λ, β^{q}, X(A)_{2}→X(A′)_{2}),

where (A, ι, λ) are as in (2.1.2.2-4), β^{q} is a level K^{q}-structure as in (2.1.3.3), and X(A)_{2}→X(A′)_{2} is a q-isogeny of q-divisible O_{𝔮^{(2)}}-modules of height n and dimension 1, the whole assumed to satisfy the determinant condition (2.1.2.6).

Here is an alternative definition of the moduli problem. Define X(A)_{1} = e_{1} · X_{q}(A) by analogy with X(A)_{2} above, where e_{1} is an elementary idempotent in O_{𝔮^{(1)}}^{#}. We choose the pair (e_{1}, e_{2}) in such a way that the polarization λ places X(A)_{1} and X(A)_{2} in Cartier duality. We consider diagrams of the form

(2.1.4.1)  \begin{aligned}& & A′′ \\ & ^{ϕ} ↙ & & ↘^{ϕ′} \\ A & & & & A′ \end{aligned}

where (A, ι, λ, β^{q}), (A′, ι′, λ′, (β′)^{q}) are as above. In particular λ : A ⟶^{∼} Â, λ′ : A′ ⟶^{∼} Â′, are principal polarizations. We assume ϕ and ϕ′ are isogenies of degree q^{n}, with the following property. Let ϕ̂ : Â→Â′′ and ϕ̂′ : Â′→Â′′ be the dual isogenies to ϕ, ϕ′, respectively. Then Ker(ϕ̂′) (resp. Ker(ϕ̂)) is generated over O_{𝔮^{(2)}}^{#}

(resp. O_{𝔮^{(1)}}^{#}) by a subgroup Ĉ_{2} ⊂ X(A′)_{2} (resp. Ĉ_{1} ⊂ X(A)_{1}) of order q. We let

C_{1}, C_{2} ⊂ X(A′′) be the Cartier duals of Ĉ_{1} and Ĉ_{2}, respectively. Then ϕ (resp. ϕ′) induces an isomorphism ϕ_{2} : X(A′′)_{2} ⟶^{∼} X(A)_{2} (resp. ϕ_{1} : X(A′′)_{1} ⟶^{∼} X(A′)_{1}) so that ϕ′ ∘ (ϕ_{2})^{−1} (resp. ϕ ∘ (ϕ_{1})^{−1}) defines a q-isogeny r : X(A)_{2}→X(A′)_{2} (resp. r′ : X(A′)_{1}→X(A)_{1}) with kernel canonically isomorphic to C_{2} (resp. C_{1}).

It is assumed that ϕ and ϕ′ are compatible with the O^{#}-structure, in the sense that the induced isomorphisms of rational q-adic Tate modules are assumed to be O^{#}-linear. Finally, the two isogenies are assumed to be dual to each other, in the sense that

(2.1.4.2)  ϕ̂′ ∘ λ′ ∘ ϕ′ = ϕ̂ ∘ λ ∘ ϕ : A′′ → Â′′.

Using these conditions one reconstructs A′′ and (A′, ι′, λ′, (β′)^{q}) from (A, ι, λ, β^{q}) and r : X(A)_{2}→X(A′)_{2} in such a way that ι′ and (β′)^{q} correspond to ι and β^{q} via the isogenies. Indeed, we can define A′′ to be the quotient of A′ by the O_{𝔮^{(2)}}^{#}-

module generated by r(X(A)_{2}[q]). The map ϕ′ is defined so that A′→A′′→^{ϕ′}A′ is multiplication by q, viewed as an element of O^{#}_{𝔮^{(2)}}. Then ker(ϕ) is defined to be ker(ϕ̂′ ∘ λ′ ∘ ϕ′) ∩ A′′[𝔮^{(1)}].

It follows easily from (2.1.4.2) that

(2.1.4.3)  ker(ϕ′) = { \atop ker(ϕ)} (Cartier duality).

Indeed, letting f : A′′→A′′ denote the map in (2.1.4.2), we see that

e_{1} · ker(f) × e_{2} · ker(f) = ker(f) ∩ X(A′′)_{1} × ker(f) ∩ X(A′′)_{2}
= e_{1} · ker(ϕ) × e_{2} · ker(ϕ′)
= e_{1} · (λ ∘ ϕ)^{−1} ker(ϕ̂′) × e_{2} · (λ′ ∘ ϕ′)^{−1} ker(ϕ̂)

This implies ker(ϕ̂) ⟶^{∼} ker(ϕ′), which is equivalent to (2.1.4.3). In particular,

(2.1.4.4)  C_{2} ⟶^{∼} Ĉ_{1} ⟶^{∼} e_{2} · ker(ϕ), C_{1} ⟶^{∼} Ĉ_{2} ⟶^{∼} e_{1} · ker(ϕ′),

where C_{1} and C_{2} are defined as above.

When K^{q} is not assumed small, we can realize 𝔖_{K_{0,Q}} as a quotient of a moduli scheme by a finite group, as above. In the same way, we obtain a modular description of 𝔖_{K_{1,Q}}: 2.1.4.5. For K^{q} sufficiently small, the scheme 𝔖_{K_{1,Q}} is an open and closed subscheme of the moduli scheme parametrizing quintuples

(A, ι, λ, β^{q}, γ : O_{K}/𝔮^{(1)} ↪ X(A′)_{1}),

where (A, ι, λ) are as in (2.1.2.2-4), β^{q} is a level K^{q}-structure as in (2.1.3.3), and γ is an embedding over T of the constant group scheme O_{K}/𝔮^{(1)} in the q-divisible O_{𝔮^{(1)}}- module X(A′)_{1}, compatibly with the O_{𝔮^{(1)}}-structures, the whole assumed to satisfy the determinant condition (2.1.2.6). By Cartier duality, γ defines an embedding γ^{∗} : μ_{q}→X(A)_{2}. Remark-Definition 2.1.4.6 The subgroup generated by γ (resp. γ^{∗}) is the one denoted C_{1} (resp. C_{2}) above. Thus taking the subgroup generated by γ and forgetting the generator defines a map of functors to 𝔖_{K_{0,Q}}. The quotient group K_{0,Q}/K_{1,Q} = ∏_{w∈Q} k(w)^{×} acts on the data (γ_{1,q} | q ∈ Q), hence as a group of automorphisms of 𝔖_{K_{1,Q}}, with quotient 𝔖_{K_{0,Q}}. However, the datum (2.1.2.5) corresponds, not to γ, but rather to γ^{−1} (a map from a subgroup of X(A′)_{1} to O_{K}/𝔮^{(1)}. Now g ∈ K_{0,Q}/K_{1,Q} acts on the datum (2.1.2.5) by sending β to g ∘ β. It follows that, for w ∈ Q, a ∈ k(w)^{×} acts on γ by sending b ↦ γ(b) to b ↦ γ(a^{−1}b). For a ∈ k(w)^{×}, we thus define the diamond operator < a >∈ Aut(𝔖_{K_{1,Q}}/𝔖_{K_{0,Q}}) to be the image of a^{−1} ∈ K_{0,Q}/K_{1,Q}. Then identifying γ with the point P = γ(1), the diamond operator < a > takes P to a · P. The group of diamond operators is just K_{0,Q}/K_{1,Q}, most frequently when Q consists of a single element. 2.1.5. Now let 𝔭 be a prime of K of residue characteristic p at which D^{#} splits. Assume the rational prime p is unramified in K. Then G(ℚ_{p}) admits hyperspecial maximal compact subgroups. Suppose K_{p} is a hyperspecial maximal compact. Then Kottwitz has shown that 𝔖_{K} extends to a smooth proper scheme over Spec O_{𝔭}, representing the functor A_{K} defined as in 2.1.2, but with the additional hypothesis that the polarization λ of (2.1.2.4) is of degree prime to the residue characteristic (cf. [HT2, K2]). In particular, the representation of Gal(\overline{ℚ}/K) on the ℓ-adic cohomology is unramified at 𝔭, provided ℓ ≠ p.

As observed by Carayol (cf. [Ca]), the hypothesis that p be unramified in K is in fact unnecessary, but the functor defined by 2.1.2 needs to be modified. Details can be found in [HT2,IV.2]

2.2. Modular definition of Hecke and U operators. 2.2.1. We define Γ_{1,𝔮} ⊂ GL(n, ℤ_{q}) as in §1.2. We consider the double coset U_{𝔮} = Γ_{1,𝔮} · t(ϖ)_{1}Γ_{1,𝔮} ⊂ GL(n, ℚ_{q}). Here and in what follows we are identifying GL(n, ℚ_{q}) = U(D^{#})_{𝔮} with D_{𝔮^{(2)}}^{#,×}. The identification with D_{𝔮^{(1)}}^{#,×} identifies U_{𝔮} with the double coset Γ′_{1,𝔮} · (t(ϖ)_{1})^{−1}Γ′_{1,𝔮}, where Γ′_{1,𝔮} consists of invertible matrices over

ℤ_{q} congruent to ( \begin{matrix}1 & 0 \\ ∗ & ∗_{n−1} \end{matrix} ) (mod q).

One verifies immediately that

(2.2.1.1)  U_{𝔮} = ∐ Γ_{1,𝔮} · t(ϖ)_{1} · ( \begin{matrix}1 & b \\ 0 & I_{n−1} \end{matrix} ) ,

where b = (b_{1}, …, b_{n−1}) runs through (ℤ_{q}/qℤ_{q})^{n−1}. Write W = ℚ_{q}^{n}. Let Λ = ℤ_{q}^{n} ⊂ W be the standard lattice, and let λ denote the row vector (q^{−1}, 1, …, 1), so that

Γ_{1,𝔮} = {g ∈ GL(W) | g(Λ) = Λ, g(λ) = λ (mod ()Λ)}.

Let Λ′ be the lattice generated by Λ and λ. Then the description (2.2.1.1) shows that, in the standard right action on lattices, we have

(2.2.1.2)  (Λ, λ) · U_{𝔮} = {(Λ′, λ′) |qλ′ = λ (mod Λ))}.

Now consider the correspondence U_{𝔮} on 𝔖_{K_{1,Q}} × 𝔖_{K_{1,Q}}, defined in terms of the data 2.1.4.5 as follows: Let T be a scheme over E(G, X) and let x be a T - valued point of 𝕊_{K_{1}}^{o}. Then x corresponds to a quintuple (A, ι, λ, β^{q}, γ : O_{K}/𝔮^{(1)} ↪ X(A′)_{1}). Of course γ is determined by the 𝔮^{(1)}-torsion point P = γ(1). Let A′ be the abelian variety associated to A and to the isogeny ρ : X(A)_{2}→X(A′)_{2} = X(A)_{2}/Im(γ^{∗}), as in (2.1.3.6), where γ^{∗} : μ_{q}→X(A)_{2} is the Cartier dual to γ, as in 2.1.4.5. With respect to the Weil pairings defined by λ and λ′, we have

(2.2.1.3)  (P, γ^{∗}(ζ)) = ζ for any ζ ∈ μ_{q}.

We consider the set G^{∗} of homomorphisms (γ′)^{∗} : μ_{q}→X(A′)_{2} such that, for any ζ ∈ μ_{q},

(2.2.1.4)  (γ′)^{∗}(ζ) = ρ(P̃), qP̃ = γ^{∗}(ζ)

Then U_{𝔮} associates to x the set of quintuples

(2.2.1.5)  {(A′, ι′, λ′, (β′)^{q}, γ′)}.

where γ′ runs through the set of maps O_{K}/𝔮^{(1)} ↪ X(A′)_{1} such that (γ′)^{∗} ∈ G^{∗}.

Lemma 2.2.2. The correspondence U_{𝔮} is induced by the Hecke operator U_{𝔮} by the standard recipe [K2, p. 393].

Proof. This is clear from (2.2.1.2).

2.2.3. For use in §3, we make explicit the diagram (2.1.4.1) associated to the image of the point (A′, ι′, λ′, (β′)^{q}, γ′) in 𝔖_{K_{0,Q}}. We have replaced A by A′, and then A′ is replaced by an abelian variety B′, say, so that the diagram is

(2.2.3.1)  \begin{aligned}& & B′′ \\ & ^{ϕ} ↙ & & ↘^{ϕ′} \\ A′ & & & & B′ \end{aligned}

satisfying the analogue of (2.1.4.2).

2.3. λ-adic representations on the cohomology.

We return to the situation of §1. Let SC, Q, 𝔯, K_{1,Q}, and V be as in 1.2. The primes in Q are denoted w rather than 𝔮, and the primes above them in K are denoted w_{1} and w_{2}. As in §2.2, the identification Γ_{w} ≃ GL(n, ℚ_{q}) for w ∈ Q is made via w_{2}, so that K_{0,Q} is upper triangular parahoric. Let π ∈ Coh(G, V), in the notation of §1, and assume:

(2.3.1)  π^{K_{1,Q}} ≠ 0

Then M_{ℓ}^{n−1}[π_{f}], defined as in (1.4.2), is an n-dimensional \overline{ℚ}_{ℓ}-vector space with a natural action of Gal(\overline{K}/K), which we henceforth denote G_{K}. Let E(π)_{λ} be as in 1.4, and write M_{λ}[π_{f}] for M_{λ}^{n−1}[π_{f}]. We let

r_{ρ}^{#}(π) : G_{K}→Aut(M_{λ}[π_{f}])

denote the natural action.

This is not quite the representation we want to associate to π. In [H1,p 101] we define a Hecke character α(π) = ξ ∘ r_{μ}. More precisely, in [loc. cit.] we defined a Hecke character denoted ν(Gπ), which in the present notation should be equal to

ν(Gπ) = α(π)^{−1} · | • |_{A}^{− \frac{n−1}{2}} ;

but the formula was incorrect; the correct formula is given below. We let α(π)_{λ} denote the corresponding λ-adic character of G_{K} – indeed, the field of coefficients of α(π) is contained in E, so this makes sense – and we let

(2.3.2)  r_{ρ}(π) = r_{ρ}^{#}(π) ⊗ (α_{π})_{λ},

acting on M_{λ}[π_{f}]. The correct formula for α(π) can be found in [H2, Errata to H1]:

(2.3.2.1)  α(π)(a) = ξ_{π}N_{K/K_{0}}(a),

where ξ_{π} is the central character of π and 𝔾_{m,K_{0}} is regarded as the subgroup Z_{GL(n,K_{0})} ∩ G of G, the intersection taking place in GL(n, K).

Next, we let G_{E} denote Gal(\overline{E}/E), and define

ρ(π) = Ind_{K/E}r_{ρ}(π)

to be the induced 2n-dimensional representation of G_{E}, acting on the E(π)_{λ}-module Ind_{K/E}M_{λ}[π_{f}].

For any prime v of ℚ that splits in K_{0} and is unramified in K, and such that π_{v} is unramified we let ϕ_{v,π} : T_{v} → E(π) denote the character by which the local unramified Hecke algebra acts on π_{v}^{K_{v}}. Let S(π) be the set of all such unramified primes, let T^{S} be the corresponding global Hecke algebra, as in §1.5, and let ϕ_{π} : T^{S} → E(π) be the corresponding character; ϕ_{π} gives the natural action of T^{S(π)} on π^{K}. If w is a prime of E dividing some prime in S(π), we let

(2.3.3)  ϕ_{π}(P_{w})(X) = 1 + ∑_{i=1}^{n}(−1)^{i}ϕ(π)(T_{i,w})X^{i}

in the notation of (1.5.2).

Theorem 2.3.4. Suppose π contains a fixed vector for a hyperspecial maximal compact subgroup of G_{q(𝔯)} (i.e., not only a fixed vector for K_{q(𝔯)}, as implied by (2.3.1). Let S^{bad} be the set of primes of K dividing primes in Q or of residue characteristic ℓ. Then the representation r_{ρ}(π) is unramified outside S^{bad}. Moreover, for all but finitely many primes w of E dividing rational primes in S(π), there is a prime w_{1} of K dividing w and such that the arithmetic Frobenius F rob_{w_{1}} satisfies

ϕ_{π}(P_{w})(F rob_{w_{1}}) = 0.

Remark. The choice of w_{1} in the above theorem is determined as in 1.5 by the choice of identification of G′_{w} with a general linear group. The base change of π to K is conjugate self-dual, and one verifies that, if w_{2} is the other prime dividing w, then ϕ_{π}(P_{w})(q^{n−1}F rob_{w_{2}}^{−1}) = 0. Bear in mind also that the natural action on the cohomology of the Shimura variety is that of the Galois group of \overline{ℚ} over the reflex field, and that there is an implicit identification of the reflex field with K.

An alternative way of phrasing this theorem is in terms of partial L-functions: there is a finite set S of finite primes such that we have the equality of Euler products

(2.3.4.1)  L^{S}(s, r_{ρ}(π)) = L^{S}(s − \frac{n − 1}{2}, BC_{K}(π)).

Here the right-hand side is the standard L-function with the unitary (Langlands) normalization; the superscript ^{S} indicates that factors at S have been removed. As mentioned above, we have normalized r_{ρ} to make (2.3.4.1) true; cf. the discussion on p. 100 ff. of [H1].

Proof. The proof of this theorem is mainly due to Kottwitz, and is contained in [K2]. Specifically, Kottwitz proves there that r_{ρ}(π) is unramified outside S^{bad}, except possibly at primes of K ramified over ℚ. Moreover, Kottwitz determines the characteristic polynomial of Frobenius in [K1] for almost all unramified primes. His original formulation includes an exponent for the (unknown) multiplicity. This exponent was removed by Taylor a few years later (cf. [H1, pp. 102-103]).

Theorem 2.3.5. Let w ∈ Q of residue characteristic q. Let w_{1} be the prime of K above w and let Z_{w} ⊂ G_{E} denote a decomposition group. Suppose π_{w} is in case (b) of Proposition 1.4.3. Let (α, β_{1}, …, β_{n−1}) be the corresponding n-tuple of characters. Then r_{ρ}(π)|_{Z_{w}} breaks up as a direct sum

r_{ρ}(π)|_{Z_{w}} ⟶^{∼} A ⊕ B.

Here B is an unramified representation and the inertia subgroup I_{w} of Z_{w} acts on A via the restriction to I_{w} of the character associated to α via local class field theory.

One of the main results of [HT2] is the generalization of Theorem 2.3.4 to determine the restriction of r_{ρ}(π) to the decomposition group of an arbitrary prime of K not dividing ℓ, up to semisimplification. In particular, Theorem 2.3.5 is a weak version, in a very special case, of what is proved in [HT2]. We give another proof in this special case, based on a different analysis of the bad reduction. The proof of Theorem 2.3.5 requires a lengthy detour through the theory of moduli of p-divisible groups, and is the subject of §3.

3. Regularity of certain moduli problems

3.0. Conventions. We will now study the Shimura varieties 𝔖_{K} at certain places of bad reduction. The techniques of the present section undoubtedly generalize, but for the time being we will make a series of restrictive hypotheses. The notation is as at the beginning of 2.1. We fix a level subgroup K ⊂ G(A_{f}) that satisfies (1.2.3), for q(𝔯) sufficiently large. Let q be a rational prime that splits completely in K and such that K = K_{q} × K^{q}, with K_{q} ⊂ G(ℚ_{q}) a maximal compact subgroup.

We assume the division algebra D^{#} to be split at all primes of K dividing q, and we identify K_{q} as in 2.1.3 with

(3.0.1)  ℤ_{q}^{×} × ∏ GL(n, O_{v}),

the product being taken over divisors v of q in E.

We let 𝔮 be the divisor of q in E chosen in §2.1.3 and let 𝔮^{(1)} and 𝔮^{(2)} be the divisors of 𝔮 in K, numbered as in (2.1.3.2). We identify G′_{𝔮} with GL(n, K_{𝔮^{(2)}}). Inside its compact open subgroup GL(n, O_{𝔮^{(2)}}), which we identify with GL(n, ℤ_{q}), we define the open subgroups Γ_{0,𝔮} and Γ_{0,𝔮} as in §1.2. Similarly, we define K_{0,q} and K_{1,q} as subgroups of G_{q}. We let

K_{0}(𝔮) = K_{0,q} × K^{q}; K_{1}(𝔮) = K_{1,q} × K^{q}

be the corresponding subgroups of K. There is a natural isomorphism

(3.0.2)  𝔽_{q}^{×} ⟶^{∼} K_{0}(𝔮)/K_{1}(𝔮) ≅ Aut(𝔖_{K_{1}}(𝔮)/𝔖_{K_{0}}(𝔮)).

We denote by < a >∈ Aut(𝔖_{K_{1}}(𝔮)/𝔖_{K_{0}}(𝔮)) the image of a ∈ 𝔽_{q}^{×} under the isomorphism (3.0.2). We will later have occasion to consider subgroups intermediate between K_{0}(𝔮) and K_{1}(𝔮).

On the other hand, we will be studying the reduction of Shimura varieties over O_{𝔮^{(1)}} ⊂ K_{𝔮^{(1)}}. We will identify O_{𝔮^{(1)}} with O_{𝔮}, which is isomorphic to ℤ_{q}, and write S_{𝔮} = Spec(O_{𝔮}). N.B.: the divisors in K of 𝔮 used in the geometry and in the group theory are conjugate to one another! This is an inevitable consequence of our choice of Shimura datum.

Since the prime 𝔮 will be fixed throughout this discussion, we will write K_{0} = K_{0}(𝔮) and K_{1} = K_{1}(𝔮) when there is no danger of confusion. We write 𝔖_{K_{1}} and 𝔖_{K_{0}} for the corresponding Shimura varieties over E(G, X). We let O_{q} = O_{E} ⊗_{ℤ} ℤ_{q}, so that O_{q} ≅ ∏_{v|q} O_{v}.

3.1. Modular interpretations of 𝔖_{K} and 𝔖_{K_{0}}.

As indicated in §2.1.5, the Shimura variety 𝔖_{K} has good reduction at all primes of K dividing q. Let 𝕊_{K} denote a smooth model of 𝔖_{K} over S_{𝔮} and let p = p_{K} : 𝕊_{K} → S_{𝔮} be the natural map. Under our hypothesis that K satisfies (1.2.4) for q(𝔯) sufficiently large, we have seen in 2.1.5 that p is a smooth projective morphism [K2, §5]. Moreover, 𝕊_{K} is (an open and closed subscheme of) a moduli space for abelian varieties with a certain PEL type. Indeed, with our choice of 𝔮, the moduli problem (2.1.2.2-6) is well-defined and representable on the category of schemes over S_{𝔮}, and 𝕊_{K} is the corresponding moduli scheme. Note, however, that Kottwitz’ determinant condition (2.1.2.6) must be interpreted as an equality of polynomials, cf. [K2,p. 390].

Let i : Spec(k(𝔮)) ≅ Spec(𝔽_{q}) ↪ S_{𝔮} denote the special point of S_{𝔮}. The following proposition is a special case of a theorem of Wedhorn.

Proposition 3.1.1 [We1]. Let \overline{i} : y → S_{𝔮} be a geometric point lying over i, and let 𝕊_{K}_{y} = y ×_{S_{𝔮}} 𝕊_{K} denote the corresponding geometric special fiber. The ordinary locus is open and dense in every irreducible component of 𝕊_{K}_{y}.

Here by “ordinary locus” we mean the subscheme whose geometric points correspond to ordinary abelian varieties with structure (2.1.2.2-6).

The moduli problem defining 𝔖_{K_{0}} is rather more intricate to describe. Our conventions will be those of Rapoport and Zink in [RZ2]. We will first define the moduli problem in their notation, then provide a brief translation into more familiar language. Let V = R_{K/ℚ}D^{#}, K_{q} = K ⊗_{ℚ} ℚ_{q}, E_{q} = E ⊗_{ℚ} ℚ_{q}, V_{q} = V ⊗_{ℚ} K_{q}. Let q^{(1)} and q^{(2)} denote the two primes of K_{0} dividing q, and let

(3.1.2)  K_{q} = K ⊗_{ℚ} ℚ_{q} ⟶^{∼} K_{q^{(1)}} × K_{q^{(2)}}

be the natural decomposition, with K_{q^{(i)}} ≅ ∏_{v|q^{(i)}} K_{v}, i = 1, 2, the product taken over primes of K. There is a corresponding decomposition V_{q} ⟶^{∼} V_{q^{(1)}} × V_{q^{(2)}}, and the hermitian form < a, b >= T r_{D^{#}}(ac̃(b)) (reduced trace) on V induces a duality between V_{q^{(1)}} and V_{q^{(2)}}:

(3.1.3)  V_{q^{(2)}} ⟶^{∼} Hom_{E_{q}}(V_{q^{(1)}}, E_{q}).

As in §2, we let 𝔮^{(1)} and 𝔮^{(2)} be the primes of K dividing 𝔮, with 𝔮^{(1)} above q^{(1)} and 𝔮^{(2)} above q^{(2)}.

By a lattice in V_{q^{(i)}}, i = 1, 2, we mean a free O_{q}^{#}-submodule. The subgroup K_{q} fixes a pair of lattices Λ_{i} ⊂ V_{q^{(i)}}, i = 1, 2, that are placed in duality with respect to (3.1.3). The subgroup K_{0,q} is the stabilizer in K_{q} of a unique sublattice Λ′_{1} ⊂ Λ_{1} of index q; we let Λ′_{2} ⊃ Λ_{2} denote its dual in V_{q^{(2)}}. In general, if Λ ⊂ V_{q^{(1)}} is a lattice, we write Λ^{⊥} for its dual in V_{q^{(2)}}. Let L denote the collection of all pairs of lattices (Λ, Λ^{⊥}), with Λ ⊂ V_{q^{(1)}}, Λ^{⊥} ⊂ V_{q^{(2)}}, such that Λ is fixed by K_{0,q}. Then L is an example of what Rapoport and Zink call a multichain of O_{q}-lattices in V_{q} [RZ2, Def. 3.10]. More precisely, the set Λ ⊕ Λ^{⊥} ⊂ V_{q} is the multichain, in the sense of [RZ2], but in the present case the two notions are equivalent.

For future reference, we let L_{1} denote the pair of lattices Λ_{2} ⊂ Λ′_{2} in V_{q^{(2)}}.

We will work in the category AV of abelian varieties up to prime-to-q isogeny. This is the category obtained from the category of abelian varieties by formally adjoining the inverses of all isogenies with kernel of order prime to q. It is a category whose sets of morphisms are naturally modules over the localization ℤ_{(q)} of ℤ at q. If A is an abelian variety we let A_{(q)} denote the corresponding object in AV. If A, A′ are abelian varieties, an isogeny ρ : A_{(q)} → A′_{(q)} is a morphism such that nρ is an isogeny from A to A′ for some integer n > 0 prime to q. The q-part of the kernel of ρ is well-defined and independent, up to canonical isomorphism, of the choice of representatives A and A′ in the category of abelian varieties; we call it the q-kernel of ρ.

A quasi-isogeny ρ : A_{(q)}→A′_{(q)} is a diagram of the form

A_{(q)} ⟵^{r_{1}} A′′_{(q)} ⟶^{r_{2}} A′_{(q)}

where the two arrows are isogenies. We define

height(ρ) = height(r_{2}) − height(r_{1}).

Let AV_{O_{q}^{#}} denote the category whose objects are abelian varieties A_{(q)} up to prime-to-q isogeny, given with an embedding

O_{q}^{#} = O^{#} ⊗ ℤ_{(q)} ↪ End A_{(q)}.

The morphisms in AV_{O_{q}^{#}} are those morphisms in AV that respect the O_{q}^{#}-action. The q-divisible group X_{q}(A) of A ∈ AV_{O_{q}^{#}} depends only on A_{(q)}. We define the q-divisible O_{𝔮^{(i)}}-module X(A)_{i} = e_{i} ·X_{q}(A), i = 1, 2, as in 2.1.3. Recall that X(A)_{2} is of height n and dimension 1.

An L-set of abelian varieties is a covariant functor from the category L (maps given by inclusions) to the category AV_{O_{q}^{#}} satisfying several periodicity hypotheses [RZ2, Def. 6.5]. In the present situation, it corresponds to a pair (A, A′) of abelian varieties in the same isogeny class, determined up to prime-to-q isogeny, with PEL structures (2.1.2.2-4) and a K^{q}-level structure. We are also given a pair of isogenies ρ : X(A)_{2} → X(A′)_{2}, ρ_{1} : X(A′)_{1} → X(A)_{1}, in each case with kernels isomorphic over O_{K} to O_{𝔮^{(i)}}/𝔮^{(i)}-modules C_{i}, respectively, each of rank 1. It is assumed that Ker(ρ_{2}) is the Cartier dual of Ker(ρ_{1}) with respect to the polarization (cf. 2.1.3.6). We can express this by saying that there is a quasiisogeny A ⟵^{r_{2}} A′′ ⟶^{r_{1}} A′ with r_{1} and r_{2} both isogenies of height n of O^{#}-modules, such that r_{1} (resp. r_{2}) induces an isomorphism on X(•)_{1} (resp. on X(•)_{2}). In particular, A′ is uniquely determined by C_{2}, or equivalently by ρ, and by the condition that A and A′ both admit actions by the same maximal order in O^{#} ⊗ ℤ_{q}. Thus we occasionally drop A′ from the notation.

Rapoport and Zink [RZ2, Def. 6.9] define a functor A_{K^{q}}(L), a point of which over the S_{𝔮}-scheme T is given by a quintuple as in (2.1.3.6). In the language of [RZ2], this consists of

3.1.4.1. An L-set (A, A′, ρ : X(A)_{2} → X(A′)_{2}) of objects of AV_{O_{q}^{#}}(T), as above;

3.1.4.2. A ℚ-homogeneous principal polarization λ_{A} of A;

3.1.4.3. An O_{K,(q)}-equivariant K^{q}-level structure

β^{q} : H_{1}(A, A_{f}^{q}) ≃ V ⊗ A_{f}^{q} (mod K^{q})

that respects the bilinear forms on both sides up to a constant in (A_{f}^{q})^{×}.

We also use the term L-set to refer to the triple (X(A)_{2}, X(A′)_{2}, ρ). The above data have to satisfy the determinant condition (2.1.2.6), which is adequate because q is assumed unramified in K. As remarked in [RZ2, p. 279], we then have

Proposition 3.1.5. The functor A_{K^{q}}(L) is representable by a projective scheme over S_{𝔮}, which we also denote A_{K^{q}}(L). The generic fiber A_{K^{q}}(L)_{K_{𝔮}} of A_{K^{q}}(L) contains (𝔖_{K_{0}})_{K_{𝔮}} as an open and closed subscheme.

The second statement is as in §2.1.

In particular, we can define 𝕊_{K_{0}} to be the scheme-theoretic closure of (𝔖_{K_{0}})_{K_{𝔮}} in A_{K^{q}}(L). It is again an open and closed subscheme and the local properties of A_{K^{q}}(L) determine those of 𝕊_{K_{0}}.

3.2. Local models of 𝔖_{K_{0}}.

For any A ∈ AV let X(A) denote its associated q-divisible group, and define X(A) as above. The determinant condition (2.1.2.6) translates into the condition that 3.2.1. (a) The connected part X(A)_{2}^{o} of X(A)_{2} is a rank-one formal O_{K,𝔮^{(2)}}-module. In particular, X(A)_{2,𝔮^{(2)}}^{o} is a one-dimensional formal group.

(b) Let 𝔮′ ≠ 𝔮 be a prime of E dividing q, and let 𝔮^{′,(i)} be the divisor of 𝔮′ in K dividing q^{(i)}, i = 1, 2. Then the localization X(A)_{𝔮^{′,(i)}} of XA at O_{K,𝔮^{′,(i)}} is of multiplicative type (resp. étale) for i = 1 (resp. i = 2).

Let 𝕊_{K_{0}}^{s} denote the special fiber of 𝕊_{K_{0}}. Let x be a geometric point of 𝕊_{K_{0}}^{s}; i.e., a morphism Spec(\overline{κ}) → 𝕊_{K_{0}} of S_{𝔮}-schemes, where \overline{κ} is the algebraic closure of the residue field k(𝔮) of O_{𝔮^{(1)}}, which we identify with O_{𝔮}. Rapoport and Zink have shown that the infinitesimal structure of the moduli space A_{K^{q}}(L), and hence of the open and closed subscheme 𝕊_{K_{0}}, is determined in a neighborhood of x by the deformation theory of the q-divisible O_{K,(q)}-module(s) attached to x.

We follow the discussion in [RZ2,6.12], to which we refer for a discussion of the crystalline data. The point x corresponds to a quintuple

(A_{x}, ι_{x}, λ_{x}, β_{x}^{q}, ρ_{x} : X(A_{x})_{2} → X(A′_{x})_{2})

of data (2.1.3.6). Let K_{0} be the field of fractions of the Witt ring W (\overline{κ}), with canonical Frobenius operator σ, and write \overline{S}_{q} = Spec(W (\overline{κ})). Let N be the isocrystal associated to A_{x} by the covariant Dieudonné functor. There is an isomorphism of polarized K ⊗ K_{0}-modules

(3.2.2)  N ⟶^{∼} V ⊗ K_{0}

The crystalline Frobenius operator ϕ on N corresponds under this isomorphism to the operator b ⊗ σ for a (unique) b ∈ G(K_{0}). To these data we can then associate a (crystalline) PEL type

(3.2.3)  (∗, O_{K,(q)}, V, b, μ, L),

where μ is the cocharacter of G defined in §𝔠. The data (3.2.3) determine a formal scheme M̆ which is a moduli space, on the category Nilp_{S_{𝔮}} of S_{𝔮} schemes on which q is nilpotent, for L-sets (X, X′) of q-divisible O_{K,(q)}-module(s) with PEL structure analogous to (2.1.2.2-4), endowed with quasi-isogenies

(3.2.4)  ρ : (X, X′) ⟶ (X(A_{x}), X(A′_{x})).

For any such L-set (X, X′) of q-divisible O_{K,(q)}-modules, there is a decomposition

(3.2.5)  X ⟶^{∼} X_{1} × X_{2}

and a quasi-isogeny

(3.2.6)  λ_{2} : X_{2}→X̂_{1}

corresponding to (3.1.2) and (3.1.3), respectively, with compatible decomposition and quasi-isogeny for the primed q-divisible group. The map λ_{2} identifies the action of O_{K,q^{(2)}} on the left-hand side with that of O_{K,q^{(1)}} on the right, via the isomorphism (complex conjugation) c : K_{q^{(2)}}→K_{q^{(1)}}. Moreover, these structures, together with the O_{K,(q)}-action and the pair of quasi-isogenies

ρ_{1} : X_{1}→X(A_{x})_{1}; ρ_{2} : X_{2}→X(A_{x})_{2},

deduced from (3.2.4), determine the object (X, X′) uniquely. Here X(A_{x})_{1} is defined as in 2.1.3 and X(A_{x})_{2} is defined analogously.

We state a weak version of the Rapoport-Zink theorem on p-adic uniformization [RZ2, Theorem 6.23].

Theorem 3.2.7. Let x be a geometric point of 𝕊_{K_{0}}^{s}. There is a locally closed connected subscheme T (x) ⊂ 𝕊_{K_{0}} containing the image of x, and a connected formal subscheme M̆^{0} ⊂ M̆, admitting an étale morphism

Θ : M̆^{0} ⟶ T̂(x)

of formal schemes over Spf(W (\overline{κ})). Here T̂(x) is the formal completion of 𝕊_{K_{0}} along T (x).

The statement in [RZ2] concerns the whole isogeny class containing x, which is a union, generally infinite, of closures of subschemes of the form T (x). We are only concerned with the étale local structure of 𝕊_{K_{0}} in a neighborhood of x, hence the above formulation suffices for our purposes.

It follows that the nature of the singularities of of 𝕊_{K_{0}}^{s}, if any, is encoded by the singularities of the connected components of the formal scheme M̆. Let X^{∗}(G) denote the group of ℚ_{p}-rational characters of G. Rapoport and Zink construct a natural morphism χ : M̆→Δ of formal schemes over Spf(W (\overline{κ})) where Δ is the constant group scheme over Spf(W (\overline{κ})) [RZ2,3.52] associated to the discrete group Hom(X^{∗}(G), ℤ) (homomorphisms of abelian groups). By construction χ is a fibration. It can be defined by replacing the group G by its abelianization G^{ab} and by replacing the data (b, μ) in (3.2.3) by their images in G^{ab}, cf. the proof of Prop. 1.21 in [RZ2]; however, we will not use this fact.

Now we return to the decomposition (3.2.5). The map

(3.2.8)  (X, X′; ρ) ↦ (X_{2}, X′_{2}; ρ_{2})

defines a morphism

f_{1} : M̆ ⟶ M̆_{1}

where M̆_{1} is the moduli space of EL type, in the terminology of [RZ2,§3] (see below), parametrizing L_{1}-sets (X_{1}, X′_{1}) of q-divisible O_{K,q^{(1)}}-modules of height n and dimension 1, rigidified by a quasi-isogeny ρ_{1} to the fixed L_{1}-set (X_{2}(A_{x}), X_{2}(A′_{x})). Here by L_{1}-set we just mean a pair (X_{2}, X′_{2}) as above connected by an isogeny X_{2} → X′_{2} of degree q (cf. the definitions following (3.1.3)). By the discussion preceding Theorem 3.2.7, we see that this morphism is just the forgetful map “forget λ_{2}”. The set of possible λ_{2}’s is a principal homogeneous space under ℚ_{q}^{×}/ℤ_{q}^{×}, thus the map f_{1} is étale. Moreover, if M̆^{0} ⊂ M̆ is a connected component, f induces an isomorphism M̆^{0} ⟶^{∼} f(M̆^{0}). Thus we have

Corollary 3.2.9. In Theorem 3.2.7 M̆ can be replaced by M̆_{1}.

Now we are reduced to studying the singularities of M̆_{1}. We begin by recalling the definition of the EL data defining M̆_{1}. We are given the ℚ_{p}-algebra K_{q^{(2)}}, its maximal order O_{K,q^{(2)}}, the free rank n K_{q^{(2)}}-module V_{q^{(2)}}, and its rank one submodule V_{0,q^{(2)}}. To these data is associated the group G_{2} = GL(n)/K_{q^{(2)}}. We are also given the crystalline Frobenius ϕ_{2} = ϕ|V_{q^{(2)}} ⊗ K_{0} with respect to the isomorphism (3.2.5), and hence the image b_{2} of the element b ∈ G(K_{0}) under the natural map G→G_{2} (forget the similitude factor).

However, the structure of possible singularities is independent of b_{2}. More precisely, Rapoport and Zink define a local model M̂^{loc}, a projective formal scheme over Spf(W (\overline{κ})) depending only on the data (K_{q^{(1)}}, O_{K,q^{(2)}}, V_{q^{(2)}}, V_{1,q^{(2)}}), for which the following proposition holds:

Proposition 3.2.10. ( [RZ2,Prop. 3.33]) Let x ∈ M̆_{1} be a closed point. Then x has a pointed étale neighborhood (U, y)→(M̆_{1}, x) which is formally étale over M̂^{loc}.

The construction underlying this proposition will be outlined in the course of the proof of Lemma 3.3.2. We copy the definition of the functor M^{loc} from [RZ2, Definition 3.27]. Recall that we identify O_{K,q^{(1)}} with O_{𝔮}.

Definition 3.2.11. A point of M^{loc} with values in an O_{𝔮}-scheme T is given by (a) an L_{1}-set of locally free rank one O_{T} -modules t_{Λ_{2}} → t_{Λ′_{2}} and

(b) a morphism of L_{1}-sets

ϕ′ : Λ′_{2} ⊗_{ℤ_{q}} O_{T} → t_{Λ′_{2}}; ϕ : Λ_{2} ⊗_{ℤ_{q}} O_{T} → t_{Λ_{2}},

Both ϕ′ and ϕ are surjective, and the obvious diagram is assumed commutative.

We let M̂^{loc} denote the q-adic completion of M^{loc} ×_{S_{𝔮}} Spec(W (\overline{κ})).

Proposition 3.2.12. Let s be a closed point of M̂^{loc}, and let M̂_{s}^{loc} denote the formal completion of M̂^{loc} along s. Suppose M̂^{loc} is singular at s. Then there are a complete local smooth W (\overline{κ})-algebra S and an isomorphism

M̂_{s}^{loc} ⟶^{∼} Spf(S[[u, v]]/(uv − q))

of formal schemes over Spf(W (\overline{κ})).

In particular, M̂^{loc} is regular and flat over Spf(W (\overline{κ})), and its special fiber has only singularities of the form (smooth) × (ordinary double point).

Proof. The proof proceeds by making the local equations for M^{loc} explicit. Our approach is similar to that of de Jong in [dJ]. We work over W (\overline{κ}). If T is a W (\overline{κ})- scheme, the points of M^{loc} with values in T are given by the data (a) and (b) of Definition 3.2.11. In particular, if T is in characteristic zero then the homomorphism t_{Λ_{2}} → t_{Λ′_{2}} is necessarily an isomorphism.

We choose bases {e, e_{1}, …, e_{n−1}} and {f, f_{1}, …, f_{n−1}} of Λ′_{2} and Λ_{2}, respectively, so that

e = qf, e_{i} = f_{i}, i = 1, …, n − 1.

These bases identify the projectivizations ℙ(Λ′_{2}) and ℙ(Λ_{2}) with projective space ℙ_{K_{0}}^{n−1}. Let {X, X_{1}, …, X_{n−1}} and {Y, Y_{1}, …, Y_{n−1}} be the corresponding sets of homogeneous coordinates. The quotients t_{Λ′_{2}} of Λ′_{2} ⊗_{ℤ_{q}} O_{T} defines a T -valued point of ℙ(Λ′_{2}); likewise t_{Λ_{2}} defines a T -valued point of ℙ(Λ_{2}). The compatibility condition is that these two points are identified to the same point in ℙ_{K_{0}}^{n−1}. Assume T = Spec(R) to be affine and let (x, x_{1}, …, x_{n−1}) and (y, y_{1}, …, y_{n−1}) be their expressions in the homogeneous coordinates defined above, with the x’s and y’s in R; then the condition is that (qx, x_{1}, …, x_{n−1}) and (y, y_{1}, …, y_{n−1}) are proportional. In other words, the matrix

(3.2.12.1)  ( \begin{matrix}qx & x_{1} & … & x_{n−1} \\ y & y_{1} & … & y_{n−1} \end{matrix} )

is of rank one.

The 2-by-2 minors of (3.2.12.1) provide a set of homogeneous equations for the scheme M^{loc}. The assertions of the proposition follow easily from the form of these equations, as we will see in a moment. We note for future reference the following fact:

Remark 3.2.12.2. The open subscheme M^{loc,é} ⊂ M^{loc}, defined by the condition that the homomorphism t_{Λ_{2}} → t_{Λ′_{2}} be an isomorphism, is non-singular.

It suffices to consider a geometric point z of M^{loc,é} in characteristic q. Let R be a complete local ring with residue field \overline{κ} and let the x’s and y’s be local homogeneous coordinates near z, as above; we may assume the x’s and y’s lie in R. The condition z ∈ M^{loc,é} corresponds to the condition that the rows of (3.2.12.1) be proportional with a factor of proportionality α invertible in R. Consider the inhomogeneous coordinates. There are essentially three possibilities:

(a) x_{1} is a unit, y_{1} is a unit. Let

x′_{j} = \frac{x_{j}}{x_{1}} , y′_{j} = \frac{y_{j}}{y_{1}} , j > 1, y′ = \frac{y}{y_{1}} , x′ = \frac{x}{x_{1}} .

The inhomogeneous equations are

y′ = x′; x′_{j} = y′_{j}, j > 1,

and these are always smooth.

(b) y_{i} is a unit for some i ≥ 1. Without loss of generality we may assume i = 1. Then the invertibility of α implies x_{1} is also a unit, and we are reduced to case (a).

(c) y is a unit. This is impossible because y = αqx with α invertible and x ∈ R. It is case (c) that gives rise to the singularities of M^{loc} outside M^{loc,é}.

The following theorem appears to be the natural generalization to higher dimensions of the well-known theorem of Deligne and Rapoport on the local structure of the modular curve X_{0}(q).

Theorem 3.2.13. The moduli scheme 𝕊_{K_{0}} is regular and flat over S_{𝔮}, and its special fiber is a union of smooth divisors with normal crossings. Moreover, if x is a singular point of the special fiber then the formal completion 𝕊_{K_{0},x} is isomorphic to a formal scheme of the form Spf(S[[u, v]]/(uv − q)), where S is a complete local smooth O_{𝔮}-algebra.

Proof. The properties in question are local in the étale topology. Thus it suffices to verify the theorem after base change to W (\overline{κ}). The Theorem is thus a consequence of Theorem 3.2.7, Corollary 3.2.9, Proposition 3.2.10, and Proposition 3.2.12.

The following proposition is a strengthening of Proposition 3.1.1.:

Proposition 3.2.14. Let \overline{i} : y → S_{𝔮} be a geometric point lying over i, and let 𝕊_{K_{0},y} = y ×_{S_{𝔮}} 𝕊_{K_{0}} denote the corresponding geometric special fiber. The ordinary locus is open and dense in every irreducible component of 𝕊_{K_{0},y}.

Proof. First note that the forgetful morphism 𝕊_{K_{0},y} → 𝕊_{K,y} is finite, i.e. has finite fibers. Indeed, let x be a geometric point of 𝕊_{K,y}, corresponding to an abelian variety A_{x}; then the points of 𝕊_{K_{0},y} above x correspond to the rank 1 O_{𝔮^{(2)}}/𝔮^{(2)}- subgroup schemes C_{2} ⊂ A_{x}[𝔮^{(2)}]. But the connected part of the finite flat group scheme A_{x}[𝔮^{(1)}] is of dimension one. It follows easily that the set of possible C_{2}’s is finite.

On the other hand, it follows from Theorem 3.2.13 that every irreducible component 𝕊_{K_{0},y} is of dimension n−1. Combining these two facts with Proposition 3.1.1, we see that no irreducible component of 𝕊_{K_{0},y} lies entirely over the complement of the ordinary locus in 𝕊_{K,y}. The assertion now follows from Proposition 3.1.1.

Remarks The results of this section were obtained in 1996. Analogous results for Iwahori level were known to Rapoport previously (cf. [R] which, however, only treats places where the division algebra is ramified). General parahoric levels and general signatures have since been treated by Görtz [G], and Theorem 3.2.13 is now regarded as the trivial case.

Counterexamples due to Stamm show that the analogue of Proposition 3.2.14 is not valid in the generality of [We1]. However, the present case has also been considered in [We2], where Wedhorn obtains a congruence formula related to the one proved in section 4.2, below.

3.3. Local models of 𝔖_{K_{1}}.

3.3.1. Let (A^{univ}, A^{′,univ}, ρ^{univ} : X(A^{univ})_{2}→X(A^{′,univ})_{2}) denote the universal Lset (3.1.4.1) of abelian varieties over the moduli scheme A_{K^{q}}(L). Then the kernel of ρ^{univ} is a finite flat group scheme C_{2}^{univ} of order q, isomorphic over O_{K} to an O_{𝔮^{(2)}}/𝔮^{(2)}-module of rank 1 (3.1.4). We use the same notation to denote the pullback of C_{2}^{univ} to 𝕊_{K_{0}} via the natural inclusion. Similarly, we let C_{1}^{univ} denote the kernel of the homomorphism ρ^{′,univ} : X(A^{′,univ})_{1}→X(A^{univ})_{1} described in 2.1.3.6, so that C_{1}^{univ} and C_{2}^{univ} are canonically Cartier dual (2.1.4.4).

The property “C_{2} is étale” is relatively representable over the functor A_{K^{q}}(L) and defines an open subscheme A_{K^{q}}(L)o ; é [cf. [DR,§V.1]. Likewise, the property “C_{1} is étale”, equivalent to the property “C_{2} is locally isomorphic to μ_{q} in the étale topology”, defines an open subscheme A_{K^{q}}(L)_{m}^{o} of A_{K^{q}}(L). It follows from Proposition 3.2.14 that

3.3.1.1. The union A_{K^{q}}(L)^{o} = A_{K^{q}}(L)é∪A_{K^{q}}(L)_{m}^{o} is open and dense in A_{K^{q}}(L).

We let 𝕊_{K_{0}}^{o}, 𝕊_{K_{0},é}^{o}, and 𝕊_{K_{0},m}^{o} denote the corresponding subschemes of 𝕊_{K_{0}}. Note that 𝕊_{K_{0}}^{o} contains the ordinary locus but is in general larger.

Let \overline{𝕊}_{K_{0}} denote the special fiber of 𝕊_{K_{0}}, X_{m}^{o} = \overline{𝕊}_{K_{0}} ∩ 𝕊_{K_{0},m}^{o} Xo ; é = \overline{𝕊}_{K_{0}} ∩ 𝕊_{K_{0},é}^{o},

and let X_{m} and Xé denote the closures in \overline{𝕊}_{K_{0}} of X_{m}^{o} and Xo ; é, respectively. It follows from (3.3.1.1) that

(3.3.1.2)  \overline{𝕊}_{K_{0}} = X_{m} ∪ Xé.

We will need the following strengthening of Theorem 3.2.13:

Lemma 3.3.2. Let x be a singular point of the special fiber of 𝕊_{K_{0}}, and let

(A_{x}, A′_{x}, ρ_{x} : X(A_{x})_{2}→X(A′_{x})_{2})

be the corresponding L-set of objects of AV_{O_{q}^{#}}. Then Ker ρ_{x} is isomorphic to the unipotent group scheme α_{q}.

Proof. First, suppose Ker ρ_{x} is of multiplicative type. Since A_{x}[𝔮^{(2)}] is onedimensional, it follows that x is in the ordinary locus and Ker ρ_{x} is the full qtorsion subgroup X(A)_{2}^{o}[q] of the formal group X(A)_{2}^{o} (cf. 3.2.4). Let (\overline{𝕊}_{K})^{o}, resp. (\overline{𝕊}_{K_{0}})^{o} denote the ordinary locus of \overline{𝕊}_{K}, resp. \overline{𝕊}_{K_{0}}. The former is an open and closed subscheme of the moduli space A_{K} for triples (A, λ, β) as in 3.1.4, and the functor “forget ρ : X(A)_{2}→X(A′)_{2}” from A_{K^{q}}(L) to A_{K} restricts to the canonical map f : X_{m}^{o}→(\overline{𝕊}_{K})^{o}. Then the map f admits a section:

(A, λ, β) ↦ (A, A′, ρ, λ, β),

where ρ : X_{1}→X_{2}/X(A)_{2}^{o}[q] is the tautological map and A′ is determined by A and Ker(ρ) as in the discussion preceding 3.1.4.1. Since \overline{𝕊}_{K} is smooth, it follows that X_{m}^{o} is smooth, hence x is a smooth point.

To treat the étale case we need to recall briefly how Proposition 3.2.10 is proved. Let (𝕏_{2}, 𝕏′_{2}; ρ_{2}) be the universal rigidified L_{1}-set of q-divisible O_{K,q^{(2)}}-modules over M̆_{1}, in the notation of (3.2.8). Let Nilp_{W (\overline{κ})} be the category of W (\overline{κ}) schemes on which q is locally nilpotent. Let N be the functor on Nilp_{W (\overline{κ})} such that, for S ∈ Nilp_{W (\overline{κ})}, N (S) is given by triples ((X_{2}, X′_{2}); ρ_{2}; (γ_{2}, γ′_{2})). Here ((X_{2}, X′_{2}); ρ_{2}) ∈ M̆_{1}(S) and

(γ_{2}, γ′_{2}) : (M(X_{2}), M(X′_{2})) ⟶^{∼} (Λ_{2}, Λ′_{2}) ⊗_{ℤ_{q}} O_{S}

is an isomorphism of L_{1}-sets of O_{S}-modules, where M(•) denotes the crystal associated to the q-divisible O_{K,q^{(2)}}-module •. Then N is representable by a formal scheme. The forgetful functor defines a formally smooth morphism N →M̆_{1}.

Consider the morphism ψ of functors that to the triple ((X_{2}, X′_{2}); ρ_{2}; (γ_{2}, γ′_{2})) associates the composite morphism of L_{1}-sets:

(3.3.2.1)  (Λ_{2}, Λ′_{2}) ⊗_{ℤ_{q}} O_{S} ⟶^{∼} (M(X_{2}), M(X′_{2})) → (Lie(X_{2}), Lie(X′_{2})).

Here the first arrow is given by (γ_{2}, γ′_{2})^{−1}, while the second morphism is given by the theory of Grothendieck and Messing [Me]. Then ψ defines a morphism ψ : N →M̂^{loc} (cf. [RZ2], p. 90).

Let x ∈ M̆_{1} be a closed geometric point, and assume the corresponding morphism X_{2,x}→X′_{2,x} is étale. We need to show that x is a smooth point of M̆_{1}. Rapoport and Zink consider a certain class of sections s of N →M̆_{1} on pointed étale neighborhoods (U, y) of (M̆_{1}, x), which are called rigid of the first order [RZ2,p. 91]. They show that if s : (U, y)→M̆_{1} is rigid of the first order, then ψ∘s : U→M̂^{loc} is formally étale in a Zariski open neighborhood of y ([RZ2, Prop. 3.33]). On the other hand, it is immediate from the definitions that if X_{2,x}→X′_{2,x} is étale, then s(y) ∈ M̂^{loc,é}, defined as the q-adic completion of M^{loc,é} ×_{Spec(O_{𝔮})} W (\overline{κ}). But Remark 3.2.12.2 implies that M̂^{loc,é} is smooth. This implies that x is a smooth point, and completes the proof of Lemma 3.3.2.

Proposition 3.3.3. The subschemes X_{m} and Xé are smooth divisors on 𝕊_{K_{0}}.

Proof. Let x be a singular point of \overline{𝕊}_{K_{0}} contained in X_{m}. We replace \overline{𝕊}_{K_{0}} by an appropriate étale neighborhood of x, which we may take to be isomorphic to Spec(R), R = S[u, v]/(uv − q), with notation as in Theorem 3.2.13. It follows from Theorem 3.2.13 that x lies on the intersection of two smooth irreducible components of the inverse image in Spec(R) of Xé, say D_{1} and D_{2}. Following a suggestion of de Jong, we use the Tate-Oort theory [TO] to describe the group scheme of prime order C_{1}^{univ} in a neighborhood of x. We may assume u to be a local parameter for D_{1} and v a local parameter for D_{2}. Let C denote the pullback of C_{1}^{univ} to Spec(R). Since R is affine it follows from the main theorem of [TO] that C is determined by a pair of elements x, y ∈ R with xy = q, and that C is étale on the subset of Spec(R) where x is invertible. But our hypothesis is that C_{2}^{univ} is multiplicative, hence C_{1}^{univ} is étale, on the non-singular locus of D_{1} ∪ D_{2}. Since 𝕊_{K_{0}} is regular, the subvariety defined by the vanishing of x is purely of codimension one. Thus x is invertible in R, and C is étale on Spec(R). But then it follows from Lemma 3.3.2 that x is not a singular point. The argument for Xé is analogous.

Remark 3.3.4. Let R be a ℤ_{q} algebra. The Tate-Oort theory actually classifies group schemes of order q over Spec(R) by elements a and b in R such that that ab = w_{q}, where w_{q} = νq for a certain explicit unit ν in ℤ_{q}. Our parameters above are x = ν^{−1}a, y = b. The distinction is important for the group structure but not for the local geometry. Thus we let G_{R}(x, y) denote the group over Spec(R) associated to (a = νx, b = y) in the Tate-Oort classification. Then

G_{R}(x, y) ≅ G_{R}(x′, y′) if and only if x/x′ is the (q − 1)st power of a unit in R.

The proof yields the following Corollary:

Corollary 3.3.5. Let s be a singular point of \overline{𝕊}_{K_{0}} and let Spf(S[[u, v]]/(uv−q)) be a local model for the formal completion of 𝕊_{K_{0}} along s, in the notation of Theorem 3.2.13. Then s lies in the intersection X_{m} ∩ Xé, and, up to interchanging u and v, u is a local equation for Xé and v is a local equation for X_{m}. Moreover, in the Tate-Oort classification for the pullback C of C_{1}^{univ} to Spf(S[[u, v]]/(uv − q)), we have C ≃ G_{R}(x, y), with R = S[[u, v]]/(uv − q), x = αu, y = α^{−1}v, for some α ∈ (S[[u, v]]/(uv − q))^{×}.

Proof. Only the last statement needs justification. But in the Tate-Oort theory, x (resp. y) vanishes precisely where C_{1}^{univ} is not étale (resp. multiplicative). Thus the variety defined by x is the subset Xé where C_{2}^{univ} is not multiplicative, so x = au for some a ∈ (S[[u, v]]/(uv − q))^{×}, and we conclude by recalling that xy = q = uv.

We now turn to the moduli scheme 𝔖_{K_{1}}. Let C^{+} denote the complement of the zero section of C_{1}^{univ} over the generic fiber (𝔖_{K_{0}})_{K_{𝔮}}. Then C^{+} tautologically represents the functor

(3.3.5.1)  T ↦ \{ \begin{matrix}T -valued point f : T →(𝔖_{K_{0}})_{K_{𝔮}} \\ plus a generator of f^{∗}(C_{1}^{univ}) over T \end{matrix} \}

On the other hand, just as in the elliptic modular case (3.3.5.1) is represented by 𝔖_{K_{1}} over K_{𝔮} (cf. 2.1.4.5). Thus in order to obtain a regular model for 𝔖_{K_{1}} over S_{𝔮} it will be enough to find a regular model for C^{+}. We do this explicitly using the Tate-Oort equations.

Proposition 3.3.6. (a) The moduli scheme 𝔖_{K_{1}} has a flat model 𝕊_{K_{1}} over S_{𝔮} which is a regular scheme, and whose special fiber is a divisor with normal crossings, each of whose components has multiplicity prime to q. More precisely, there is a finite flat morphism p_{1} : 𝕊_{K_{1}} → 𝕊_{K_{0}}.

(b) The inverse image Y_{m} = p_{1}^{−1}(X_{m}) is a reduced divisor on 𝕊_{K_{1}}, while Yé = p_{1}^{−1}(Xé) is a divisor with multiplicity q − 1, and the map p_{1} : Y_{é,red}→Xé is an isomorphism.

(c) Let Y_{sing} denote the singular locus of 𝕊_{K_{1}}. Then Y_{sing,red} is smooth and all its components are of codimension 1 in the special fiber.

Proof. We define 𝕊_{K_{1}} to be the normalization of 𝕊_{K_{0}} in C^{+}. All assertions are local on 𝕊_{K_{0}}, so we can work over a neighborhood of a geometric point s of the special fiber \overline{𝕊}_{K_{0}}. There are three cases to consider.

Case 1: s ∈ X_{m}^{o}. Then C^{+} is étale over 𝕊_{K_{0}} in a neighborhood of s, and the claims follow immediately from Theorem 3.2.13.

Case 2: s ∈ Xo ; é. Then C^{+} is of multiplicative type, hence is isomorphic to μ_{q} on an étale neighborhood of s. Thehe statement about multiplicities is then obvious, as is the assertion that p_{1} : Y_{é,red}→Xé is an isomorphism.

Case 3: s ∈ Xé ∩ X_{m}. We use the notation of Corollary 3.3.5. Let α be the unit introduced in the statement of Corollary 3.3.5. By the remark preceding that corollary, C ≃ G_{R}(u, v) if and only if the unit α is a (q − 1)st power in S[[u, v]]/(uv − q). Now the extension (S[[u, v]]/(uv − q))[(α)^{\frac{1}{q−1}} ] is finite and étale over S[[u, v]]/(uv − q). Since our theorem is local in the étale topology, we may assume C ≃ G_{R}(u, v).

Now we recall the Tate-Oort local equations for G_{R}(u, v). On p. 13 of [TO] are introduced parameters U and y such that, setting X_{1} = w_{q}U^{1−q}, X_{2} = U^{q−1}, and Y = U^{−1}y, and working on the formal completion of Spec S[u, v]/(uv − q), we have (3.3.7)

G_{R}(u, v) = Spf(S[Y][[X_{1}, X_{2}]]/(X_{1}X_{2} − w_{q}, Y^{q} − X_{1}Y); X_{1} = νu, X_{2} = v.

Now the factorization Y^{q} − X_{1}Y = Y (Y^{q−1} − X_{1}) defines an injective map from the ring on the right-hand side of (3.3.7) to the direct product of two rings R_{0} and R^{∗}, where

R_{0} = S[[X_{1}, X_{2}]]/(X_{1}X_{2} − w_{q}); R^{∗} = S[[X_{2}, Y]]/(Y^{q−1}X_{2} − w_{q}).

Dually, we have a surjective morphism

Spf(R_{0}) ∐ Spf(R^{∗}) → G_{R}(u, v).

The image of Spf(R_{0}) corresponds to the zero section, while the image of Spf(R^{∗}) is the scheme-theoretic closure of C^{+}. It follows that Spf(R^{∗}) is a local model for C^{+} over a neighborhood of the singular point s. Obviously Spf(R^{∗}) has the properties required, and the construction shows that p_{1} : Y_{e,red}→Xé is an isomorphism near s. Moreover, Spf(R^{∗}) is normal, and it follows just as in [DR,§V.2] that Spf(R^{∗}) is a formal local model for 𝕊_{K_{1}} near s.

This completes the proof.

The analogy with [DR] becomes clearer if we extend scalars to O_{𝔮}[ζ_{q}], where ζ_{q} denotes a primitive q-th root of 1. Let S̃_{𝔮} = Spec(O_{𝔮}[ζ_{q}])

Proposition 3.3.8. (a) The moduli scheme 𝔖_{K_{1}} ×_{K_{𝔮}} K_{𝔮}(ζ_{q}) has a flat model 𝕊̃_{K_{1}} over S̃_{𝔮} which is a regular scheme, and whose special fiber is a reduced divisor with normal crossings. More precisely, there is a finite flat morphism p̃_{1} : 𝕊̃_{K_{1}} → 𝕊_{K_{0}} ×_{S_{𝔮}} S̃_{𝔮}.

(b) The inverse images Ỹé = p̃_{1}^{−1}(Xé) and Ỹ_{m} = p̃_{1}^{−1}(X_{m}) are reduced divisors on 𝕊̃_{K_{1}}. Moreover, Ỹ_{m} ≃ Y_{m} and Ỹ_{e}^{o} = p̃_{1}^{−1}(Xo ; é) is étale over (Y o ; é)_{red} ≃ Xo ; é with Galois group isomorphic to the group of diamond operators (cf. Remark-Definition 2.1.4.6).

(c) The model 𝕊̃_{K_{1}} can be defined as the normalization of 𝕊_{K_{0}} ×_{S_{𝔮}} S̃_{𝔮} in the scheme

Isom_{𝕊_{K_{0}} ×_{S𝔮} Spec(K_{𝔮}(ζ_{q}))}(ℤ/qℤ, C_{1}).

(d) Over the complement of Ỹ_{m} (resp. Ỹé) 𝕊̃_{K_{1}} represents the functor Isom(μ_{q}, C_{1}) (resp. Isom(ℤ/qℤ, C_{1}), and these two are canonically isomorphic over the generic fiber.

Proof. The first part of the proof of Proposition 3.3.6 showed that we can take the universal group scheme C over 𝕊_{K_{0}} to be étale locally isomorphic to G_{R}(u, v). Let R_{0} = ℤ_{q}[u, v]/(uv−q). Then G_{R}(u, v) is the base change from R_{0} to R of G_{R_{0}}(u, v), and it suffices to work over the base R_{0}, a local curve over Spec(ℤ_{q}) ≅ S_{𝔮}. Then the proof of [DR,Lemma V.2.8] applies word for word to yield the result. Parts (c) and (d) are the analogues of [DR, Theorem 2.7] and [DR, Prop. 2.3], respectively, and the proofs are identical.

We write down the explicit local equations since we will need them for the following lemma. The formal completion of 𝕊̃_{K_{1}} at a singular point is isomorphic to

(3.3.8.1)  Spf(S[[ζ_{q}, x, y]]/(xy − w)),

where w is a uniformizing parameter in ℤ_{q}[ζ_{q}].

Let ω : (ℤ/qℤ)^{×} → ℤ_{q}^{×} be the Teichmüller character.

Lemma 3.3.9. The diamond operators < a > ∈ Aut(𝔖_{K_{1}}(𝔮)/𝔖_{K_{0}}(𝔮)) extend to automorphisms of 𝕊_{K_{1}} (resp. 𝕊̃_{K_{1}}) that act trivially on (Yé)_{red} (resp. on Ỹ_{m} ∩ Ỹé).

Proof. The action of the diamond operators < a > on 𝔖_{K_{1}}(𝔮), in terms of (3.3.6), is deduced from the action on C^{+} taking a section c to [a]c, where [a] denotes multiplication by a (cf. 2.1.4.6). In the model G_{R}(u, v), with equations (3.3.8) [a] takes Y to ω(a)Y [TO, p. 13]. The assertion of the lemma follows when 𝕊_{K_{1}} is replaced by its local model Spec(R^{∗}), in the notation of the proof of Proposition 3.3.7, and this implies the result for 𝕊_{K_{1}}.

In the local model 3.3.8.1 near a singular point of 𝕊̃_{K_{1}}, [a] acts by taking x to ω(a)x, y to ω^{−1}(a)y (cf. [DR, V.2.15]). This shows that [a] extends to an automorphism that fixes Ỹ_{m} ∩ Ỹé.

4. Vanishing cycles and cohomology

4.1 Calculations of vanishing cycles.

Let η = Spec(K_{𝔮^{(1)}}) be the generic point of S_{𝔮}, \overline{η} the spectrum of the algebraic closure of K_{𝔮^{(1)}}, j : \overline{η} → S_{𝔮} the natural map. We write Gal(\overline{η}/η) for Gal(\overline{K}_{𝔮^{(1)}}/K_{𝔮^{(1)}}), and let I ⊂ Gal(\overline{η}/η) denote the inertia subgroup and P ⊂ I the wild inertia subgroup. The next step is to study the ℓ-adic cohomology of 𝔖_{K_{1}}(𝔮)\overline{η} = 𝔖_{K_{1}}(𝔮) ×_{η} \overline{η} as a module for Gal(\overline{η}/η), by comparing it to the ℓ-adic cohomology of the special fiber. It follows from Proposition 3.3.7 that the model 𝕊_{K_{1}} satisfies hypotheses (i)-(iv) of [RZ1,(2.2)], and that each irreducible component of the special fiber occurs with multiplicity prime to q. Then Grothendieck’s purity conjecture [SGA 7, I, §3] follows from [RZ1, (2.21)]. It follows that the calculation in [SGA 7, I, 3.3] of the sheaves of tame vanishing cycles is valid. 4.1.1. We follow the notation of [RZ1] and let RΨ^{q}ℚ_{ℓ} denote the qth vanishing cycle sheaf. By [RZ1, (2.23)], the vanishing cycle sheaves and the tame vanishing cycle sheaves coincide under our present hypotheses. We recall the calculation of RΨ^{q}ℚ_{ℓ} in the case at hand, which we will treat abstractly.

Consider a regular scheme X, flat and of finite type over S_{𝔮}, with smooth generic fiber X_{η} and special fiber X_{s}. The special fiber is assumed to be étale locally of one of the following types: 4.1.1.1. X_{s} is smooth. This corresponds to a neighborhood of a point in Y_{m} not in Yé. Then the regularity of X implies that X is smooth. In this case RΨ^{q}ℚ_{ℓ} = 0 for q > 0, RΨ^{0}ℚ_{ℓ} = ℚ_{ℓ}. 4.1.1.2. X is of the form Spec ℤ_{q}[X, Y, Y^{−1}]/(X^{q−1}Y − q). This corresponds to a neighborhood of a point in Yé not in Y_{m}. Let X′ = Spec ℤ_{q}[x, y, y^{−1}]/(x^{q−1} − q), and define a map π : X′→X via the embedding

ℤ_{q}[X, Y, Y^{−1}]/(X^{q−1}Y − q) → ℤ_{q}[x, y, y^{−1}]/(x^{q−1} − q); Y ↦ y^{q−1}, X ↦ xy^{−1}.

The morphism π is étale with Galois group 𝔽_{q}^{×} ≅ μ_{q−1}, the group of (q − 1)st roots of unity. Here ζ ∈ μ_{q−1} acts by multiplying both x and y by ζ. The special fiber X_{s} is a non-reduced divisor with multiplicity q − 1; the associated reduced divisor X_{s,red} is defined by the equation X = 0 and is smooth; it is isomorphic to Spec 𝔽_{q}[Y, Y^{−1}], which we view as (𝔾_{m})_{𝔽_{q}}.

We first compute the vanishing cycle sheaves RΨ^{q}ℚ_{ℓ} in the étale neighborhood X′ of X. Let O = ℤ_{q}[ζ_{q}] = ℤ_{q}[x]/(x^{q−1} − q). Then we may write

(4.1.1.2.1)  X′ = Spec O ×_{Spec ℤ_{q}} Y,

where Y = Spec ℤ_{q}[y, y^{−1}]. The second factor is smooth over ℤ_{q}, hence (RΨ^{q}ℚ_{ℓ})_{X′} is the pullback from RΨ^{q} for the finite flat morphism Spec O→Spec ℤ_{q}. The morphism is of relative dimension zero, hence RΨ^{q} = 0 for q > 0, and it is elementary to see that RΨ^{0} is the group algebra ℚ_{ℓ}[μ_{q−1}], with the inertia group of O over ℤ_{q} acting as μ_{q−1}.

It follows that RΨ^{q}ℚ_{ℓ} = 0 for q > 0, and that RΨ^{0}ℚ_{ℓ} is a lisse ℓ-adic sheaf of rank q−1 on X_{s,red} that becomes constant over X′_{s,red}. Moreover, since Gal(X′/X) acts as the inertia group on the first factor of (4.1.1.2.1), one sees easily that the canonical action of Gal(X′/X) on (RΨ^{0}ℚ_{ℓ})_{X′_{s,red}} identifies the latter with the group algebra ℚ_{ℓ}[Gal(X′/X)]. It follows that

(4.1.1.2.2)  RΨ^{0}ℚ_{ℓ} ⟶^{∼} π_{∗}ℚ_{ℓ}.

The inertia group μ_{q−1} still acts on RΨ^{0}, and we have seen that the lift of this action to X′_{s,red} coincides with the action of Gal(X′/X). We write

(4.1.1.2.3)  RΨ^{0}ℚ_{ℓ} = ⊕_{χ}RΨ^{0}ℚ_{ℓ}[χ],

the decomposition with respect to characters of the inertia group. We write L[χ] for the rank one local system RΨ^{0}ℚ_{ℓ}[χ]. Let χ_{0} denote the trivial character. Consider the embedding

j : X_{s,red} = (𝔾_{m})_{𝔽_{q}} ↪ 𝔸^{1} = Spec 𝔽_{q}[Y]

as the complement of the origin Y = 0. The morphism π is totally ramified along Y = 0. It follows that

(4.1.1.2.4)  R^{0}j_{∗}L[χ] = j_{!}L[χ]; R^{q}j_{∗}L[χ] = 0, q > 0 (χ ≠ χ_{0});

(4.1.1.2.5)  R^{0}j_{∗}L[χ_{0}] = ℚ_{ℓ}; R^{q}j_{∗}L[χ_{0}] = 0, q > 0.

More generally, suppose X = Spec R[X, Y, Y^{−1}]/(X^{q−1}Y − q), where R is a smooth ℤ_{q}-algebra of finite type. Then X is the fiber product over

X = Spec R ×_{Spec ℤ_{q}} Spec ℤ_{q}[X, Y, Y^{−1}]/(X^{q−1}Y − q),

where the first factor is smooth. We define

j_{X} = 1 × j : X_{s,red} = Spec R ×_{Spec ℤ_{q}} (𝔾_{m})_{𝔽_{q}}→Spec R ×_{Spec ℤ_{q}} Spec 𝔽_{q}[Y].

Letting X_{2} denote the second factor above p_{2} denote the projection X→X_{2}, factor, we see that

(4.1.1.2.6)  RΨ^{q}ℚ_{ℓ} = p_{2}^{∗}RΨ_{X_{2}}^{q}ℚ_{ℓ},

where RΨ_{X_{2}} denotes the vanishing cycle sheaves for the map from X_{2} to Spec ℤ_{q}. In particular, RΨ^{q}ℚ_{ℓ} vanishes for q > 0, while RΨ^{0}ℚ_{ℓ} breaks up under the action of the inertia subgroup of Gal(\overline{ℚ}_{q}/ℚ_{q}) as the sum of rank one local systems L[χ]: L[χ_{0}] = ℚ_{ℓ}, whereas L[χ] for non-trivial χ satisfies the analogue of (4.1.1.2.4):

(4.1.1.2.7)  R^{0}j_{X,∗}L[χ] = j_{X,!}L[χ]; R^{q}j_{X,∗}L[χ] = 0, q > 0 (χ ≠ χ_{0}).

In other words, j_{X,!}L[χ] is the intersection complex (pure perverse sheaf) on Spec R ×_{Spec ℤ_{q}} Spec 𝔽_{q}[Y] associated to the local system L[χ] on the open subscheme X_{s,red}. 4.1.1.3. X is of the form Spec R[X, Y]/(X^{q−1}Y − q), where R is a smooth ℤ_{q}- algebra of finite type. This corresponds to a neighborhood of a point in Yé ∩ Y_{m}. We want to calculate the stalks of RΨ^{q}ℚ_{ℓ} at a geometric point \overline{x} of the singular locus X_{sing} of the special fiber, defined by X = Y = 0. In this case we simply quote the result from [SGA 7, loc. cit]: RΨ^{q}ℚ_{ℓ} = 0, q > 1; (RΨ^{0}ℚ_{ℓ})\overline{x} = ℚ_{ℓ}, (RΨ^{1}ℚ_{ℓ})\overline{x} = ℚ_{ℓ}(−1), with trivial action of the inertia group on ℚ_{ℓ}, the (−1) denoting Tate twist.

We now apply the above calculations to the cohomology of 𝕊_{K_{1}}. We write Y_{a} = Yé ∩ Y_{m}. Let i_{m} : Y_{m}→𝕊_{K_{1}}, i_{e} : Yé→𝕊_{K_{1}}, and i_{a} : Y_{a}→𝕊_{K_{1}} be the natural maps. Let Y o ; é denote the complement of Y_{a} in Yé, and let j_{e} : (Y o ; é)_{red}→(Yé)_{red} be the open immersion. Then

Proposition 4.1.1.4. The vanishing cycle sheaves RΨ^{q}ℚ_{ℓ} are calculated as follows. Let I denote the inertia subgroup of Gal(\overline{ℚ}_{q}/ℚ_{q}). Then the action of I on RΨ^{q}ℚ_{ℓ} factors through the map to μ_{q−1} given by the action on ℚ[ζ_{q}]. For a character χ of μ_{q−1}, let [χ] denote the χ-isotypic component, and let χ_{0} denote the trivial character. Then

(i) RΨ^{0}ℚ_{ℓ}[χ_{0}] = ℚ_{ℓ}.

(ii) RΨ^{1}ℚ_{ℓ} = RΨ^{1}ℚ_{ℓ}[χ_{0}] is a rank one local system supported on Y_{a}, locally isomorphic at any point of Y_{a} to ℚ_{ℓ}(−1).

(iii) For χ ≠ χ_{0}, RΨ^{0}ℚ_{ℓ}[χ] is the extension by zero of a rank one lisse sheaf L[χ] supported on Y o ; é. Moreover, the natural map j_{e,!}RΨ^{0}ℚ_{ℓ}[χ]→Rj_{e,∗}RΨ^{0}ℚ_{ℓ}[χ] is a quasi-isomorphism.

(iv) Finally, RΨ^{q}ℚ_{ℓ} = 0 for q > 1.

Proof. Everything follows immediately from (4.1.1.1-3) except the global triviality of RΨ^{0}ℚ_{ℓ}[χ_{0}]. But there is always an injection ℚ_{ℓ}→RΨ^{0}ℚ_{ℓ}[χ_{0}], so (i) follows from the fact that all stalks of RΨ^{0}ℚ_{ℓ}[χ_{0}] are one-dimensional.

Since the tame vanishing cycle sheaves are concentrated in two degrees, the vanishing cycle spectral sequence degenerates to a long exact sequence

… →H^{i}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ})→H^{i}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ})

→H^{i−1}(𝕊_{K_{1},\overline{κ}}, RΨ^{1}ℚ_{ℓ})→H^{i+1}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ})→ ….

Using (4.1.1.4.ii), we rewrite this (4.1.2)

… →H^{i}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ})→H^{i}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ})→H^{i−1}((Y_{a})_{red}, RΨ^{1}ℚ_{ℓ}[χ_{0}])→ ….

We deduce from (i) and (iii) of (4.1.1.4) that the first term in turn is calculated by a long exact sequence

(4.1.3) … H^{i−1}((Y_{a})_{red}, ℚ_{ℓ})→H^{i}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ})

→H^{i}(Y_{m}, ℚ_{ℓ}) ⊕ H^{i}((Yé)_{red}, ℚ_{ℓ}) ⊕ ⨁_{χ≠χ_{0}} H_{c}^{i}(Y o ; é, L[χ])→ ….

Here and in (4.1.2) we have replaced Yé and Y_{a} by the associated reduced schemes, since the étale cohomology is insensitive to nilpotents.

The diamond operators act on 𝔖_{K_{1}}(𝔮)\overline{η} as well as on Yé and Y_{m}, and thus induce compatible actions on the spaces in the exact sequences (4.1.2) and (4.1.3). These are determined as follows:

Lemma 4.1.4. The diamond operators < a > act on the outer terms of the exact sequence (4.1.3) as follows: The action is trivial on H^{•}((Yé)_{red}, ℚ_{ℓ}), H^{•}((Y_{a})_{red}, ℚ_{ℓ}), and on H^{i−1}((Y_{a})_{red}, RΨ^{1}ℚ_{ℓ}[χ_{0}]) and acts via χ on L[χ].

Proof. By Lemma 3.3.10 the diamond operators act trivially on (Yé)_{red} and (Y_{a})_{red}, so it suffices to determine their action on RΨ^{0}ℚ_{ℓ} and on RΨ^{1}ℚ_{ℓ}.

We first determine the action on RΨ^{0}ℚ_{ℓ}. By the calculation in (4.1.1.2), we see it suffices to determine the action of the diamond operators on H^{0}(Spec(\overline{ℚ}_{q} ⊗_{ℚ_{q}} ℚ_{q}[ζ_{q}], ℚ_{ℓ}), via the identification of ℚ_{q}[ζ_{q}] with the generic fiber of μ_{q} and the latter with C_{1}^{univ} in (4.1.1.1-2). But the diamond operators on μ_{q} are tautologically given by the cyclotomic character.

As for the action on RΨ^{1}ℚ_{ℓ}, this is again local. But locally the calculation in (4.1.1.3) shows that RΨ^{1}ℚ_{ℓ} is a constant sheaf, so the triviality of the action of the diamond operators is clear.

Corollary 4.1.5. Suppose χ ≠ χ_{0}, and denote by ^{<>=χ} the χ-isotypic component for the action of the diamond operators. Then for any i, there is a canonical isomorphism of Gal(\overline{η}/η)-modules

H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ} ⊕ H_{c}^{i}((Y o ; é)_{red}, L[χ]) ⟶^{∼} H^{i}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ})^{<>=χ}

Proof. Indeed, in (4.1.3), the diamond operators act trivially on the term H^{i−1}((Y_{a})_{red}, ℚ_{ℓ}) and coincide with inertia on L[χ], inducing an isomorphism

H^{i}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ})^{<>=χ} ⟶^{∼} H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ} ⊕ H_{c}^{i}(Y o ; é, L[χ]).

Similarly, the diamond operators act trivially on the H^{i−1}((Y_{a})_{red}, RΨ^{1}ℚ_{ℓ}[χ_{0}]) term in (4.1.3).

Over S̃_{𝔮} = Spec(O_{𝔮}[ζ_{q}]) the special fiber of 𝕊̃_{K_{1}} is a reduced divisor with normal crossings (Prop. 3.3.8). Let RΨ̃^{q}ℚ_{ℓ} denote the corresponding vanishing cycle sheaves. It follows that there is a canonical isomorphism

(4.1.6)  RΨ̃^{0}ℚ_{ℓ} ≃ ℚ_{ℓ}.

The following lemma is immediate from the above calculation.

Lemma 4.1.7. We identify (Y o ; é)_{red} ≃ Xo ; é via the morphism p_{1}. Then the morphism p̃_{1} : Ỹ_{e}^{o}→(Y o ; é)_{red} of Proposition 3.3.8 defines a canonical isomorphism

p̃_{1,∗}RΨ̃^{0}ℚ_{ℓ} ≃ p̃_{1,∗}ℚ_{ℓ} ≃ j_{e}^{∗}(RΨ^{0}ℚ_{ℓ}).

Moreover, this isomorphism is equivariant with respect to the diamond operators.

Proof. It is easy to see that the étale covering X′ of X used in the calculation of 4.1.1.2 is represented by Isom_{X}(μ_{q}, C_{1}). It follows from Prop. 3.3.8 (d) that the calculation globalizes to give the indicated isomorphism. The diamond operators act as Aut(μ_{q}) ≃ μ_{q−1} on X′, and the final assertion follows from the discussion in 4.1.1.2.

Corollary 4.1.8. Let χ ≠ χ_{0} be a non-trivial character of the diamond operators. Then the canonical maps

H_{c}^{i}((Y o ; é)_{red}, L[χ]) → H_{c}^{i}(Ỹ_{e}^{o}, ℚ_{ℓ})^{<>=χ} → H^{i}(Ỹé, ℚ_{ℓ})^{<>=χ}

are isomorphisms.

Proof. The first isomorphism is a consequence of the preceding lemma. The second isomorphism follows from the fact that the diamond operators act trivially on the complement Ỹ_{m} ∩ Ỹé of Ỹ_{e}^{o} in Ỹé.

4.2. The congruence formula and proof of Theorem 2.3.5.

As explained in §2.2, the double coset U_{𝔮} defines a correspondence U_{𝔮} on 𝔖_{K_{1}}(𝔮)\overline{η}×𝔖_{K_{1}}(𝔮)\overline{η}. Its modular interpretation in characteristic zero has been described in §2.2. An alternative description will be useful in defining its reduction (mod 𝔮). Consider the finite flat group scheme X(A)_{2}[q^{2}] over 𝔖_{K_{1,Q}}, and let m_{q} : X(A)_{2}[q^{2}]→X(A)_{2}[q] denote multiplication by q. Let P_{ζ} = γ^{∗}(ζ), ζ ∈ μ_{q}, as in the discussion in 2.2. The inverse image m_{q}^{−1}(P_{ζ}) ⊂ X(A)_{2}[q^{2}] defines an étale covering of 𝔖_{K_{1}}(𝔮)\overline{η}, whose image under ρ, ρ(m_{q}^{−1}(P_{ζ})) ⊂ X(A′)_{2}[q] is a well-defined étale covering of 𝔖_{K_{1}}(𝔮)\overline{η}. Over 𝔖_{K_{1}}(𝔮)\overline{η}×𝔖_{K_{1}}(𝔮)\overline{η}, U_{𝔮} is in one-to-one correspondence with the set of

(4.2.1)  {(A, ι, λ, β^{q}, P; A′, ι′, λ′, (β′)^{q}, P′)}

with notation as in (2.2.2). Here P = γ(1) ∈ X(A′)_{1}[q], P′ = γ′(1) ∈ X((A′)′)_{1}[q], where (A′)′ bears the same relation to A′ and Im((γ′)^{∗}) ∈ X(A′)_{2} that A′ bears to A and Im(γ^{∗}). Recall that P and P′ determine P_{ζ} = γ^{∗}(ζ) and P′_{ζ} = (γ′)^{∗}(ζ) as in (2.2.1.3), and (P_{ζ}, P′_{ζ}) ⊂ X(A)_{2}[q] × X(A′)_{2}[q] is defined by the closed relation

(4.2.2)  P′_{ζ} ∈ ρ(m_{q}^{−1}(P_{ζ})).

This subset is independent of the choice of ζ and defines a closed relation on the set of (P, P′) ∈ X(A′)_{1}[q] × X((A′)′)_{1}[q].

Let s : 𝔖_{K_{1,Q}}→X(A′)_{1}[q] denote the canonical section sending (A, ι, λ, β^{q}, P) to P. Then (4.2.1) is contained in s(𝔖_{K_{1}}(𝔮)\overline{η}) × s(𝔖_{K_{1}}(𝔮)\overline{η}). We use the same notation U_{𝔮} to denote the scheme-theoretic closure of (4.2.1) in s(𝕊_{K_{1}})×s(𝕊_{K_{1}}). Let p_{i} : U_{𝔮} → 𝕊_{K_{1}} i = 1, 2, denote projections on the ith factor.

Over the open subset Y o ; é, defined to be the complement of Y_{a} in Yé, the morphisms p_{1} and p_{2} are both étale. Moreover, it is straightforward to see that p_{2}(p_{1}^{−1}(Y o ; é)) ⊂ Y o ; é. In particular, the modular description (4.2.1) holds without change over Y o ; é. We let Ué denote the restriction of U_{𝔮} to Y o ; é×Y o ; é.

The situation is more interesting over Y_{m}^{o}. First, suppose (A, ι, λ, β^{q}, P) is a (T -valued) point of Y_{m}^{o}. Then the étale group C_{1} generated by P corresponds (by Cartier duality) to a subgroup C_{2,P} of X(A)_{2} isomorphic to μ_{q}. The isogeny F_{2} : X(A)_{2}→X(A′)_{2} = X(A)_{2}/C_{P} is just the geometric Frobenius map. Indeed, one verifies that A′ is nearly isomorphic to the transform A^{(q)} of A under Frobenius. In fact, this is true at the level of 𝔮-divisible groups: writing

X(A′) = [X(A′)_{1} ⊕ X(A′)_{2}]^{n},

we have canonical isomorphisms (4.2.3)

X(A′)_{2} ⟶^{∼} X(A^{(q)})_{2}; m_{q} · X(A′)_{1} ⟶^{∼} X(A)_{1}/r′(X(A′)_{1}[q]) ⟶^{∼} X(A^{(q)})_{1}

where the first isomorphism is given by F_{2}, the second is obtained by factorizing m_{q} : X(A′)_{1}→X(A′)_{1} via the morphism r′ : X(A′)_{1}→X(A)_{1} of 2.1.4, and the third is a consequence of our choice of F_{2}. Moreover, the point P′ of (4.2.1) satisfies the relation

P′ ∈ m_{q}.(r′)^{−1}(P)

derived from (4.2.2), as one verifies by looking at the multiplicative formal group. Here the isogeny r′ : X((A′)′)_{1}→X(A′)_{1} is again as in 2.1.4, but this time relative to A′! We note that, identifying m_{q} ·X(A′)_{1} with X(A′)_{1}, P′ can be identified with the image of P under the Frobenius map

X(A)_{1} × X(A)_{2} → X(A^{(q)})_{1} × X(A^{(q)})_{2}.

It follows that

(4.2.4)  (A, ι, λ, β^{q}, P) → {(A, ι, λ, β^{q}, P; A′, ι′, λ′, (β′)^{q}, P′}

is the graph of a morphism U_{m}^{+} : Y_{m}^{o}→Y_{m}^{o} that defines a section of p_{1} over Y_{m}^{o}. Here the quadruple (A′, ι′, λ′, (β′)^{q}) is defined by A and the isogeny F_{2} : X(A)_{2}→X(A′)_{2} as in 2.1.4.

Next, let U_{m}^{−} = p_{1}^{−1}(Y_{m}^{o}) ∩ p_{2}^{−1}(Y o ; é). Then we have

(4.2.5)  p_{1}^{−1}(Y_{m}^{o}) = U_{m}^{+} ⊔ U_{m}^{−}.

In particular, U_{m}^{+} (resp. U_{m}^{−}) is the graph of a correspondence on Y_{m}^{o} × Y_{m}^{o} (resp. Y_{m}^{o} × Y o ; é). We denote their closures in the special fiber of 𝕊_{K_{1}}×𝕊_{K_{1}} by the same notation.

The morphism U_{m}^{+} is not quite equal to geometric Frobenius on Y_{m}^{o}. Recall from 3.2.1(b) that the q-divisible O_{K,q}-module X(A) breaks up over the divisors of q in K. The components corresponding to divisors of q^{(1)} (resp. q^{(2)}) other than 𝔮^{(1)} (resp. 𝔮^{(2)}) are of multiplicative type (resp. étale). Thus geometric Frobenius on Y_{m}^{o} acts as multiplication by q on the q^{(1)}-factors of X(A) (other than 𝔮^{(1)}) and as the identity on the q^{(2)}-factors (other than 𝔮^{(2)}). Moreover, the maximal connected subgroup X(A)_{1}^{0} is purely of multiplicative type. Thus geometric Frobenius replaces X(A)_{1} by X(A)_{1}/X(A)_{1}^{0}[q]. But it follows easily from (2.1.4.2) that

X(A)_{1}/X(A)_{1}^{0}[q] = X(A′′)/X(A′′)[q].

We combine these observations as follows. Let [q^{−1}]_{1} denote the image of q^{−1} in GL(n, K_{0,q^{(1)}}), which we embed diagonally in GL(n, K_{q^{(1)}}). This is a central element of G(A_{f}), hence induces automorphisms of all Shimura varieties at finite level that extend trivially to 𝔖_{K_{1}}(𝔮)\overline{η}. Bearing in mind that inclusions of lattices induce maps of abelian varieties in the opposite direction, we conclude that Proposition 4.2.6 (Congruence formula). We have the identity

[q^{−1}]_{1} · U_{m}^{+} = F rob,

the geometric Frobenius map on the scheme Y_{m}^{o} over k(𝔮^{(1)}).

4.2.7. We need to explain how to relate the cohomological correspondence defined by U_{𝔮} in characteristic zero to the finite correspondences calculated above over Y o ; é and Y_{m}^{o}. This is somewhat delicate, since the base scheme 𝕊_{K_{1}} has a singular special fiber and we have not determined the structure of U_{𝔮} over the singular locus. Fortunately, we are only interested in the action on RΨ^{0}. It is convenient to write

(4.2.7.1)  RΨ^{0}ℚ_{ℓ} = ℚ_{ℓ} ⊕ ⨁_{χ≠χ_{0}} j_{m,!}L[χ].

We consider the first map of (4.1.2):

b_{i} : H^{i}(𝕊_{K_{1},\overline{κ}}, RΨ^{0}ℚ_{ℓ}) → H^{i}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ}).

We need to find a cohomological correspondence U_{𝔮}^{0} on RΨ^{0}ℚ_{ℓ} such that

(4.2.7.2)  b_{i} ∘ U_{𝔮}^{0} = U_{𝔮} ∘ b_{i}.

We define U_{𝔮}^{0} on the separate summands on the right-hand side of (4.2.7.1). On the summand ℚ_{ℓ} one just takes the natural cohomological correspondence (pullback followed by push-forward) defined by the proper correspondence U_{𝔮}. Recall from Corollary 4.1.5 that, if χ ≠ χ_{0}, then

(4.2.7.3)  H^{i}(𝕊_{K_{1},\overline{κ}}, ℚ_{ℓ})^{<>=χ} = H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ}.

Now U_{𝔮} commutes with the diamond operators, hence fixes the χ-eigenspace. We have seen that U_{𝔮} restricts to a sum of two correspondences U_{m}^{+} + U_{m}^{−} on Y_{m}, and the restriction of U_{𝔮}^{0} to H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ} is evidently the sum of the induced cohomological correspondences, say U_{m}^{+}[χ] and U_{m}^{−}[χ].

It remains to define U_{𝔮}^{0} on the L[χ]. In fact, it is enough to define U_{𝔮}^{0} on H_{c}^{i}((Y o ; é)_{red}, L[χ]). Extending scalars to S̃_{𝔮}, we can use the model 𝕊̃_{K_{1}}, and define a correspondence Ũ_{𝔮} on 𝕊̃_{K_{1}} × 𝕊̃_{K_{1}} as above. We have seen that RΨ̃^{0}ℚ_{ℓ} = ℚ_{ℓ} (4.1.6), hence we can define Ũ_{𝔮}^{0} again as the natural cohomological correspondence on RΨ̃^{0}ℚ_{ℓ}. This commutes with the action of the diamond operators, hence for each χ defines an operator on H^{i}(𝕊̃_{K_{1},\overline{κ}}, ℚ_{ℓ})^{<>=χ}. Since the diamond operators act trivially on Ỹé ∩ Ỹ_{m}, it follows as before that, for χ ≠ χ_{0}, this breaks up as

H^{i}(Ỹ_{m}, ℚ_{ℓ})^{<>=χ} ⊕ H^{i}(Ỹé, ℚ_{ℓ})^{<>=χ} = H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ} ⊕ H_{c}^{i}((Y o ; é)_{red}, L[χ]),

by Proposition 3.3.8 (b) and Corollary 4.1.8. It is clear that Ũ_{𝔮}^{0} induces U_{m}^{+}[χ] ⊕ U_{m}^{−}[χ] on the first factor. On the second factor it suffices to know that Ũ_{𝔮} fixes Ỹé. By proper base change, Ũ_{𝔮}^{0} satisfies (4.2.7.2). Hence Lemma 4.2.8. Fix an integer i. Let χ ≠ χ_{0} be a non-trivial character of the diamond operators. The correspondence U_{𝔮} acts on

H^{i}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ})^{<>=χ} ≅ H^{i}(Y_{m}, ℚ_{ℓ})^{<>=χ} ⊕ H_{c}^{i}(Y o ; é, L[χ])

via the matrix

( \begin{matrix}[q^{−1}]_{1}^{−1} · F rob & 0 \\ 0 & D(χ) \end{matrix} )

where [q^{−1}]_{1} is the automorphism defined above, F rob is geometric Frobenius, lifted to K_{𝔮}[ζ_{q}], and D(χ) ∈ Aut(H^{i}(Ỹé, ℚ_{ℓ})^{<>=χ}).

Proof. Everything is clear except the 0 in the upper right-hand corner. But it follows from Lemma 4.1.4 and Proposition 4.1.1 that the inertia group I acts trivially on H^{i}(Y_{m}, ℚ_{ℓ}) and coincides with the diamond operators on H^{i}(Y o ; é, L[χ]). The correspondence U_{𝔮} is defined over S_{𝔮}, hence commutes with inertia. The Proposition follows.

4.2.9. Proof of Theorem 2.3.5. We can now complete the proof of Theorem 2.3.5. We are given an automorphic representation π that contributes to H^{n−1}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ}) but has no K_{0}(𝔮)-fixed vector; i.e., we are in case (b) of Proposition 1.4. Let {α, β_{1}, …, β_{n−1}} be the characters defined there, so that α is tamely ramified and the β_{i} are unramified. Let χ denote the restriction of α to the inertia subgroup; thus χ coincides with the action of K_{0}(𝔮)/K_{1}(𝔮) on the K_{1}(𝔮)-fixed vectors in π. In particular, and bearing in mind the discussion in 2.1.4.6, π contributes to H^{n−1}(𝔖_{K_{1}}(𝔮)\overline{η}, ℚ_{ℓ})^{<>=χ^{−1}}. Define M_{λ}[π_{f}] as in §2.3. The two components Y_{m} and Yé are invariant under prime-to-q isogenies, hence M_{λ}[π_{f}] breaks up as above as the

sum of A^{#} = H^{n−1}(Y_{m}, ℚ_{ℓ})^{<>=χ^{−1}}[π_{f}] and B^{#} = H_{c}^{n−1}(Y o ; é, L[χ^{−1}])[π_{f}]. Inertia

at 𝔮 acts on the first factor trivially and on the second factor by the character χ^{−1}. Then

r_{ρ}(π) ⟶^{∼} A^{#} ⊗ α(π) ⊕ B^{#} ⊗ α(π) = A ⊕ B.

But

α(π) = ξ_{π} ∘ N_{K/K_{0}} = α · ∏_{i} β_{i}.

On the one hand

α(π)(ϖ) = ξ_{π}(N_{K/K_{0}}(ϖ)) = ξ_{π}([q^{−1}]_{1})^{−1}

On the other hand, the restriction to inertia of ξ_{π} is the same as that of α, namely χ. It follows from Lemma 4.2.8 that 4.2.9.1. Geometric Frobenius coincides with U_{𝔮} on A. Moreover, inertia acts as χ on A. 4.2.9.2. The representation at 𝔮^{(1)} on B is unramified.

These two assertions complete the proof of Theorem 2.3.5. 4.2.10. In particular, inertia at 𝔮^{(1)} acts on the determinant of r_{ρ}(π) by χ^{dim A}. But the determinant of r_{ρ}(π) is given by ξ_{π} at all unramified places, hence at 𝔮^{(1)} by Chebotarev density. It follows that

(4.2.10.1)  χ^{dim A−1} = χ_{0}.

REGULAR MODELS OF CERTAIN SHIMURA VARIETIES 35

Now suppose

(4.2.10.2)  q ≡ 1 (mod ℓ), ℓ > n, χ|_{Δ_{q}} ≠ 1,

where Δ_{q} denotes the ℓ-Sylow subgroup of 𝔽_{q}^{×}. It follows from (4.2.10.1) that ℓ|(dim A − 1). Under hypotheses (4.2.10.2), this is only possible if dim A = 1. Thus Corollary 4.2.10.3. Under the hypotheses (4.2.10.2), the spaces A and B in the statement of Theorem 2.3.5 are of dimension 1 and n − 1, respectively.

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