On the Rigid Cohomology of Certain Shimura Varieties

We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic representation away from $l$ and finite number of rational primes above which the CM field or the automorphic representation ramify. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.


Introduction
Our main theorem is as follows (see corollary 7.14).
Theorem A. Let p denote a rational prime and let ı : Q p ∼ → C. Suppose that E is a CM (or totally real) field and that π is a cuspidal automorphic representation of GL n (A E ) such that π ∞ has the same infinitesimal character as an irreducible algebraic representation ρ π of RS E Q GL n . Then there is a unique continuous semisimple representation r p,ı (π) : G E −→ GL n (Q p ) such that, if q = p is a prime above which π and E are unramified and if v|q is a prime of E, then r p,ı (π) is unramified at v and r p,ı (π)| ss W Ev = ı −1 rec Ev (π v | det | (1−n)/2 v ).
Here rec Ev denotes the local Langlands correspondence for E v . It may be possible to extend the local-global compatibility to other primes v. Ila Varma is considering this question.
The key point is that we make no self-duality assumption on π. In the presence of such a self-duality assumption ('polarizability', see [BLGGT]) the existence of r p,ı (π) has been known for some years (see [Sh1] and [CH]). In almost all polarizable cases r p,ı (π) is realized in the cohomology of a Shimura variety, and in Date: November 26, 2014. We would all like to thank the Institute for Advanced Study for its support and hospitality. This project was begun, and the key steps completed, while we were all attending the special IAS special year on 'Galois representations and automorphic forms'. M all polarizable cases r p,ı (π) ⊗2 is realized in the cohomology of a Shimura variety (see [Ca]). In contrast, according to unpublished computations of one of us (M.H.) and of Laurent Clozel, in the non-polarizable case the representation r p,ı (π) will never occur in the cohomology of a Shimura variety. Rather we construct it as a p-adic limit of representations which do occur in the cohomology of Shimura varieties.
We sketch our argument. We may easily reduce to the case of an imaginary CM field F which contains an imaginary quadratic field in which p splits. For all sufficiently large integers N, we construct a 2n-dimensional representation R p (ı −1 (π|| det || N ) ∞ ) such that for good primes v we have , as a p-adic limit of (presumably irreducible) p-adic representations associated to polarizable, regular algebraic cuspidal automorphic representations of GL 2n (A F ). It is then elementary algebra to reconstruct r p,ı (π).
We work on the quasi-split unitary similitude group G n associated to F 2n . Note that G n has a maximal parabolic subgroup P + n,(n) with Levi component L n,(n) ∼ = GL 1 × RS F Q GL n . (We will give all these groups integral structures.) We set Π(N) = Ind Gn(A p,∞ ) P + n,(n) (A p,∞ ) (1 × ı −1 (π|| det || N ) p,∞ ). Then our strategy is to realize Π(N), for sufficiently large N, in a space of overconvergent p-adic cusp forms for G n of finite slope. It is a space of forms of a weight for which we expect no classical forms. Once we have done this, we can use an argument of Katz (see [Ka1]) to find congruences modulo arbitrarily high powers of p to classical (holomorphic) cusp forms on G n (of other weights). (Alternatively it is presumably possible to construct an eigenvariety in this setting, but we have not carried this out.) One can attach Galois representations to these classical cusp forms by using the trace formula to lift them to polarizable, regular algebraic, discrete automorphic representations of GL 2n (A F ) (see e.g. [Sh2]) and then applying the results of [Sh1] and [CH].
We learnt the idea that one might try to realize Π(N) in a space of overconvergent p-adic cusp forms for G n (of finite slope) from Chris Skinner. The key problem was how to achieve such a realization. To sketch our approach we must first establish some more notation.
To a neat open compact subgroup U of G n we can associate a Shimura variety X n,U /Spec Q. It is a moduli space for abelian n[F : Q]-folds with an isogeny action of F and certain additional structures. It is not proper. It has a canonical normal compactification X min n,U and, to certain auxiliary data ∆, one can attach a smooth compactification X n,U,∆ which naturally lies over X min n,U and whose boundary is a simple normal crossings divisor. To a representation ρ of L n,(n) (over Q) one can attach a locally free sheaf E U,ρ /X n,U together with a canonical (locally free) extension E U,∆,ρ to X n,U,∆ , whose global sections are holomorphic automorphic forms on G n 'of weight ρ and level U'. (The space of global sections does not depend on ∆.) The product of E U,∆,ρ with the ideal sheaf of the boundary of X n,U,∆ , which we denote E sub U,∆,ρ , is again locally free and its global sections are holomorphic cusp forms on G n 'of weight ρ and level U' (and again the space of global sections does not depend on ∆).
To the schemes X n,U , X min n,U and X n,U,∆ one can naturally attach dagger spaces X † n,U , X min, † n,U and X † n,U,∆ in the sense of [GK]. These are like rigid analytic spaces except that one consistently works with overconvergent sections. If U is the product of a neat open compact subgroup of G n (A ∞,p ) and a suitable open compact subgroup of G n (Q p ), then one can define admissible open sub-dagger spaces ('the ordinary loci') X ord, † n,U ⊂ X † n,U and X min,ord, † n,U ⊂ X min, † n,U and X ord, † n,U,∆ ⊂ X † n,U,∆ . By an overconvergent cusp form of weight ρ and level U one means a section of E sub U,ρ over X ord, † n,U,∆ . (Again this definition does not depend on the choice of ∆.) We write G (m) n for the semi-direct product of G n with the additive group with Q-points Hom F (F m , F 2n ), and P (m),+ n,(n) for the pre-image of P + n,(n) in G (m) n . We also write L (m) n,(n) for the semi-direct product of L n,(n) with the additive group with Q-points Hom F (F m , F n ), which is naturally a quotient of P (m),+ n,(n) . (Again we will give these groups integral structures.) To a neat open compact subgroup U ⊂ G (m) n (A ∞ ) with projection U ′ in G n (A ∞ ) one can attach a (relatively smooth, projective) Kuga-Sato variety A (m) n,U /X n,U ′ . For a cofinal set of U it is an abelian scheme isogenous to the m-fold self product of the universal abelian variety over X n,U ′ . To certain auxiliary data Σ one can attach a smooth compactification A (m) n,U,Σ of A (m) n,U whose boundary is a simple normal crossings divisor; which lies over X min n,U ; and which, for suitable Σ depending on ∆, lies over X n,U ′ ,∆ . Thus A n,U ֒→ A n,U,Σ ↓ ↓ X n,U ′ ֒→ X n,U ′ ,∆ || ↓ X n,U ′ ֒→ X min n,U ′ .
We define A More specifically cohomology with compact support towards the toroidal boundary, but not towards the non-ordinary locus. Hence our notation. However we have not bothered to verify that this group only depends on ordinary locus in the special fibre. The theory of Shimura varieties provides us with sufficiently canonical lifts that this will not matter to us. Our proof that for N sufficiently large Π(N) occurs in the space of overconvergent p-adic cusp forms for G n proceeds by evaluating H i c−∂ (A (m),ord n,U , Q p ) in two ways. Firstly we use the usual Hodge spectral sequence. The higher direct images from A (m) n,U,Σ to X n,U ′ ,∆ of the tensor product of the ideal sheaf of the boundary and the sheaf of differentials of any degree with log poles along the boundary, is canonically filtered with graded pieces sheaves of the form E sub U ′ ,∆,ρ . Thus H i c−∂ (A (m),ord n,U , Q p ) can be computed in terms of the groups H j (X ord, † n,U,∆ , E sub U,∆,ρ ) A crucial observation for us is that for j > 0 this group vanishes (see theorem 5.4 and proposition 6.12). This observation seems to have been made independently, at about the same time, by Andreatta, Iovita and Pilloni (see [AIP1] and [AIP2]). It seems quite surprising to us. It is false if one replaces E sub U,∆,ρ with E can U,∆,ρ . Its proof depends on a number of apparently unrelated facts, including: • X min,ord, † n,U is affinoid. • The stabilizer in GL n (O F ) of a positive definite hermitian n × n matrix over F is finite. • Certain line bundles on self products A of the universal abelian variety over X n ′ ,U ′ (for n ′ < n) are relatively ample for A/X n ′ ,U ′ . This observation implies that H i c−∂ (A (m),ord n,U , Q p ) can be computed by a complex whose terms are spaces of overconvergent cusp forms. Hence it suffices for us to show that, for N sufficiently large, Π(N) occurs in  n,(n) (A)/U ′ (R × >0 × (U(n) [F + :Q] R × >0 )), with U(n) denoting the usual n × n compact unitary group. We deduce that We will write simply T (n),U ′ for T (n),U ′ , a locally symmetric space associated to L n,(n) ∼ = GL 1 × RS F Q GL n . If ρ is a representation of L n,(n) over C, then it gives rise to a locally constant sheaf L ρ,U ′ over T (n),U ′ . We set a smooth L n,(n) (A ∞ )-module. The space T (m) (n),U ′ is an (S 1 ) nm[F :Q] -bundle over the locally symmetric space T (0) (n),U ′ and if π (m) denotes the fibre map then R j π (m) * C ∼ = L ∧ j Hom F (F m ,F n ) ∨ ⊗ Q C,U ′ , where L n,(n) acts on Hom F (F m , F n ) via projection to RS F Q GL n . Moreover the Leray spectral sequence (n) , C) degenerates at the second page. (This can be seen by considering the action of the centre of L n,(n) (A ∞ ).) Thus it suffices to show that for all sufficiently large N, we can find non-negative integers j and m and an irreducible constituent ρ of ∧ j Hom F (F m , F n ) ∨ ⊗ Q C such that the representation 1 × (π|| det || N ) p,∞ occurs in H i Int (T (n) , L ρ ) for some i ∈ Z >0 . Clozel [Cl] checked that (for n > 1) this will be the case as long as 1 × (π|| det || N ) ∞ has the same infinitesimal character as some irreducible constituent of ∧ j Hom F (F m , F n ) ⊗ Q C, i.e. if ρ π ⊗ (N F/Q • det) N occurs in ∧ j Hom F (F m , F n ) ⊗ Q C. From Weyl's construction of the irreducible representations of GL n , for large enough N this will indeed be the case for some m and j.
We remark it is essential to work with N sufficiently large. It is not an artifact of the fact that we are working with Kuga-Sato varieties rather than local systems on the Shimura variety. We can twist a local system on the Shimura variety by a power of the multiplier character of G n . However the restriction of the multiplier factor of G n to L n,(n) ∼ = GL 1 × RS F Q GL n factors through the GL 1 -factor and does not involve the RS F Q GL n factor.
We learnt from the series of papers [HZ1], [HZ2], [HZ3], the key observation that |S(∂A (m) n,U,Σ )| has a nice geometric interpretation involving the locally symmetric space for L n, (n) and that this could be used to calculate cohomology.
Although the central argument we have sketched above is not long, this paper has unfortunately become very long. If we had only wanted to construct r p,ı (π) for all but finitely many primes p, then the argument would have been significantly shorter as we could have worked only with Shimura varieties X n,U which have good integral models at p. The fact that we want to construct r p,ı (π) for all p adds considerable technical complications and also requires appeal to the recent work [La4]. (Otherwise we would only need to appeal to [La1] and [La2].) Another reason this paper has grown in length is the desire to use a language to describe toroidal compactifications of mixed Shimura varieties that is different from the language used in [La1], [La2] and [La4]. We do this because at least one of us (R.T.) finds this language clearer. In any case it would be necessary to establish a substantial amount of notation regarding toroidal compactifications of Shimura varieties, which would require significant space. We hope that the length of the paper, and the technicalities with which we have to deal, won't obscure the main line of the argument.
After we announced these results, but while we were writing up this paper, Scholze found another proof of theorem A, relying on his theory of perfectoid spaces. His arguments seem to be in many ways more robust. For instance he can handle torsion in the cohomology of the locally symmetric varieties associated to GL n over a CM field. Scholze's methods have some similarities with ours. Both methods first realize the Hecke eigenvalues of interest in the cohomology with compact support of the open Shimura variety by an analysis of the boundary and then show that they also occur in some space of p-adic cusp forms. We work with the ordinary locus of the Shimura variety, which for the minimal compactification is affinoid. Scholze works with the whole Shimura variety, but at infinite level. He (very surprisingly) shows that at infinite level, as a perfectoid space, the Shimura variety has a Hecke invariant affinoid cover.
Notation. If G → → H is a surjective group homomorphism and if U is a subgroup of G we will sometimes use U to also denote the image of U in H.
If f : X → Y and f ′ : Y → Z then we will denote by f ′ • f : X → Z the composite map f followed by f ′ . In this paper we will use both left and right actions. Suppose that G is a group acting on a set X and that g, h ∈ G. If G acts on X on the left we will write gh for g • h. If G acts on X on the right we will write hg for g • h.
If G is a group (or group scheme) then Z(G) will denote its centre. We will write S n for the symmetric group on n letters. We will write U(n) for the group of n × n complex matrices h with t h c h = 1 n .
If G is an abelian group we will write G[∞] for the torsion subgroup of G, G[∞ p ] for the subgroup of elements of order prime to p, and G TF = G/G[∞]. We will write T G = lim ←N G[N] and T p G = lim ←p |N G [N]. We will also write V G = T G ⊗ Z Q and V p G = T p G ⊗ Z Q.
If A is a ring, if B is a locally free, finite A-algebra, and if X/Spec B is a quasi-projective scheme; then we will let RS B A X denote the restriction of scalars (or Weil restriction) of X from B to A. (See for instance section 7.6 of [BLR].) By a p-adic formal scheme we mean a formal scheme such that p generates an ideal of definition.
If X is an A-module and B is an A-algebra, we will sometimes write X B for X ⊗ A B. We will also use X to denote the abelian group scheme over A defined by for all A-algebras B.
If Y is a scheme and if G 1 , G 2 /Y are group schemes then we will let Hom(G 1 , G 2 ) denote the Zariski sheaf on Y whose sections over an open W are Hom (G 1 | W , G 2 | W ).
If in addition R is a ring then we will let Hom(G 1 , G 2 ) R denote the tensor product Hom(G 1 , G 2 ) ⊗ Z R and we will let Hom (G 1 , G 2 ) R denote the R-module of global sections of Hom(G 1 , G 2 ) R . If Y is noetherian this is the same as Hom (G 1 , G 2 ) ⊗ Z R, but for a general base Y it may differ. If S is a simplicial complex we will write |S| for the corresponding topological space. To a scheme or formal scheme Z, we will associate to it a simplicial complex S(Z) as follows: Let {Z j } j∈J denote the set of irreducible components of Z. Then J will be the set of vertices of S(Z) and a subset K ⊂ J will span a simplex if and only if j∈K Z j = ∅.
If F is a field then G F will denote its absolute Galois group. If F is a number field and F 0 ⊂ F is a subfield and S is a finite set of primes of F 0 , then we will denote by G S F the maximal continuous quotient of G F in which all primes of F not lying above an element of S are unramified.
Suppose that F is a number field and that v is a place of F . If v is finite we will write ̟ v for a uniformizer in F v and k(v) for the residue field of v. We will write | | v for the absolute value on F associated to v and normalized as follows: We will write D −1 F for the inverse different of O F . If w ∈ Z >0 and p is a prime number then by a Weil p w -number we mean an element α ∈ Q which is an integer away from p and such that for each infinite place v of Q we have |α| v = p w .
Suppose that v is finite and that is a continuous representation, which in the case v|l we assume to be de Rham. Then we will write WD(r) for the corresponding Weil-Deligne representation of of the Weil group W Fv of F v (see for instance section 1 of [TY].) If π is an irreducible smooth representation of GL n (F v ) over C we will write rec Fv (π) for the Weil-Deligne representation of W Fv corresponding to π by the local Langlands conjecture (see for instance the introduction to [HT]). If π i is an irreducible smooth representation of GL n i (F v ) over C for i = 1, 2 then there is an irreducible smooth representation π 1 ⊞ π 2 of GL n 1 +n 2 (F v ) over C satisfying rec Fv (π 1 ⊞ π 2 ) = rec Fv (π 1 ) ⊕ rec Fv (π 2 ).
Suppose that G is a reductive group over F v and that P is a parabolic subgroup of G with unipotent radical N and Levi component L. Suppose also that π is a smooth representation of L(F v ) on a vector space W π over a field Ω of characteristic 0. We will define Ind G(Fv) P (Fv) π to be the representation of G(F v ) by right translation on the set of locally constant functions ϕ : G(F v ) −→ W π such that ϕ(hg) = π(h)ϕ(g) for all h ∈ P (F v ) and g ∈ G(F v ). In the case Ω = C we also define n-Ind G(Fv) P (Fv) π = Ind G(Fv) P (Fv) π ⊗ δ 1/2 P where δ P (h) 1/2 = | det(ad (h)| Lie N )| 1/2 v . If G is a linear algebraic group over F then the concept of a neat open compact subgroup of G(A ∞ F ) is defined for instance in section 0.6 of [Pi].
1. Some algebraic groups and automorphic forms.
For the rest of this paper fix the following notation. Let F + be a totally real field and F 0 an imaginary quadratic field, and set F = F 0 F + . Write c for the non-trivial element of Gal (F/F + ). Also choose a rational prime p which splits in F 0 and choose an element δ F ∈ O F,(p) with tr F/F + δ F = 1 (which is possible as p is unramified in F/F + ).
Fix an isomorphism ı : Q p ∼ → C. Fix a choice of √ p ∈ Q p by ı √ p > 0. If v is a prime of F and π an irreducible admissible representation of GL m (F v ) over Q l define rec Fv (π) = ı −1 rec Fv (ıπ) a Weil-Deligne representation of W Fv over Q l .
Let n be a non-negative integer. We will often attach n as a subscript to other notation, when we need to record the particular choice of n we are working with, but, at other times when the choice of n is clear, we may drop it from the notation.
1.1. Three algebraic groups. Write Ψ n for the n × n-matrix with 1's on the anti-diagonal and 0's elsewhere, and set J n = 0 Ψ n −Ψ n 0 ∈ GL 2n (Z).
Let Λ n = (D −1 F ) n ⊕ O n F , and define a perfect pairing , n : Λ n × Λ n −→ Z by x, y n = tr F/Q t xJ n c y.
We will write V n for Λ n ⊗ Q. Let G n denote denote the group scheme over Z defined by G n (R) = {(g, µ) ∈ Aut ((Λ n ⊗ Z R)/(O F ⊗ Z R)) × R × : t gJ n c g = µJ n }, for any ring R, and let ν : G n → GL 1 denote the multiplier character which sends (g, µ) to µ. Then G n is a quasi-split connected reductive group scheme over Z[1/D F/Q ] (where D F/Q denotes the discriminant of F/Q) and splits over O F nc [1/D F/Q ] (where F nc denotes the normal closure of F/Q). In particular G 0 will denote GL 1 and ν : G 0 → GL 1 is the identity map. If n > 0 set Then there is a natural map G n −→ C n (g, µ) −→ (µ, µ −n det g).
If n = 0 we set C 0 = G m and let G 0 −→ C 0 denote the map ν. In either case this map identifies C n with G n /[G n , G n ].
We will write Λ n,(i) for the submodule of Λ n consisting of elements whose last 2n − i entries are 0, and V n,(i) for Λ n,(i) ⊗ Q. If W is a submodule of Λ n we will write W ⊥ for its orthogonal complement with respect to , n . Thus Λ ⊥ n,(i) is the submodule of Λ n consisting of vectors whose last i entries are 0. Also write Λ (m) n = Hom (O m F , Z) ⊕ Λ n , and set V  n . Suppose that R is a ring and that X is an O F ⊗ Z R-module. We will write Herm X for the R-module of R-bilinear pairings ( , ) : X × X −→ R which satisfy (1) (ax, y) = (x, c ay) for all a ∈ O F and x, y ∈ X; (2) (x, y) = (y, x) for all x, y ∈ X. If z ∈ Herm X we will sometimes denote the corresponding pairing ( , ) z . If S is an R-algebra we have a natural map There is a natural isomorphism under which an element (z, f, w) of the right hand side corresponds to ((x, y), (x ′ , y ′ )) (z,f,w) Set N (m) n (Z) to be the set of pairs for all x, y ∈ O m F . We define a group scheme N (m) n /Spec Z by setting N (m) n (R) to be the set of pairs (f, z) ∈ N (m) n (Z) ⊗ Z R with group law given by where by f , f ′ n − f ′ , f n we mean the hermitian form (x, y) −→ f (x), f ′ (y) n − f ′ (x), f (y) n .
Note that (f, z) −1 = (−f, −z). Thus there is an exact sequence If m 1 ≥ m 2 then we get a natural map G (m 1 ) n → G (m 2 ) n . Note that G (m) n ∼ = G (m) n /Herm (m) . Let B n denote the subgroup of G n consisting of elements which preserve the chain Λ n,(n) ⊃ Λ n,(n−1) ⊃ ... ⊃ Λ n,(1) ⊃ Λ n,(0) and let N n denote the normal subgroup of B n consisting of elements with ν = 1, which also act trivially on Λ n,(i) /Λ n,(i−1) for all i = 1, ..., n. Let T n denote the group consisting of the diagonal elements of G n and let A n denote the image of G m in G n via the embedding that sends t onto t1 2n . Over Q we see that T n is a maximal torus in a Borel subgroup B n of G n , and that N n is the unipotent radical of B n . Moreover A n is a maximal split torus in the centre of G n .
Also let Φ n ⊂ X * (T n,/Ω ) denote the set of roots of T n on Lie G n ; let Φ + n ⊂ Φ n denote the set of positive roots with respect to B n and let ∆ n ⊂ Φ + n denote the set of simple positive roots. We will write ̺ + n for half the sum of the elements of Φ + n . If R ⊂ R is a subring then X * (T n,/Ω ) + R will denote the subset of X * (T n,/Ω ) R consisting of elements which pair non-negatively with the corootα ∈ X * (T n,/Ω ) corresponding to each α ∈ ∆ n . We will write simply X * (T n,/Ω ) + for X * (T n,/Ω ) + Z . If λ ∈ X * (T n,/Ω ) + we will let ρ n,λ (or simply ρ λ ) denote the irreducible representation of G n with highest weight λ. When ρ λ is used as a subscript we will sometimes replace it by just λ.
There is a natural identification This gives rise to the further identification We will use this to identify X * (T n,/Ω ) with a quotient of Ω) . Under this identification X * (T n,/Ω ) + is identified to the image of the set of (a 0 , (a τ,i )) ∈ Z ⊕ (Z 2n ) Hom (F,Ω) with a τ,1 ≥ a τ,2 ≥ ... ≥ a τ,2n for all τ . If R is a subring of R and H an algebraic subgroup of G (m) n we will write H(R) + for the subgroup of H(R) consisting of elements with positive multiplier. Thus (This map depends on identifications F ⊗ F + ,τ R ∼ = C, but the image of the map does not, and this image is all that will concern us.) Then U n,∞ is a maximal compact subgroup of G n (R) (and even of G We will write p n for the set of elements of Lie G n (R) of the form We give the real vector space p n a complex structure by letting i act by We decompose p n ⊗ R C = p + n ⊕ p − n by setting p ± n = (p n ⊗ R C) i 0 ⊗1=±1⊗i . We also set q n = p − n ⊕ Lie (U n,∞ A n (R)) ⊗ R C. It is a parabolic sub-algebra of (Lie G n (R)) ⊗ R C with unipotent radical p − n and Levi component Lie (U n,∞ A n (R)) ⊗ R C. We will write Q n for the parabolic subgroup of G n × Q C with Lie algebra q n . Note that Q n (C) ∩ G n (R) = U 0 n,∞ A n (R). Let H + n (resp. H ± n ) denote the set of I in G n (R) with multiplier 1 such that I 2 = −1 2n and such that the symmetric bilinear form I , n on Λ n ⊗ Z R is positive definite (respectively positive or negative definite). Then G n (R) (resp. G n (R) + ) acts transitively on H ± n (resp. H + n ) by conjugation. Moreover J n ∈ H + n has stabilizer U 0 n,∞ A n (R) 0 and so we get an identification of H ± n (resp. H + n ) with is an open embedding and gives H ± n the structure of a complex manifold. The action of G n (R) is holomorphic and the complex structure induced on the tangent space T J H ± n ∼ = p n is the complex structure described in the previous paragraph. If ρ is an algebraic representation of Q n on a C-vector space W ρ , then there is a holomorphic vector bundle E ρ /H ± n together with a holomorphic action of G n (R), defined as the pull back to H ± of (G n (C) × W ρ )/Q n (C), where • h ∈ Q n (C) sends (g, w) to (gh, h −1 w), • and where h ∈ G n (R) sends [(g, w)] to [(hg, w)]. If N 2 ≥ N 1 ≥ 0 are integers we will write U p (N 1 , N 2 ) n for the subgroup of G n (Z p ) consisting of elements whose reduction modulo p N 2 preserves and acts trivially on Λ n / (Λ n,(n) ). We will also set 1.2. Maximal parabolic subgroups. We will write P + n,(i) (resp. P (m),+ n,(i) , resp. P (m),+ n,(i) ) for the subgroup of G n (resp. G (m) n , resp. G (m) n ) consisting of elements which (after projection to G n ) take Λ n,(i) to itself. We will also write N + n,(i) (resp. N (m),+ n,(i) , resp. N (m),+ n,(i) ) for the subgroups of P + n,(i) (resp. P (m),+ n,(i) , resp. P (m),+ n,(i) ) consisting of elements which act trivially on Λ n,(i) and Λ ⊥ n,(i) /Λ n,(i) and Λ n /Λ ⊥ n,(i) . Over Q the groups P + n,(i) (resp. P (m),+ n,(i) , resp. P (m),+ n,(i) ) are maximal parabolic subgroups of G n (resp. G (m) n , resp. G (m) n ) containing the pre-image of B n . The groups N + n,(i) (resp. N (m),+ n,(i) , resp. N (m),+ n,(i) ) are their unipotent radicals. In some instances it will be useful to replace these groups by their 'Hermitian part'. We will write P n,(i) for the normal subgroup of P + n,(i) consisting of elements which act trivially on Λ n /Λ ⊥ n,(i) . We will also write P (m) n,(i) for the normal subgroup n,(i) . We will let N n,(i) = N + n, (i) and N n,(i) . Over Q these are the unipotent radicals of P n,(i) (resp. P (m) n,(i) , resp. P (m) n,(i) ). We have an isomorphism P n,(i) ∼ = G (i) n−i . To describe it let Λ ′ n,(i) denote the subspace of Λ n consisting of vectors with their first 2n − i entries 0, so that Λ ′ n,(i) by letting g ∈ G n−i act as ν(g) on Λ n,(i) , as g on Λ n−i ∼ = Λ ⊥ n,(i) ∩ (Λ ′ n,(i) ) ⊥ and as 1 on Λ ′ n,(i) , i.e.
We define N n,(i) −→ Hom n−i . We will describe the second of these isomorphisms. Suppose f ∈ Hom (i) n−i and g ∈ Hom (m) n−i . Also suppose that z ∈ 1 2 Herm (i) and w ∈ 1 2 Herm (m) and u ∈ Let h(f, z) denote the element of P n,(i) corresponding to (f, z) ∈ N n,(i) . Think of g as a map n,(i) ) ∼ = Herm i+m and that Z(N (m) n,(i) ) ∼ = Herm i+m /Herm m . Write L n,(i),lin for the subgroup of P + n,(i) consisting of elements with ν = 1 which preserve Λ ′ n,(i) ⊂ Λ n and act trivially on Λ ⊥ n,(i) /Λ n,(i) . We set N(L (m) n,(i),lin ) to be the additive group scheme over Z associated to n,(i) . Note that P + n,(i) = L n,(i),lin ⋉ P n,(i) and P . We let L n,(i),herm denote the subgroup of P n,(i) consisting of elements which preserve Λ ′ n,(i) . Thus L n,(i),herm ∼ = G n−i . In particular ν : L n,(n),herm Over Q it is a Levi component for P n,(i) and P n,(i),lin . Over Q we see that L n,(i) is a Levi component for each of P + n,(i) and P n,(i) . We will occasionally write P (m),− n,(i) (resp. L − n,(i),herm ) for the kernel of the map P (m) n,(i) → C n−i (resp. L n,(i),herm → C n−i ). We will write R n,(n),(i) for the subgroup of L n,(n) mapping Λ ′ n,(i) to itself. We will write N(R n,(n),(i) ) for the subgroup of R n,(n),(i) which acts trivially on Λ ′ n, (i) and (Λ ′ n,(i) ) ⊥ /Λ ′ n,(i) and Λ n /(Λ ′ n,(i) ) ⊥ . We will also write R (m) n,(n) for the semi-direct product If m ′ ≤ m we will fix Z m → → Z m ′ to be projection onto the last m ′ -coordinates and define Q m,(m ′ ) for the subgroup of GL m consisting of elements preserving the kernel of this map. We also define Q ′ m,(m ′ ) to be the subgroup of Q m,(m ′ ) consisting of elements which induce 1 Z m ′ on Z m ′ . Thus there is an exact sequence (m) . We will also write A n,(i),lin (resp. A n,(i),herm ) for the image of the map from G m to L n,(i),lin (resp. L n,(i),herm ) sending t to t1 i (resp. (t 2 , t1 2(n−i) )). Moreover write A n,(i) for A n,(i),lin × A n,(i),herm . The group A n,(i) (resp. A n,(i),lin , resp. A n,(i),herm ) is the maximal split torus in the centre of L n,(i) (resp. L n,(i),lin , resp. L n,(i),herm ).
Again suppose that Ω is an algebraically closed field of characteristic 0. Let Φ (n) ⊂ Φ n denote the set of roots of T n on Lie L n,(n) , and set Φ + (n) = Φ + n ∩ Φ (n) and ∆ (n) = ∆ n ∩ Φ (n) . We will write ̺ n,(n) for half the sum of the elements of Φ + (n) . If R ⊂ R then X * (T n,/Ω ) + (n),R will denote the subset of X * (T n,/Ω ) R consisting of elements which pair non-negatively with the corootα ∈ X * (T n,/Ω ) corresponding to each α ∈ ∆ (n) . We write X * (T n,/Ω ) + (n) for X * (T n,/Ω ) + (n),Z . If λ ∈ X * (T n,/Ω ) + (n) we will let ρ (n),λ denote the irreducible representation of L n,(n) with highest weight λ. When ρ (n),λ is used as a subscript we will sometimes replace it by just (n), λ.
Note that Lie P n,(n) (C) and q n are conjugate under G n (C) and hence we obtain an identification ('Cayley transform') of (Lie U n,∞ A n (R)) ⊗ R C and Lie L n,(n) (C), which is well defined up to conjugation by L n,(n) (C). Similarly Q n and P n,(n) (C) are conjugate in G n × Q C. Thus L n,(n) (C) can be identified with Q n modulo its unipotent radical, canonically up to L n,(n) (C)-conjugation. Thus if ρ is an algebraic representation of L n,(n) over C, we can associate to it a representation of Q n and of q n , and hence a holomorphic vector bundle E ρ /H ± n with G n (R)action.
The isomorphism L n,(n) ∼ = GL 1 ×RS O F Z GL n gives rise to a natural identification L n,(n) × Spec Ω ∼ = GL 1 × GL Hom (F,Ω) n , and hence to identifications To compare this parametrization of X * (T n,/Ω ) with the one introduced in section 1.1 note that the map .

Thus the map
A section is provided by the map In particular we see that X * (T n,/Ω ) + ⊂ X * (T n,/Ω ) + (n) is identified with the set of Note that 2(̺ n − ̺ n,(n) ) = (n 2 [F + : Q], (−n) τ,i ). We write Std for the representation of L n,(n) on Λ n /Λ n,(n) over Z, and if τ : F ֒→ Q we write Std τ for the representation of L n,(n) on (Λ n /Λ n,(n) ) If Ω is an algebraically closed field of characteristic 0 and if τ : F ֒→ Ω we will sometimes write Std τ for the representation of L n,(n) on (Λ n /Λ n,(n) ) ⊗ O F ,τ Ω.
We hope that context will make clear the distinction between these two slightly different meaning of Std τ . We also let KS denote the unique representation of L n,(n) over Z such that Note that over Q the representation Std ∨ τ is irreducible and in our normalizations We will let ς p ∈ L n,(n),herm (Q p ) ∼ = Q × p denote the unique element with multiplier p −1 . Set U p (N) n,(i) = ker(L n,(i),lin (Z p ) → L n,(i),lin (Z/p N Z)) and n,(i),lin (Z/p N Z)). Also set ). In the case i = n these groups do not depend on N 2 , so we will write simply U p (N 1 ).
For the study of the ordinary locus we will need a variant of G n (A ∞ ) and G (m) n,(n) (Z p )). Its maximal sub-semigroup that is also a group is n ) for any N ′ ≥ N. The group does not depend on the choice of N ′ .
1.3. Base change. We will write B GLm for the subgroup of upper triangular elements of GL m and T GLm for the subgroup of diagonal elements of B GLm .
We will also let G 1 n denote the group scheme over O F + defined by n for the subgroup of G 1 n consisting of upper triangular matrices and T 1 n for the subgroup of B 1 n consisting of diagonal matrices. There is a natural projection B 1 n → → T 1 n obtained by setting the off diagonal entries of an element of B 1 n to 0. Suppose that q is a rational prime. Let u 1 , ..., u r denote the primes of F + above Q which split u i = w i c w i in F and let v 1 , ..., v s denote the primes of F + above q which do not split in F . Then Suppose that Π is an irreducible smooth representation of G n (Q q ) then We define BC (Π) w i = Π w i and BC (Π) cw i = Π c,∨ w i . Note that this does not depend on the choice of primes w i |u i . We will say that Π is unramified at However n-Ind The next lemma is easy to prove. Lemma 1.1. Suppose that ψ ⊗ π is an irreducible smooth representation of If v is split over F + and Π is an irreducible sub-quotient of the normalized induction n-Ind In this paragraph let K be a number field, m ∈ Z >0 , and write U K,∞ for a maximal compact subgroup of GL m (K ∞ ). We shall (slightly abusively) refer to an admissible ) module as an admissible G n (A)-module (resp. L n,(i) (A)-module, resp. GL m (A K )module). By a square-integrable automorphic representation of G n (A) (resp. L n,(i) (A), resp. GL m (A K )) we shall mean the twist by a character of an irreducible admissible G n (A)-module (resp. L n,(i) (A)-module, resp. GL m (A K )-module) that occurs discretely in the space of square integrable automorphic forms on the double coset space . By a cuspidal automorphic representation of G n (A) (resp. L n,(i) (A), resp. GL m (A K )) we shall mean an irreducible admissible G n (A)sub-module (resp. L n,(i) (A)-sub-module, resp. GL m (A K )-sub-module) of the space of cuspidal automorphic forms on G n (A) (resp. L n,(i) (A), resp. GL m (A K )).
Proposition 1.2. Suppose that Π is a square integrable automorphic representation of G n (A) and that Π ∞ is cohomological. Then there is an expression 2n = m 1 n 1 + ... + m r n r with m i , n i ∈ Z >0 and cuspidal automorphic representations Proof: This follows from the main theorem of [Sh2] and the classification of square integrable automorphic representations of GL m (A F ) in [MW]. Corollary 1.3. Keep the assumptions of the proposition. Then there is a continuous, semi-simple, algebraic (i.e. unramified almost everywhere and de Rham above p) representation with the following property: If v is a prime of F above a rational prime q such that • either q splits in F 0 , • or F and Π are unramified above q, .
Proof: Combine the proposition with for instance theorem 1.2 of [BLGHT] and theorem A of [BLGGT2]. (These results are due to many people and we simply choose these particular references for convenience.) 1.4. Spaces of Hermitian forms. There is a natural pairing We further define sw : Note that if F/F + is ramified above 2 then S(O m F ) can have 2-torsion, but that S(O m F,(p) ) is torsion free. (Either p > 2 or by assumption F/F + is not ramified above 2.) There is a perfect duality where e 1 , ..., e m denotes the standard basis of O m F . If R ⊂ R then we will denote by Herm >0 X (resp. Herm ≥0 X ) the set of pairings ( , ) in Herm X such that (x, x) > 0 (resp. ≥ 0) for all x ∈ X − {0}. We will denote by S(F m ) >0 the set of elements a ∈ S(F m ) such that for each τ : F ֒→ C the image of a under the map is positive definite, i.e. all the roots of its characteristic polynomial are strictly positive real numbers. Then S(F m ) >0 is the set of elements of S(F m ) whose pairing with every element of Herm >0 F m is strictly positive; and Herm >0 F m is the set of elements of Herm F m whose pairing with every element of S(F m ) >0 is strictly positive.
Suppose that W ⊂ V n is an isotropic F -direct summand. We set If m = 0 we will drop it from the notation. There is a natural map This establishes an isomorphism and hence an isomorphism We extend this to an action of G (m) n (Q) as follows: We will write C >0 (W ) = Herm >0 Vn/W ⊥ ⊗ Q R and C ≥0 (W ) = Herm ≥0 Vn/W ⊥ ⊗ Q R . We will also write C (m),>0 (W ) (resp. C (m),≥0 (W )) for the pre-image of C >0 (W ) (resp. C ≥0 (W )) in C (m) (W ). Moreover we will set (W ) and that the pre image of (0) is (0). Also note that if W ′ ⊂ W then there is a closed embedding Finally note that the action of G (m) n (Q) takes C (m),≻0 (W ) (resp. C (m),>0 (W ), resp. C (m),≥0 (W )) to C (m),≻0 (gW ) (resp. C (m),>0 (gW ), resp. C (m),≥0 (gW )).
In fact L (m) n,(i) (R) acts transitively on π 0 (L n,(i),herm (R)) × C (m),>0 (V n,(i) ). For this paragraph let ( , ) 0 ∈ C >0 (V n,(i) ) denote the pairing on (V n /V ⊥ n,(i) ⊗ Q R) 2 induced by J n , n . Then the stabilizer of 1 × We define C (m) to be the topological space (This is sometimes referred to as the 'conical complex'.) Thus as a set (W ).
(The second factor acts on each C (m),≻0 (W ) by scalar multiplication.) We have homeomorphisms (Use the fact, strong approximation for unipotent groups, that n (A ∞ ) and if g −1 Ug ⊂ U ′ then the right translation map ) corresponds to the coproduct of the right translation maps It has a left action of G Proof: There is a natural surjection. We must check that it is also injective. The right hand side equals which is clearly isomorphic to the left hand side.
We set There does not seem to be such a simple description of for i = n. However we do have the lemma below.
Lemma 1.5. There is a natural homeomorphism We can replace the second P [This follows from the fact that primes above p on F + are unramified in F , which implies that n )× π 0 (G n (R)) × C (m),>0 (V n,(i) )).
Abusing notation slightly, we will write Corollary 1.6. .
If Y is a locally compact, Hausdorff topological space then we write H i as N runs over positive integers and U p runs over neat open compact subgroups of L (m) n,(n) (A p,∞ ). With these definitions we have the following corollary.
Interior cohomology has the following property which will be key for us.
Lemma 1.8. Suppose that G is a locally compact, totally disconnected topologyical group. Suppose that for any sufficiently small open compact subgroup U ⊂ G we are given a compact Hausdorff space Z U and an open subset Y U ⊂ Z U . Suppose moreover that whenever U, U ′ are sufficiently small open compact subgroups of G and g ∈ G with g −1 Ug ⊂ U ′ , then there is a proper continuous map Also suppose that g • h = hg whenever these maps are all defined and that, if g ∈ U then the map g : Z U → Z U is the identity.
If Ω is a field, set and 1.5. Locally symmetric spaces. In this section we will calculate H i Int (T (m) (n) , Q p ) in terms of automorphic forms on L n,(n) (A).
If m = 0 we will write T (n) for T (0) (n) . Let Ω denote an algebraically closed field of characteristic 0. If ρ is a finite dimensional algebraic representation of L n,(n) on a Ω-vector space W ρ then we define a locally constant sheaf L ρ,U /T (n),U as The system of sheaves L ρ,U has a right action of L n,(n) (A ∞ ). We define The natural map L Lemma 1.9.
(1) The maps π (m) are real-torus bundles (i.e. (S 1 ) r -bundles for some r), and in particular are proper maps.
(2) There are L (m) In particular the action of L Using the identification of spaces (but not of groups) that comes from the group product L n,(n) (A ∞ )-action is by right translation on the second factor. The left action of L n,(n) (Q) is via conjugation on the first factor and left translation on the second.
The first part of the lemma follows, and we see that n,(n) (A ∞ ) equivariantly identified with the locally constant sheaf Proof: There is an L , Ω). If α ∈ Q × >0 ⊂ Z(L n,(n),lin )(A ∞ ), then α acts on E i,j 2 via α j . We deduce that all the differentials (on the second and any later page) vanish, i.e. the spectral sequence degenerates on the second page. Moreover the α → α j eigenspace in . (This standard argument is sometimes referred to as 'Lieberman's trick'.) As the maps π (m) are proper, there is also a L (m) , Ω) and α ∈ Q × >0 ⊂ Z(L n,(n),lin )(A ∞ ) acts on E i,j c,2 via α j . Again we see that the spectral sequence degenerates on the second page, and that the α → α j eigenspace in H i+j . The lemma follows.
Corollary 1.11. Suppose that ρ is an irreducible representation of L n,(n),lin over Ω, which we extend to a representation of L n,(n) by letting it be trivial on L n,(n),herm . Let d = N F/Q • det : L n,(n),lin → G m . Then for all N sufficiently large there are j(N), m(N) ∈ Z ≥0 such that, for all i, , Ω).
Proof: It follows from Weyl's construction of the irreducible representations of GL n that, for N sufficiently large, ρ ⊗ d −N is a direct summand of for certain non-negative integers m τ (N). Hence for N sufficiently large and m(N) = max{m τ (N)} the representation ρ ⊗ d −N is also a direct summand of Lemma 1.12. Suppose that ρ is an irreducible algebraic representation of L n,(n) on a finite dimensional C-vector space.
Proof: The first part results from [Bo], more precisely from combining theorem 5.2, the discussion in section 5.4 and corollary 5.5 of that paper. The second part results from [Cl], see the proof of theorem 3.13, and in particular lemma 3.14, of that paper.
Combining this lemma and corollary 1.11 we obtain the following consequence.
Corollary 1.13. Suppose that n > 1 and that ρ is an irreducible algebraic representation of L n,(n),lin on a finite dimensional C-vector space. Suppose also that π is a cuspidal automorphic representation of L n,(n),lin (A) so that π ∞ has the same infinitesimal character as ρ ∨ and that ψ is a continuous character of Q × \A × /R × >0 . Then for all sufficiently large integers N there are integers m(N) ∈ Z ≥0 and i(N) ∈ Z >0 , and a L n,(n) (A ∞ )-equivariant embedding , C).

Tori, torsors and torus embeddings.
Throughout this section let R 0 denote an irreducible noetherian ring (i.e. a noetherian ring with a unique minimal prime ideal). In the applications of this section elsewhere in this paper it will be either Q or Z (p) or Z/p r Z for some r. We will consider R 0 endowed with the discrete topology so that Spf R 0 ∼ = Spec R 0 .
2.1. Tori and torsors. If S/Y is a torus (i.e. a group scheme etale locally on Y isomorphic to G N m for some N) then we can define its sheaf of characters X * (S) = Hom (S, G m ) and its sheaf of cocharacters X * (S) = Hom (G m , S). These are locally constant sheaves of free Z-modules in the etale topology on Y . They are naturally Z-dual to each other. More generally if S 1 /Y and S 2 /Y are two tori then Hom (S 1 , S 2 ) is a locally constant sheaf of free Z-modules in the etale topology on Y . In fact Hom (S 1 , S 2 ) = Hom (X * (S 1 ), X * (S 2 )) = Hom (X * (S 2 ), X * (S 1 )).
By a quasi-isogeny (resp. isogeny) from S 1 to S 2 we shall mean a global section of the sheaf Hom (S 1 , S 2 ) Q (resp. Hom (S 1 , S 2 )) with an inverse in Hom (S 2 , S 1 ) Q . We will write [S] isog for the category whose objects are tori over Y quasi-isogenous to S and whose morphisms are isogenies. The sheaves X * (S) Q and X * (S) Q only depend on the quasi-isogeny class of S so we will write X * ( If y is a geometric point of Y then we define and with the transition map from MN to N being multiplication by M. (The Tate modules of S.) Also define V S y = T S y ⊗ Z Q and If Y is a scheme over Spec Z (p) then Now suppose that S is split, i.e. isomorphic to G N m for some N. By an S-torsor T /Y we mean a scheme T /Y with an action of S, which locally in the Zariski topology on Y is isomorphic to S. By a rigidification of T along e : Y ′ → Y we mean an isomorphism of S-torsors e * where L T (χ) is a line bundle on U. If Z is any open subset of Y and if χ ∈ X * (S)(Z) then there is a unique line bundle L T (χ) on Z whose restriction to any connected open subset U ⊂ Z is L T (χ| U ). Multiplication gives isomorphisms Note that if U has infinitely many connected components then it may not be the case that T | U = Spec χ∈X * (S)(U ) L T (χ). The map gives a bijection between isomorphism classes of G m -torsors and isomorphism classes of line bundles on Y . The inverse map sends L to If α : S → S ′ is a morphism of split tori and if T /Y is an S-torsor we can form a pushout α * T , an S ′ torsor on Y defined as the quotient There is a natural map T → α * T compatible with α : S → S ′ . If α is an isogeny then α * T = (ker α)\T. If T 1 and T 2 are S-torsors over Y we define . If T is an S-torsor on Y we define an S-torsor T ∨ /Y by taking T ∨ = T as schemes but defining an S action . on T ∨ by the map that sends (t 1 , t 2 ) to the unique section s of S with st 1 = t 2 .
2.2. Log structures. We will call a formal scheme where A i is a finitely generated R 0 -algebra and I i is an ideal of A i whose inverse image in R 0 is (0). By a log structure on a scheme X (resp. formal scheme X) we mean a sheaf of monoids M on X (resp. X) together with a morphism is an isomorphism. We will refer to a scheme (resp. formal scheme) endowed with a log structure as a log scheme (resp. log formal scheme). By a morphism of log schemes (resp. morphism of log formal schemes) ) we shall mean a morphism φ : X → Y (resp. φ : X → Y) and a map We will consider R 0 endowed with the trivial log structure (O × Spec R 0 , 1) (resp. (O × Spf R 0 , 1)). We will call a log formal scheme (X, M, α)/Spf R 0 suitable if X/Spf R 0 is suitable and if, locally in the Zariski topology, M/α * O × X is finitely generated. (In the case of schemes these definitions are well known. We have not attempted to optimize the definition in the case of formal schemes. We are simply making a definition which works for the limited purposes of this article.) If X/Spec R 0 is a scheme of finite type and if Z ⊂ X is a closed sub-scheme which is flat over Spec R 0 , then the formal completion X ∧ Z is a suitable formal scheme. Let i ∧ denote the map of ringed spaces X ∧ Z → X. If (M, α) is a log structure on X, then we get a map is a morphism of schemes with log structures over Spec R 0 then there is a right exact sequence If the map (φ, ψ) is log smooth then this sequence is also left exact and the sheaf Ω 1 X/Y (log M/N ) is locally free. As usual, we write . By a coherent sheaf of differentials on a formal scheme X/Spf R 0 we will mean a coherent sheaf Ω/X together with a differential d : O X → Ω which vanishes on R 0 . By a coherent sheaf of log differentials on a log formal scheme (X, M, α)/Spf R 0 we shall mean a coherent sheaf Ω/X together with a differential, which vanishes on R 0 , d : O X −→ Ω, and a homomorphism dlog : M −→ Ω such that α(m)dlog m = d(α(m)). By a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) we shall mean a coherent sheaf of differentials (Ω, d) (resp. a coherent sheaf of log differentials (Ω, d, dlog )) such that for any other coherent sheaf of differentials (Ω ′ , d ′ ) (resp. a coherent sheaf of log differentials (Ω ′ , d ′ , dlog ′ )) there is a unique map f : Note that if a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) exists, it is unique up to unique isomorphism.
Lemma 2.1. Suppose that R 0 is a discrete, noetherian topological ring.
(1) A universal sheaf of coherent differentials Ω 1 X/Spf R 0 exists for any suitable formal scheme X/Spf R 0 .
(2) If X/Spec R 0 is a scheme of finite type and if Z ⊂ X is flat over R 0 then exists for any suitable log formal scheme (X, M, α)/Spf R 0 .
(4) Suppose that X/Spec R 0 is a scheme of finite type, that Z ⊂ X is flat over R 0 and that (M, α) is a log structure on X such that Zariski locally M/α −1 O × X is finitely generated. Then is an affine open in X, where A is a finitely generated R 0 -algebra and I is an ideal of A with inverse image (0) in R 0 . Then there exists a universal finite module of differentials Ω 1 U for U, namely the coherent sheaf of O U -modules associated to (Ω 1 A/R 0 ) ∧ I . (See sections 11.5 and 12.6 of [Ku].) We must show that if U ′ ⊂ U is open then Ω 1 U | U ′ is a universal finite module of differentials for U ′ . For then uniqueness will allow us to glue the coherent sheaves Ω 1 U to form Ω 1 X . So suppose that (Ω ′ , d ′ ) is a finite module of differentials for U ′ . We must show that there is a unique map of O U ′ -modules We may cover U ′ by affine opens of the form Spf (A g ) ∧ I and it will suffice to find, for each g, a unique Thus we may assume that U ′ = Spf (A g ) ∧ I . But in this case we know Ω 1 U ′ exists, and is the coherent sheaf associated to the first part follows. The second part also follows from the proof of the first part.
For the third part, because of uniqueness, it suffices to work locally. Thus we may assume that there are finitely many sections m 1 , ..., m r ∈ M(X), which together with α −1 O × X generate M. Then we define Ω 1 (X, M,α) to be the cokernel of the map O ⊕r It is elementary to check that this has the desired universal property. The fourth part is also elementary to check.
is a map of suitable log formal schemes over Spf R 0 then we set Corollary 2.2. Suppose that R 0 is a discrete, noetherian topological ring; that (X, M, α) → (Y, N , β) is a map of log schemes over Spec R 0 ; and that Z ⊂ X and W ⊂ Y are closed sub-schemes flat over Spec R 0 which map to each other under X → Y . Suppose moreover that X and Y have finite type over Spec R 0 and that M/α −1 O × X and N /β −1 O × Y are locally (in the Zariski topology) finitely generated. Then Proof: This follows from the lemma and from the exactness of completion.
If Y is a scheme we will let denote the union of the coordinate hyperplanes in Aff n Y . Now suppose that X → Y is a smooth map of schemes of relative dimension n. By a simple normal crossing divisor in X relative to Y we shall mean a closed subscheme D ⊂ X such that X has an affine Zariski-open cover {U i } such that each U i admits an etale map In the case that Y is just the spectrum of a field we will refer simply to a simple normal crossing divisor in X.
Suppose that Y is locally noetherian and separated, and that the connected components of Y are irreducible. If S is a finite set of irreducible components of D we will set It is smooth over Y . We will also set If E is an irreducible component of D (s) then the set S(E) of irreducible components of D containing E has cardinality s. If ≥ is a total order on the set of irreducible components of D, we can define a delta set S(D, ≥), or simply S(D), as follows. (For the definition of delta set, see for instance [Fr]. We can, if we prefer to be more abstract, replace S(D, ≥) by the associated simplicial set.) The n cells consist of all irreducible components of D (n+1) . If E is such an irreducible component and if i ∈ {0, ..., n} then the image of E under the face map d i is the unique irreducible component of which contains E. Here S(E) i equals S(E) with its (i + 1) th smallest element removed. The topological realization |S(D, ≥)| does not depend on the total order ≥, so we will often write |S(D)|.
If D is a simple normal crossing divisor in X relative to Y we define a log structure M(D) on X by setting We record a general observation about log de Rham complexes and divisors with simple normal crossings, which is probably well known. We include a proof because it is of crucial importance for our argument.
Lemma 2.3. Suppose that Y is a smooth scheme of finite type over a field k and that Z ⊂ Y is a divisor with simple normal crossings. Let Z 1 , ..., Z m denote the distinct irreducible components of Z and set There is a double complex being the sum of the maps The natural inclusions give rise to a map of complexes . For fixed r the simple complexes Proof: Only the last assertion is not immediate. So consider the last assertion. We can work Zariski locally, so we may assume that the complex is pulled back from the corresponding complex for the case Y = Spec k[X 1 , ..., X d ] and Z is given by X 1 X 2 ...X m = 0. In this case we take Z j to be the scheme X j = 0, for j = 1, ..., m. In this case where T runs over r element subsets of {1, ..., d}. On the other hand where T runs over r element subsets of {1, ..., d} − S. Thus it suffices to show that, for each subset T ⊂ {1, ..., d} the sequence ..,d} , and so we only need treat the case m = d and T = ∅.
If µ is a monomial in the variables X 1 , ..., X m , let R(µ) denote the subset of {1, ..., m} consisting of the indices j for which X j does not occur in µ. Then our complex is the direct sum over µ of the complexes where A µ = k if R(µ) = ∅ and = (0) otherwise. So it suffices to prove this latter complex exact for all µ. If R(µ) = ∅ then it becomes If we suppress the first k, this is the complex that computes the simplicial cohomology with k-coefficients of the simplex with #R(µ) vertices. Thus it is exact everywhere except S⊂R(µ), #S=1 k and the kernel of k is just k. The desired exactness follows.
2.3. Torus embeddings. We will now discuss relative torus embeddings. We will suppose that Y /Spec R 0 is flat and locally of finite type. To simplify the notation, for now we will restrict to the case of a split torus S/Y with Y connected. We will record the (trivial) generalization to the case of a disconnected base below. Thus we can think of X * (S) and X * (S) as abelian groups, rather than as locally constant sheaves on Y , i.e. we replace the sheaf by its global sections over Y . We will let T /Y denote an S-torsor.
By a rational polyhedral cone σ ⊂ X * (S) R we mean a non-empty subset consisting of all R ≥0 -linear combinations of a finite set of elements of X * (S), but which contains no complete line through 0. (We include the case σ = {0}. The notion we define here is sometimes called a 'non-degenerate rational polyhedral cone'.) By the interior σ 0 of σ we shall mean the complement in σ of all its proper faces. (We consider σ as a face of σ, but not a proper face.) We call σ smooth if it consists of all R ≥0 -linear combinations of a subset of a Z-basis of X * (S). Note that any face of a smooth cone is smooth. Then we define σ ∨ to be the set of elements of X * (S) R which have non-negative pairing with every element of σ and σ ∨,0 to be the set of elements of X * (S) R which have strictly positive pairing with every element of σ − {0}. Moreover we set Suppose that Σ 0 is a non-empty set of faces of σ such that If Σ 0 contains all the faces of σ other than {0} we will write ∂T σ for ∂ Σ 0 T σ . If σ ′ is a face of σ then under the open embedding By a fan in X * (S) R we shall mean a non-empty collection Σ of rational polyhedral cones σ ⊂ X * (S) R which satisfy We call Σ smooth if each σ ∈ Σ is smooth. We will call Σ full if every element of Σ is contained in an element of Σ with the same dimension as X * (S) R . Define We call Σ ′ a refinement of Σ if each σ ′ ∈ Σ ′ is a subset of some element of Σ and each element σ ∈ Σ is a finite union of elements of Σ ′ .
(1) If Σ is a fan and Σ ′ ⊂ Σ is a finite cardinality sub-fan then there is a refinement Σ of Σ with the following properties: • any element of Σ which is smooth also lies in Σ; • any element of Σ contained in an element of Σ ′ is smooth; • and if σ ′ ∈ Σ − Σ then σ ′ has a non-smooth face lying in Σ ′ .
Proof: The first part is proved just as for finite fans by making a finite series of elementary subdivisions by 1 cones that lie in some element σ ′ ∈ Σ ′ but not in any of its smooth faces. See for instance section 2.6 of [Fu].
For the second part, consider the S the set of pairs ( Σ, ∆) where Σ is a refinement of Σ and ∆ is a sub-fan of Σ such that • every smooth element of Σ lies in Σ; • and if σ ∈ Σ is contained in an element of ∆ then σ is smooth. It suffices to show that S contains an element ( Σ, ∆) with ∆ = Σ.
If ( Σ, ∆) ∈ S and σ ∈ Σ we define Σ(σ) to be the set of elements of Σ contained in σ. We define a partial order on S by decreeing that ( Σ, ∆) ≥ ( Σ ′ , ∆ ′ ) if and only if the following conditions are satisfied: and let Σ denote the set of cones σ ′ which lie in Σ ′ for all sufficiently large elements of ( Σ ′ , ∆ ′ ) ∈ S ′ . If σ ∈ Σ then we can choose ( Σ ′ , ∆ ′ ) ∈ S ′ so that the number of faces of σ in ∆ ′ is maximal.
By Zorn's lemma S has a maximal element ( Σ, ∆). We will show that ∆ = Σ, which will complete the proof of the lemma. Suppose not. Choose σ ∈ Σ − ∆. Set ∆ ′ to be the union of ∆ and the faces of σ. Let Σ ′ be a refinement of Σ such that • any element of Σ which is smooth also lies in Σ ′ ; • any element of Σ ′ contained in σ is smooth; • and if σ ′ ∈ Σ − Σ ′ then σ ′ has a non-smooth face contained in σ.
To a fan Σ one can attach a connected scheme T Σ that is separated, locally (on T Σ ) of finite type and flat over Y of relative dimension dim R X * (S) R , together with an action of S and an S-equivariant dense open embedding which restricts to the identity on T : its restriction to T σ ′ equals the map By boundary data for Σ we shall mean a proper subset Σ 0 ⊂ Σ such that Σ−Σ 0 is a fan. (Note that Σ 0 may not be closed under taking faces.) If Σ 0 is boundary data we define Thus ∂ Σ 0 T Σ has an open cover by the sets In the special case Σ 0 = Σ − {{0}} we will write ∂T Σ for ∂ Σ 0 T Σ and I ∂T Σ for I ∂ Σ 0 T Σ . Then T = T Σ − ∂T Σ . We will write M Σ → O T Σ for the log structure corresponding to the closed embedding ∂T Σ ֒→ T Σ . We will write Ω 1 If Σ 0 is boundary data for Σ we will set • finite if it has finite cardinality; • locally finite if for every rational polyhedral cone τ ⊂ |Σ 0 | (not necessarily in Σ 0 ) the intersection τ ∩ |Σ 0 | 0 meets only finitely many elements of Σ 0 .
(We remark that although this condition may be intuitive in the case |Σ 0 | 0 = |Σ 0 |, in other cases it may be less so.) Let Σ continue to denote a fan and Σ 0 boundary data for Σ. If σ ∈ Σ we write This is an example of boundary data for Σ. If σ ∈ Σ 0 then If dim σ = 1 then ∂ 0 σ T Σ = ∂T σ . Keep the notation of the previous paragraph. We define S(σ) to be the split torus with co-character group X * (S) divided by the subgroup generated by σ ∩ X * (S), and T (σ) to be the push-out of T to S(σ). We also define Σ(σ) to be the set of images in X * (S(σ)) R of elements of Σ(σ). It is a fan for To verify this suppose that x ∈ τ and y ∈ τ ′ with x−y ∈ σ R . Then x−y = z −w with z, w ∈ σ. Thus x + w = y + z ∈ τ ∩ τ ′ and x + σ R = (x + w) + σ R .] If σ ∈ Σ 0 we will sometimes write Σ 0 (σ) for Σ(σ), as it depends only on Σ 0 and not on Σ. Then Thus ∂ σ T Σ is separated, locally (on the source) of finite type and flat over Y . The closed subscheme ∂ σ T Σ has codimension in T Σ equal to the dimension of σ.
The schemes ∂ σ 1 T Σ , ..., ∂ σs T Σ intersect if and only if σ 1 , ..., σ s are all contained in some σ ∈ Σ. In this case the intersection equals ∂ σ T Σ for the smallest such σ. We set If Y is irreducible then T Σ and each ∂ σ T Σ is irreducible. Moreover the irreducible components of ∂T Σ are the ∂ σ T Σ as σ runs over one dimensional elements of Σ. If Σ is smooth then we see that S(∂T Σ ) is the delta complex with cells in bijection with the elements of Σ − {{0}} and with the same 'face relations'. In particular it is in fact a simplicial complex and If Σ is a fan, then by line bundle data for Σ we mean a continuous function ψ : |Σ| → R, such that for each cone σ ∈ Σ, the restriction ψ| σ equals some Note that there are natural isomorphisms We have the following examples of line bundle data.
(1) O T Σ is the line bundle associated to ψ ≡ 0.
(2) If Σ is smooth then I ∂T Σ is the line bundle associated to the unique such function ψ Σ which for every one dimensional cone σ ∈ Σ satisfies Suppose that α : S → → S ′ is a surjective map of split tori over Y . Then X * (α) : X * (S ′ ) ֒→ X * (S) and X * (α) : X * (S) → X * (S ′ ), the latter with finite cokernel. We call fans Σ for X * (S) and Σ ′ for X * (S ′ ) compatible if for all σ ∈ Σ the image X * (α)σ is contained in some element of Σ ′ . In this case the map α : T → α * T extends to an S-equivariant map We will write If α is an isogeny, if Σ and Σ ′ are compatible, and if every element of Σ ′ is a finite union of elements of Σ, then we call Σ a quasi-refinement of Σ ′ . In that case the map T Σ → T Σ ′ is proper.
is injective and the torsion subgroup of the kernel is finite with order invertible on Y .
We will call pairs (Σ, Σ 0 ) and (Σ ′ , Σ ′ 0 ) of fans and boundary data for S and S ′ , respectively, compatible if Σ and Σ ′ are compatible and if no cone of Σ 0 maps into any cone of Σ ′ − Σ ′ 0 . In this case We will call them strictly compatible if they are compatible and Σ − Σ 0 is the set of cones in Σ mapping into some element of Σ ′ − Σ ′ 0 . Lemma 2.6. Suppose that α : S → → S ′ is a surjective map of split tori, that T /Y is an S-torsor, and that (Σ, Σ 0 ) and (Σ ′ , Σ ′ 0 ) are strictly compatible fans with boundary data for S and S ′ respectively. Then locally on T Σ there is a strictly positive integer m such that Proof: We may work locally on Y and so we may suppose that Y = Spec A is affine and that each L T (χ) is trivial. It also suffices to check the lemma locally on T Σ . Thus we may suppose that Σ consists of a cone σ and all its faces. Let σ ′ denote the smallest element of Σ ′ containing the image of σ. Then we may further suppose that Σ ′ consists of σ ′ and all its faces. We may further suppose that σ ∈ Σ 0 and σ ′ ∈ Σ ′ 0 , else there is nothing to prove. Then Thus it suffices to show that for some positive integer m we have Note that if a ∈ σ and χ(a) = 0 then (X * (α)(χ 1 ))(a) = 0. Thus we can find ǫ > 0 such that Suppose that (Σ, Σ 0 ) and (Σ ′ , Σ ′ 0 ) are strictly compatible. We will say that If α is an isogeny, if Σ is a quasi-refinement of Σ ′ , and if (Σ, Σ 0 ) and (Σ ′ , Σ ′ 0 ) are strictly compatible, then we call (Σ, Σ 0 ) a quasi-refinement of (Σ ′ , Σ ′ 0 ). In this case Σ 0 is open and finite over Σ ′ 0 . Lemma 2.7. Suppose that α : S → → S ′ is a surjective map of split tori, that T /Y is an S-torsor, and that (Σ, Σ 0 ) and (Σ ′ , Σ ′ 0 ) are strictly compatible fans with boundary data for S and S ′ respectively. If Σ 0 is locally finite and Σ 0 is open over Choose a point P ∈ σ 0 such that We deduce that it suffices to check that The left hand side certainly contains the right hand side, so it suffices to prove that for all

Thus it suffices to find an open set
Moreover in order to find such a U ∋ P it suffices to find one satisfying each property independently and take their intersection.
One can find an open set U ∋ P satisfying the first property because To find U ∋ P satisfying the second condition we just need to avoid the faces of X * (α) −1 τ ′ which do not contain P .
It remains to check that we can find an open U ∋ P satisfying the last condition.
By a partial fan we will mean a collection Σ 0 of rational polyhedral cones satisfying • if σ 1 , σ 2 ∈ Σ 0 , and if σ ⊃ σ 2 is a face of σ 1 , then σ ∈ Σ 0 . (Again note that Σ 0 may not be closed under taking faces.) In this case we will let Σ 0 denote the set of faces of elements of Σ 0 . Then Σ 0 and Σ 0 − Σ 0 are fans, and Σ 0 is boundary data for Σ 0 . [For suppose that τ i is a face of σ i ∈ Σ 0 for i = 1, 2. Then σ 1 ∩ σ 2 is a face of σ 1 and so τ 1 ∩ σ 2 = τ 1 ∩ (σ 1 ∩ σ 2 ) is a face of σ 1 ∩ σ 2 and hence of σ 2 . Thus τ 1 ∩ τ 2 = τ 2 ∩ (τ 1 ∩ σ 2 ) is a face of τ 2 .] If Σ is a fan and Σ 0 is boundary data for Σ, then Σ 0 is a partial fan, and Σ ⊃ Σ 0 . Thus If Σ 0 and Σ ′ 0 are partial fans we will say that Σ 0 refines Σ ′ 0 if every element of Σ 0 is contained in an element of Σ ′ 0 and if every element of Σ ′ 0 is a finite union of elements of Σ 0 . In this case Σ 0 also refines Σ ′ 0 . If Σ 0 is a partial fan we will set • full if every element of Σ 0 which is not a face of any other element of Σ 0 , has the same dimension as S; • finite if it has finite cardinality; • locally finite if for every rational polyhedral cone τ ⊂ |Σ 0 | (not necessarily in Σ 0 ) the intersection τ ∩ |Σ 0 | 0 meets only finitely many elements of Σ 0 .
If Σ 0 is smooth, so is Σ 0 . Suppose that Σ 0 is a partial fan. If Σ ⊃ Σ 0 is a fan then the natural maps are isomorphisms, and we will denote these schemes/formal schemes ∂ Σ 0 T and T ∧ Σ 0 respectively. Moreover the log structures induced on T ∧ Σ 0 by M Σ 0 and by M Σ are the same and we will denote them M ∧ We will also use the following notation. • By line bundle data for Σ 0 we mean a continuous functions ψ : |Σ 0 | → R, such that for each cone σ ∈ Σ 0 , the restriction ψ| σ equals some ψ σ ∈ X * (S). This is the same as line bundle data for the fan Σ 0 , and we will write L ∧ ψ for the line bundle on We have the following examples of line bundle data. ( is the line bundle associated to ψ ≡ 0. (2) If Σ 0 is smooth then I ∧ ∂,Σ 0 is the line bundle associated to the unique such function ψ Σ 0 which for every one dimensional cone σ ∈ Σ 0 satisfies Suppose that α : S → → S ′ is a surjective map of tori, and that Σ 0 (resp. Σ ′ 0 ) is a partial fan for S (resp. S ′ ). We call Σ 0 and Σ ). The following lemma follows immediately from lemma 2.5.
We will call Σ 0 and Σ ′ 0 strictly compatible if they are compatible and if an element of Σ 0 lies in Σ 0 if and only if it maps to no element of if only finitely many elements of Σ 0 map into any element of Σ ′ 0 . If α is an isogeny, if Σ 0 and Σ ′ 0 are strictly compatible, and if every element of Σ ′ 0 is a finite union of elements of Σ 0 , then we call Σ 0 a quasi-refinement of Σ ′ 0 . In this case Σ 0 is open and finite over Σ ′ 0 . The next lemma follows immediately from lemma 2.6 and 2.7.

Cohomology of line bundles.
In this section we will compute the cohomology of line bundles on formal completions of torus embeddings. We will work throughout over a base scheme Y which is connected, separated, and flat and locally of finite type over Spec R 0 . We start with some definitions. We continue to assume that S/Y is a split torus, that T /Y is an S-torsor, that Σ 0 is a partial fan and that ψ is line bundle data for Σ 0 . If σ ∈ Σ 0 then we set For m > 0 we define X Σ 0 ,ψ,σ,m to be the set of sums of an element of X Σ 0 ,ψ,σ,0 and an element of Note the examples: ( Also note that if Σ 0 is finite then, for m large enough, H j Σ 0 ,ψ,m (χ)(U) does not depend on m. We will denote it simply H j Σ 0 ,ψ (χ)(U). It equals the cohomology of the Cech complex We follow the argument of section 3.5 of [Fu].
(See the first paragraph of section 3.5 of [Fu].) Thus the i th term of our Cech complex becomes Thus it suffices to show that the Cech complex with i th term

To this end choose an injective resolution
as sheaves of abelian groups on |Σ 0 | 0 , and consider the double complex We compute the cohomology of the corresponding total complex in two ways. Firstly the j th cohomology of the complex M). (See theorem 4.1, proposition 5.3 and theorem 5.5 of chapter II of [Br].) This vanishes for j > 0, and so the cohomology of our total complex is the same as the cohomology of the Cech complex with i th term Thus it suffices to identify the cohomology of our double complex with is exact for all j. Let I j denote the sheaf of discontinuous sections of I j , i.e. I j (V ) denotes the set of functions which assign to each point of x ∈ V an element of the stalk I j x of I j at x. Then I j is a direct summand of I j so it suffices to show that In general we will let H i Lemma 2.12. Let Y be a connected, locally noetherian, separated scheme, let S/Y be a split torus, let T /Y be an S-torsor, let Σ 0 be a partial fan for S, let ψ be line bundle data for Σ 0 , and let π ∧ Σ 0 denote the map T ∧ Σ 0 → Y . Suppose that Σ 0 is finite and open. Then (Note that R i π ∧ Σ 0 , * L ∧ ψ may not be quasi-coherent on Y . Infinite products of quasi-coherent sheaves may not be quasi-coherent.) Proof: The left hand side is the sheaf associated to the pre-sheaf ) and the right hand side is the sheaf associated to the pre sheaf Thus it suffices to establish isomorphisms compatibly with restriction, for U = Spec A, with A noetherian and Spec A connected. Write . It has the same underlying topological space as ∂ Σ 0 T Σ 0 . We will first compute using the affine cover of ∂ Σ 0 ,m T Σ 0 by the open sets T σ for σ ∈ Σ 0 . This gives rise to a Cech complex with terms As Σ 0 is finite, we see that is a finitely generated A-module, and hence, for fixed m and i, we see that the groups H i Σ 0 ,ψ,m (χ)(U) = (0) for all but finitely many χ. In particular Moreover, combining this observation with the fact that {H i Σ 0 ,ψ,m (χ)(U)} satisfies the Mittag-Leffler condition, we see that the system satisfies the Mittag-Leffler condition. Hence from proposition 0.13.3.1 of [EGA3] we see that , and the present lemma follows from lemma 2.11. Lemma 2.13. Let Y be a connected, locally noetherian, separated scheme, S/Y be a split torus, let T /Y be an S-torsor, let Σ ∞ be a partial fan for S, let Proof: The left hand side is the sheaf associated to the pre-sheaf ) and the right hand side is the sheaf associated to the pre-sheaf Thus it suffices to establish isomorphisms compatibly with restriction, for U = Spec A, with A noetherian and Spec A connected.
We can compute H i (T ∧ Σ∞ | U , L ∧ ψ ) as the cohomology of the Cech complex with i th term Note that as soon as the faces of σ in Σ j equals the faces of σ in Σ ∞ then satisfies the Mittag-Leffler condition (with j varying but i fixed). From theorem 3.5.8 of [W] we see that there is a short exact sequence . Applying lemma 2.12 and the fact that lim ← and lim ← 1 in the category of abelian groups commute with arbitrary products, the present lemma follows. (It follows easily from definition 3.5.1 of [W] and the exactness of infinite products in the category of abelian groups, that lim ← and lim ← 1 commute with arbitrary products in the category of abelian groups.) We now turn to two specific line bundles: and, in the case that Σ 0 is smooth, I ∧ ∂,Σ 0 . Lemma 2.14. Let Y be a connected, locally noetherian, separated scheme, let S/Y be a split torus, let T /Y be an S-torsor, let Σ 0 be a partial fan for S, and let π ∧ Σ 0 denote the map T ∧ Σ 0 → Y . Suppose that Σ 0 is finite and open and that |Σ 0 | 0 is convex.
(2) If in addition Σ 0 is smooth then Proof: The first part follows from lemma 2.12 because Y 0 (χ) ∩ |Σ 0 | 0 is empty if χ ∈ |Σ 0 | ∨ and otherwise, being the intersection of two convex sets, it is convex.
For the second part we have that To this end, consider the sets We will describe a deformation retraction is just projection to the first factor.) As {x ∈ |Σ 0 | 0 : χ(x) ≤ 0} is empty or convex, it would follow that Y ψ Σ 0 (χ) ∩ |Σ 0 | 0 is empty or contractible and the second part of the corollary would follow.
To define H it suffices to define, for each σ ∈ Σ 0 with σ ⊂ Y ′′ (χ), a deformation retraction .., w s denote the shortest non-zero vectors in X * (S) on each of the one dimensional faces of σ (so that r+s = dim σ), with the notation chosen such that χ(v i ) ≤ 0 for all i and χ(w j ) > 0 for all j. Note that 1−χ(v i ) > 0 for all i and 1 − χ(w j ) ≤ 0 for all j. We set it is easy to check that it has the desired properties and the proof of the lemma is complete.
Lemma 2.15. Let Y be a connected scheme, let α : S → S ′ be an isogeny of split tori over Y , and let Σ ′ 0 (resp. Σ 0 ) be a locally finite partial fan for S ′ (resp. S). Suppose that Y is separated and locally noetherian and that Σ ′ 0 is full. Also suppose that Σ 0 is a quasi-refinement of Σ ′ 0 , and let π ∧ denote the map If moreover Σ 0 and Σ ′ 0 are smooth then, for i > 0 we have Thus they have closed support. Their support is also Sinvariant. Thus it suffices to show that for each maximal element does not lie in the support of these sheaves. Let Σ 0 (σ ′ ) denote the subset of elements σ ∈ Σ 0 which lie in σ ′ , but in no face of σ ′ . Then Σ 0 (σ ′ ) is a partial fan and Thus the formal completion of the above four sheaves along ∂ σ ′ T Σ ′ 0 equal the corresponding sheaf for the pair Σ 0 (σ ′ ) and {σ ′ }, so that we are reduced to the case that Then it suffices to show that for i > 0 we have This follows from lemma 2.14. (Note that where ker α acts on the ξ term via ξ; and that These assertions remain true with |Σ 0 | ∨,0 replacing |Σ 0 | ∨ and |{σ ′ }| ∨,0 replacing |{σ ′ }| ∨ .) Lemma 2.16. Let Y be a connected scheme, let S/Y be a split torus, let T /Y be an S-torsor, let Σ 0 be a partial fan for S, and let π ∧ Σ 0 denote the map of ringed spaces T ∧ Σ 0 → Y . Suppose that Y is separated and locally noetherian, that Σ 0 is full, locally finite and open, and that |Σ 0 | 0 is convex.
Define recursively fans Σ (i) and boundary data Σ (i) 0 as follows. We set Σ (i 0 −1) = Σ 0 and Σ Then Σ (i) refines Σ (i−1) and we choose Σ 0 is locally finite. (The point being that the local finiteness of Σ (i−1) 0 implies that only finitely many elements of Σ (For the last of these properties use the fact that Σ 0 . This lemma then follows from lemmas 2.13 and 2.14.
2.5. The case of a disconnected base. Throughout this section we will continue to assume that Y is a separated scheme, flat and locally of finite type over Spec R 0 .
Let S be a split torus over Y and let T /Y be an S-torsor. By a rational polyhedral cone σ in X * (S) R we shall mean a locally constant sheaf of subsets is either empty or a rational polyhedral cone, • and the locus where σ = ∅ is non-empty and connected. We call this locus the support of σ.
• if σ, σ ′ ∈ Σ then σ ∩ σ ′ is either empty or a face of σ and σ ′ . Thus to give a fan in X * (S) R is the same as giving a fan in X It is a fan for X * (S) R (U).
We call Σ smooth (resp. full, resp. finite) if each Σ(U) is. We define a locally constant sheaf |Σ| of subsets of X * (S) R by setting for U any connected open subset of Y . We will call |Σ| (resp. |Σ| * ) convex if |Σ|(U) (resp. |Σ| * (U)) is for each connected open U ⊂ Y . We also define locally constant sheaves of subsets |Σ| ∨ and |Σ| ∨,0 of X * (S) R by setting for each open, connected U. Any fan Σ has a smooth refinement Σ ′ such that every smooth cone σ ∈ Σ also lies in Σ ′ .
To a fan Σ one can attach a scheme T Σ flat and separated over Y and locally (on T Σ ) of finite type over Y , together with an action of S and an S-equivariant embedding T ֒→ T Σ . It has an open cover {T σ } σ∈Σ , with each T σ relatively affine over Y . Over a connected open U ⊂ Y this restricts to the corresponding picture which restricts to the identity on T .
By boundary data we shall mean a proper subset If Σ 0 is boundary data, then we can associate to it a closed sub-sheme In the case that Σ 0 is the set of elements of Σ of dimension bigger than 0 we shall simply write If σ ∈ Σ has positive dimension and if Σ 0 denotes the set of elements of Σ which have σ for a face, then we will write .., σ s are all contained in some σ ∈ Σ. In this case the intersection equals ∂ σ T Σ for the smallest such σ. We set If the connected components of Y are irreducible then each ∂ σ T Σ is irreducible. Moreover the irreducible components of ∂T Σ are the ∂ σ T Σ as σ runs over one dimensional elements of Σ. If Σ is smooth then we see that S(∂T Σ ) is the delta complex with cells in bijection with the elements of Σ with dimension bigger than 0 and with the same 'face relations'. In particular it is in fact a simplicial complex and By a partial fan in X * (S) we mean a collection Σ 0 of rational polyhedral cones in X * (S) such that • Σ 0 does not contain (0) ⊂ X * (S)(U) R for any open connected U; • if σ 1 , σ 2 ∈ Σ 0 then σ 1 ∩ σ 2 is either empty or a face of σ 1 and of σ 2 ; • if σ 1 , σ 2 ∈ Σ 0 and if σ ⊃ σ 2 is a face of σ 1 , then σ ∈ Σ 0 .
In this case we will let Σ 0 denote the set of faces of elements of Σ 0 . It is a fan, and Σ 0 is boundary data for Σ 0 . By boundary data Σ 1 for Σ 0 we shall mean a subset Σ 1 ⊂ Σ 0 such that if σ ∈ Σ 0 contains σ 1 ∈ Σ 1 , then σ ∈ Σ 1 . In this case Σ 1 is again a partial fan and boundary data for Σ 0 . We say that a partial fan Σ 0 for X * (S) refines a partial fan Σ ′ 0 for X * (S) if every element of Σ 0 lies in an element of Σ ′ 0 and if every element of Σ ′ 0 is a finite union of elements of Σ 0 . If Σ 0 is a partial fan we define locally constant sheaves of subsets |Σ 0 |, |Σ 0 | * , We will call Σ 0 smooth (resp. full, resp. open, resp. finite, resp. locally finite) If Σ 0 is a partial fan we will write for the log structure induced by M Σ 0 . We make the following definitions.
for the sheaf (of abelian groups) on Y such that for any connected open subset Suppose that α : S → S ′ is a surjective map of split tori over Y . Then X * (α) : X * (S ′ ) ֒→ X * (S) and X * (α) : X * (S) R → → X * (S ′ ) R . We call fans Σ for X * (S) and Σ ′ for X * (S ′ ) compatible if for all σ ∈ Σ the image X * (α)σ is contained in some element of Σ ′ . In this case the map α : The following lemma is an immediate consequence of lemma 2.5.
. If α is an isogeny, if Σ and Σ ′ are compatible, and if every element of Σ ′ is a finite union of elements of Σ, then we call Σ a quasi-refinement of Σ ′ . In that case the map ). The following lemma follows immediately from lemma 2.8.
We will call Σ 0 and Σ ′ 0 strictly compatible if they are compatible and if an element of Σ 0 lies in Σ 0 if and only if it maps to no element of Σ ′ 0 − Σ ′ 0 . We will say that • and that Σ 0 is finite over Σ ′ 0 if only finitely many elements of Σ 0 map into any element of Σ ′ 0 . If α is an isogeny, if Σ 0 and Σ ′ 0 are strictly compatible, and if every element of Σ ′ 0 is a finite union of elements of Σ 0 , then we call Σ 0 a quasi-refinement of Σ ′ 0 . In this case Σ 0 is open and finite over Σ ′ 0 . The next lemma follows immediately from lemma 2.9.
. The next two lemmas follow immediately from lemmas 2.15 and 2.16 respectively.
Lemma 2.21. Let Y be a scheme, let α : S → S ′ be an isogeny of split tori over Y , let Σ ′ 0 (resp. Σ 0 ) be a locally finite partial fan for S ′ (resp. S). Suppose that Y is separated and locally noetherian, that Σ ′ 0 is full and that Σ 0 is locally If moreover Σ 0 and Σ ′ 0 are smooth then, for i > 0 we have Suppose that Y is separated and locally noetherian, that Σ 0 is full, locally finite and open, and that |Σ 0 | 0 is convex.
In this section we will describe the Shimura varieties associated to G n and the mixed Shimura varieties associated to G (m) n and G (m) n . We assume that all schemes discussed in this section are locally noetherian.
3.1. Some Shimura varieties. By a G n -abelian scheme over a scheme Y /Q we shall mean an abelian scheme A/Y of relative dimension n[F : Q] together with an embedding i : By a morphism (resp. quasi-isogeny) of G n -abelian schemes we mean a morphism (resp. quasiisogeny) of abelian schemes which commutes with the F -action. If (A, i) is a G nabelian scheme then we give A ∨ the structure (A ∨ , i ∨ ) of a G n -abelian scheme by setting i ∨ (a) = i(a c ) ∨ . By a quasi-polarization of a G n -abelian scheme (A, i)/Y we shall mean a quasi-isogeny λ : A → A ∨ of G n -abelian schemes, some Q ×multiple of which is a polarization. If Y = Spec k with k a field, we will let , λ denote the Weil pairing induced on the adelic Tate module V A (see section 23 of [M]).
Proof: We may suppose that k is a finitely generated field extension of Q, which we may embed into C. Then By an ordinary G n -abelian scheme over a Z (p) -scheme Y we shall mean an abelian scheme A/Y of relative dimension n[F : Q], such that for each geometric point y of Y we have #A[p](k(y)) ≥ p n[F :Q] , together with an embedding By a morphism of ordinary G n -abelian schemes we mean a morphism of abelian schemes which commutes with the O F,(p) -action. If (A, i) is an ordinary G n -abelian scheme then we give A ∨ the structure, (A ∨ , i ∨ ), of a G n -abelian scheme by setting i ∨ (a) = i(a c ) ∨ . By a prime-to-p quasi-polarization of an ordinary G n -abelian scheme (A, i)/Y we shall mean a prime-to-p quasi-isogeny λ : A → A ∨ of ordinary G n -abelian schemes, some Z × (p) -multiple of which is a prime-to-p polarization. If U is an open compact subgroup of G n (A ∞ ) then by a U-level structure on a quasi-polarized G n -abelian variety (A, i, λ) over a connected scheme Y /Spec Q with a geometric point y, we mean a π 1 (Y, y)-invariant U-orbit [η] of pairs (η 0 , η 1 ) of A ∞ -linear isomorphisms for all a ∈ F and x ∈ V n ⊗ Q A ∞ , and such that This definition is independent of the choice of geometric point y of Y . By a U-level structure on a quasi-polarized G n -abelian scheme (A, i, λ) over a general (locally noetherian) scheme Y /Spec Q, we mean the collection of a U-level structure over each connected component of Y . If [(η 0 , η 1 )] is a level structure we define ||η 0 || ∈ Q × >0 by ||η 0 ||η 0 Z = Z(1).
Now suppose that U p is an open compact subgroup of G n (A p,∞ ) and that N 1 ≤ N 2 are non-negative integers. By a U p (N 1 , N 2 )-level structure on an ordinary, prime-to-p quasi-polarized, G n -abelian scheme (A, i, λ) over a connected scheme Y /Spec Z (p) with a geometric point y, we mean a π 1 (Y, • and an isomorphism for all a ∈ O F,(p) and x ∈ p −N 1 Λ n /(p −N 1 Λ n,(n) + Λ n ).
This definition is independent of the choice of geometric point y of Y . By a U p (N 1 , N 2 )-level structure on an ordinary, prime-to-p quasi-polarized, G n -abelian scheme (A, i, λ) over a general (locally noetherian) scheme Y /Spec Z (p) , we mean the collection of a U p (N 1 , N 2 )-level structure over each connected component of )/Y is a quasi-polarized, G n -abelian scheme with U-level structure and if g ∈ G n (A ∞ ) with U ′ ⊃ g −1 Ug, then we can define a quasi-polarized, G n -abelian scheme with U ′ -level structure (A, i, λ, If (A, i, λ, [η])/S is an ordinary, prime-to-p quasi-polarized, G n -abelian scheme with U p (N 1 , N , 2), then we can define an ordinary, prime-to-p quasi-polarized, G n -abelian scheme with ( , then we can define an ordinary, prime-to-p quasi-polarized, G n -abelian scheme ) ∨ with the latter isomorphism being induced by the dual of the map A/C[p] → A induced by multiplication by p on A; where F (η p 1 ) is the composition of η p 1 with the natural map V p A ∼ → V p (A/C[p]); and where F (η p ) is the composition of η p with the natural identification ). Together these two definitions give an action of G n (A ∞ ) ord .
Lemma 3.2. Suppose that T is an O F,p -module, which is free over O F,p of rank 2n, with a perfect alternating pairing for all x, y ∈ T and a ∈ O F,p . Also suppose that T ⊂ T is a sub-O F,p -module which is isotropic for , and such that T / T is free of rank n over O F,p . Finally suppose that induced by η equals η p . Then [η] is non-empty and a single U p (N 1 , N 2 )-orbit.
Proof: Let e 1 , ..., e n denote a O F,p -basis of T / T . Note that , induces a perfect pairing between T and T / T . We recursively lift the e i to elements e i ∈ T with e i orthogonal to the O F,p span of the e j for j ≤ i. Suppose that e 1 , ..., e i−1 have already been chosen. Choose some lift e ′ i of e i . Then choose t ∈ T such that • t, x = e ′ i , x for all x ∈ i−1 j=1 O F,p e j , • and t, αe ′ i = e ′ i /2, αe ′ i for all α ∈ O c=−1 F,p . (If p = 2 some explanation is required as to why we can do this. The map Because p = 2 is unramified in F/F + , we can write β = γ − γ c for some γ ∈ D −1 F,p . Thus the second condition can be replaced by the condition t, αe ′ i = tr F/Q γα for all α ∈ O F,p . Now it is clear that the required element t exists.) Then take Thus we can write The lemma now follows without difficulty.
Proof: This follows on combining the lemmas 3.1 and 3.2.
Corollary 3.4. Suppose that Y is a scheme over Spec Q. Then there is a natural bijection between prime-to-p isogeny classes of ordinary, prime-to-p quasipolarized G n -abelian schemes with U p (N 1 , N 2 )-level structure and isogeny classes of quasi-polarized G n -abelian schemes with U p (N 1 , N 2 )-level structure. This bijection is G n (A ∞ ) ord -equivariant.
Proof: We may assume that Y is connected with geometric point y. We will show both sets are in natural bijection with the set of prime-to-p isogeny classes of 4-tuples (A, i, λ, [η]), where (A, i) is a G n -abelian variety, λ is a prime-top quasi-polarization of (A, i), and [η] is a π 1 (Y, y)-invariant U p (N 1 , N 2 )-orbit of pairs (η 0 , η 1 ), where • and η 1 : There is a natural map from this set to the set of isogeny classes of quasi-polarized G n -abelian schemes with U p (N 1 , N 2 )-level structure, which is easily checked to be a bijection. The bijection between this set and the set of prime-to-p isogeny classes of ordinary, prime-to-p quasi-polarized G n -abelian schemes with U p (N 1 , N 2 )-level structure, follows by the usual arguments (see for instance section III.1 of [HT]) from corollary 3.3.
If U is a neat open compact subgroup of G n (A ∞ ) then the functor that sends a (locally noetherian) scheme S/Q to the set of quasi-isogeny classes of polarized G n -abelian schemes with U-level structures is represented by a quasi-projective scheme X n,U which is smooth of relative dimension n 2 [F + : Q] over Q. Let [(A univ , i univ , λ univ , [η univ ])]/X n,U denote the universal equivalence class of polarized G n -abelian varieties with Ulevel structure. If U ′ ⊃ g −1 Ug then there is a map g : X n,U → X n,U ′ arising from (A univ , i univ , λ univ , [η univ ])g/X n,U and the universal property of X n,U ′ . This makes {X n,U } an inverse system of schemes with right G n (A ∞ )-action. The maps g are finite etale. If U 1 ⊂ U 2 is a normal subgroup then X n,U 1 /X n,U 2 is Galois with group U 2 /U 1 .
There are identifications of topological spaces: compatible with the right action of G n (A ∞ ). (Note that ker 1 (Q, G n ) = (0).) More precisely we associate to (g, I) ∈ G n (A ∞ )/U × H + n the torus Λ ⊗ Z R/Λ with complex structure coming from I; with polarization corresponding to the Riemann form given by , ; and with level structure coming from where ζ = lim ←N e 2πi/N ∈ Z(1). We deduce that If U p is neat then the functor that sends a scheme Y /Z (p) to the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized, G n -abelian schemes with U p (N 1 , N 2 )-level structure is represented by a scheme X ord n,U p (N 1 ,N 2 ) quasiprojective over Z (p) . Let [(A univ , i univ , λ univ , [η univ ])]/X ord n,U p (N 1 ,N 2 ) denote the universal equivalence class. If g ∈ G n (A ∞ ) ord and (U p ) ′ (N ′ 1 , N ′ 2 ) ⊃ g −1 U p (N 1 , N 2 )g, then there is a quasi-finite map arising from (A univ , i univ , λ univ , [η univ ])g/X n,U p (N 1 ,N 2 ) and the universal property of X ord (By the Serre-Tate theorem (see [Ka2]) the formal completion of X ord n,U p (N 1 ,N 2 ) at a point x in the special fibre is isomorphic to This is formally smooth as long as S (T p is torsion free. This module is torsion free because in the case p = 2 we are assuming that p = 2 is unram- is in fact flat, because the it is a quasi-finite map between locally noetherian regular schemes which are equi-dimensional of the same dimension. (See pages 507 and 508 of [KM].) On F p -fibres the map ς p X ord n,U p (N 1 ,N 2 +1) × Spec F p −→ X ord n,U p (N 1 ,N 2 ) × Spec F p is the absolute Frobenius map composed with the forgetful map 1 : X ord n,U p (N 1 ,N 2 ) → X ord n,U p (N 1 ,N 2 −1) (for any N 2 ≥ N 1 ≥ 0). Thus if N 2 > 0, then the quasi-finite, flat map has all its fibres of degree p n 2 [F + :Q] and hence is finite flat of this degree. (A flat, quasi-finite morphism f : X → Y between noetherian schemes with constant fibre degree is proper and hence, by theorem 8.11.1 of [EGA4], finite. We give the argument for properness. By the valuative criterion we may reduce to the case Y = Spec B for a DVR B with fraction field L.
is finite.
is an isomorphism.
Proof: The map has an inverse which sends [(A univ , i univ , λ univ , [η univ ])] over . Thus we will denote X ord,∧ n,U p (N 1 ,N 2 ) simply X ord n,U p (N 1 ) . Then {X ord n,U p (N ) } is a system of p-adic formal schemes with right G n (A ∞ ) ord -action. We will write X ord n,U p (N ) for the reduced sub-scheme of X ord n,U p (N ) .

Some Kuga-Sato varieties.
Recall that a semi-abelian scheme is a smooth separated commutative group scheme such that each geometric fibre is the extension of an abelian variety by a torus. To an semi-abelian scheme G/Y one can associate an etale constructible sheaf of abelian groups X * (G), the 'character group of the toric part of G'. See theorem I.2.10 of [CF]. If X * (G) is constant then G is an extension of a uniquely determined abelian scheme A G by a uniquely determined split torus S G . By an isogeny (resp. prime-to-p isogeny) of semi-abelian schemes we mean a morphism which is quasi-finite and surjective (resp. quasi-finite and surjective and whose geometric fibres have orders relatively prime to p). If Y is locally noetherian, then by a quasi-isogeny (resp. prime-to-p quasi-isogeny) α : G → G ′ we mean an element of Hom (G, G ′ ) Q (resp. Hom (G, G ′ ) Z (p) ) with an inverse in Hom (G ′ , G) Q (resp. Hom (G ′ , G) Z (p) ).
Suppose that Y /Spec Q is a locally noetherian scheme. By a G (m) Then A G is a G n -abelian scheme. By a quasi-isogeny of G (m) n -semi-abelian schemes we mean a quasi-isogeny of semi-abelian schemes for all a ∈ F , and j = X * (β) • j ′ . Note that, if y is a geometric point of Y , then j induces a map By a quasi-polarization of (G, i, j) we mean a quasi-polarization of A G .
If Y is connected and y is a geometric point of Y and if U ⊂ G n -semi-abelian scheme (G, i, j, λ) we mean a π 1 (Y, y)-invariant U-orbit of pairs (η 0 , η 1 ) where η 0 : A ∞ ∼ −→ V G m,y is an A ∞ -linear map, and where and [(η 0 , η 1 mod V S G,y )] is a U-level structure on A G . This is canonically independent of y. We define a U level structure on a G (m) n -semi-abelian scheme over a general locally noetherian scheme Y to be such a level structure over each connected component of Y . By a quasi-isogeny between two quasi-polarized, G (m) n -semi-abelian schemes with Ulevel structure and an element δ ∈ Q × such that .
• A (0) n,U = X n,U . (We will sometimes write π A (m) n /Xn for π A (m) n .) This identification is G n (A ∞ )-equivariant.
• The maps g and γ are finite etale. The maps π m.m ′ are smooth and projective. n,U 2 is Galois with group U 2 /U 1 .
n,U /X n,U ′ is an abelian scheme of relative dimension mn[F : Q].

• In general
n,U and A univ /X n,U ′ are chosen so that π * A univ depends only on A univ and not on G univ .
• If e 1 , ..., e m denotes the standard basis of F m then is an isomorphism, which does not depend on the choice of G univ . It satisfies η   [La4]; and section 3.5 of [La3].) Note that A univ (y). This does not depend on the choice of A univ . We have Now suppose that Y /Spec Z (p) is a locally noetherian scheme. By an ordinary G (m) n -semi-abelian scheme G over Y we mean a triple (G, i, j) where • G/Y is a semi-abelian scheme such that #G[p](k(y)) ≥ p n[F :Q] for each geometric point y of Y , Then A G is an ordinary G n -abelian scheme. By a prime-to-p quasi-isogeny of ordinary G (m) n -semi-abelian schemes we mean a prime-to-p quasi-isogeny of semiabelian schemes . By a prime-to-p quasi-polarization of (G, i, j) we shall mean a prime-to-p quasi- is a neat open compact subgroup, and if N 2 ≥ N 1 ≥ 0 then by a U p (N 1 , N 2 ) level structure on a prime-to-p quasi-polarized ordinary G (m) n -semi-abelian scheme (G, i, j, λ) we mean a π 1 (Y, y)-invariant U p -orbit [η] of five-tuples (η p 0 , η p 1 , C, D, η p ) consisting of • and an isomorphism . This definition is independent of the choice of geometric point y of Y . By a U p (N 1 , N 2 )-level structure on an ordinary, prime-to-p quasi-polarized, G (m) n -semi-abelian scheme (G, i, j, λ) over a general (locally noetherian) scheme Y /Spec Z (p) , we mean the collection of a U p (N 1 , N 2 )level structure over each connected component of Y .

(m)
A univ depends only on A univ and not on G univ .
is a prime-to-p quasi-isogeny Here λ(N 1 ) univ refers to the prime-to-p quasi-polarization n,U , where the first maps sends f −→ (η univ (f (e 1 )), ..., η univ (f (e m ))) and the third map sends is an isomorphism, which does not depend on the choice of G univ . It satisfies η (m)  [La4]. ) We deduce the following additional properties: • If g ∈ G is finite. If N 2 > 0 then the finite flat map A univ (y). This does not depend on the choice of A univ . We have (p) ) t=c all whose eigenvalues are positive real numbers. Thus i n (A p,∞ )) we will denote by S (m) n, U (resp. S (m),ord n, U p ) the split torus over Spec Q (resp. Spec Z (p) ) with where we think of the domain and codomain both as subspaces of Herm (m) . If γ ∈ GL m (Q) (resp. GL m (Z (p) )) and U 2 ⊃ γ U 1 (resp. U p 2 ⊃ γ U p 1 ) we get a map where again we think of the domain and codomain both as subspaces of Herm (m) . If m 1 ≥ m 2 and if U 2 (resp. U p 2 ) is the image of U 1 (resp. U p 1 ) in G These enjoy the following properties: • g 1 • g 2 = g 2 g 1 (i.e. this is a right action) and γ 1 • γ 2 = γ 1 γ 2 (i.e. this is a left action) and γ • g = γ(g) • γ.
n, U ) ⊂ S(F m ) is sufficiently divisible. Then we can can find a ∈ F m ⊗ F,c F m lifting χ such that i   n (A p,∞ ) and N 1 , N 2 run over integers with N 2 ≥ N 1 ≥ 0, there is a system of S • If m 1 ≥ m 2 and U p 2 is the image of U p 1 in G (m 2 ) (A p,∞ ), then there is a map T (m 1 ),ord n, U p compatible with the maps S  (N 1 ,N 2 ) . These enjoy the following properties: • g 1 • g 2 = g 2 g 1 (i.e. this is a right action) and γ 1 • γ 2 = γ 1 γ 2 (i.e. this is a left action) and γ • g = γ(g) • γ. • If g ∈ G  (N 11 ,N n, U p (N 1 ,N 2 ) × Spec F p equals the composition of the absolute Frobenius map with the forgetful map (for any n (A p,∞ ). Also suppose that χ ∈ X * (S .  3.4. Vector bundles. Suppose that U is a neat open compact subgroup of G n (A ∞ ). We will let Ω n,U denote the pull back by the identity section of the sheaf of relative differentials Ω 1 A univ /X n,U . This is a locally free sheaf of rank n[F : Q]. Up to unique isomorphism its definition does not depend on the choice of A univ . (Because, by the neatness of U, there is a unique isogeny between any two universal four-tuples (A univ , i univ , λ univ , [η univ ]).) The system of sheaves {Ω n,U } has an action of G n (A ∞ ). There is a natural isomorphism between Ω 1 A univ /X n,U and the pull back of Ω n,U from X n,U to A univ .
Similarly, if π : A univ → X n,U is the structural map, then the sheaf is locally free and canonically independent of the choice of A univ . These sheaves again have an action of G n (A ∞ ). We will also write Ξ n,U = O X n,U (||ν||) for the sheaf O X n,U but with the G n (A ∞ )action multiplied by ||ν||.
These maps do not depend on the choice of A univ and are G n (A ∞ )-equivariant. They further induce G n (A ∞ )-equivariant isomorphisms S(Ω n,U ) ∼ −→ Ω 1 X n,U ⊗ Ξ n,U , which again do not depend on the choice of A univ . (See for instance propositions 2.1.7.3 and 2.3.5.2 of [La1]. This is referred to as the 'Kodaira-Spencer isomorphism'.) Let E U denote the principal L n,(n) -bundle on X n,U in the Zariski topology defined by setting, for W ⊂ X n,U a Zariski open, E U (W ) to be the set of pairs The inverse system {E U } has an action of G n (A ∞ ).
Suppose that R 0 is an irreducible noetherian Q-algebra and that ρ is a representation of L n,(n) on a finite, locally free R-module W ρ . We define a locally free sheaf E U,ρ over X n,U × Spec R 0 by setting E U,ρ (W ) to be the set of L n,(n) (O W )equivariant maps of Zariski sheaves of sets Then {E U,ρ } is a system of locally free sheaves with G n (A ∞ )-action over the system of schemes {X n,U × Spec R 0 }. If g ∈ G n (A ∞ ), then the natural map g * E U,ρ −→ E U ′ ,ρ is an isomorphism.
In the case R 0 = C, the holomorphic vector bundle on X n,U (C) associated to which is equivariant for the actions of G We define the R U,ρ } is a system of locally free sheaves with both G U,ρ is canonically isomorphic to the pull-back of E U,ρ from X n,U . In general W ρ has a filtration by R n (A ∞ ) and GL m (F ) invariant filtration by local direct summands such that each graded piece is the pull back of some E U,ρ ′ from X n,U .
Similarly suppose that U p is a neat open compact subgroup of G n (A p,∞ ), and that N 2 ≥ N 1 ≥ 0 are integers. We will let Ω ord n,U p (N 1 ,N 2 ) denote the pull back by the identity section of Ω 1 A univ /X ord n,U p (N 1 ,N 2 ) . This is a locally free sheaf of rank n[F : Q]. Up to unique isomorphism its definition does not depend on the choice of A univ . (Because, by the neatness of U p , there is a unique prime-to-p isogeny between any two universal four-tuples (A univ , i univ , λ univ , [η univ ]).) The system of sheaves {Ω ord n,U p (N 1 ,N 2 ) } has an action of G n (A ∞ ) ord . There is a natural isomorphism between Ω 1 A univ /X ord n,U p (N 1 ,N 2 ) and the pull back of Ω ord n,U p (N 1 ,N 2 ) . We will also write Ξ n,U p (N 1 ,N 2 ) = O X ord n,U p (N 1 ,N 2 ) (||ν||) for the sheaf O X ord n,U p (N 1 ,N 2 ) but with the G n (A ∞ ) ord -action multiplied by ||ν||.
The line bundle (1, λ univ ) * P A univ is represented by an element of N 2 ) ).
These maps do not depend on the choice of A univ and are G n (A ∞ ) ord -equivariant. They further induce G n (A ∞ ) ord isomorphisms which again do not depend on the choice of A univ . (See for instance proposition 3.4.3.3 of [La4].) Let E ord U p (N 1 ,N 2 ) denote the principal L n,(n) -bundle on X ord n,U p (N 1 ,N 2 ) in the Zariski topology defined by setting, for W ⊂ X ord n,U p (N 1 ,N 2 ) ) a Zariski open, E ord U p (N 1 ,N 2 ) (W ) to be the set of pairs (ξ 0 , ξ 1 ), where We define the L n,(n) -action on E ord

The inverse system {E ord
U p (N 1 ,N 2 ) } has an action of G n (A ∞ ) ord . Suppose that R 0 is an irreducible noetherian Z (p) -algebra and that ρ is a representation of L n,(n) on a finite, locally free R-module W ρ . We define a locally free sheaf E ord U p (N 1 ,N 2 ),ρ over X ord n,U p (N 1 ,N 2 ) × Spec R 0 by setting E ord U p (N 1 ,N 2 ),ρ (W ) to be the set of L n,(n) (O W )-equivariant maps of Zariski sheaves of sets N 2 ),ρ } is a system of locally free sheaves with G n (A ∞ ) ord -action over the system of schemes {X ord n,U p (N 1 ,N 2 ),∆ × Spec R 0 }. The pull-back of E ord .
Suppose now that U p is a neat open compact subgroup of G (m) n (A p,∞ ) with image (U p ) ′ in G n (A p,∞ ). We will let Ω (m),ord n,U p (N 1 ,N 2 ) denote the pull back by the identity section of the sheaf of relative differentials Ω 1 G univ /A (m),ord n,U p (N 1 ,N 2 ) . This is a locally free sheaf of rank (n + m)[F : Q]. Up to unique isomorphism its definition does not depend on the choice of G univ . The system of sheaves {Ω which is equivariant for the actions of G  (W ) to be the set of pairs (ξ 0 , ξ 1 ), where and induces the canonical isomorphism Suppose that R 0 is an irreducible noetherian Z (p) -algebra and that ρ is a representation of R (m) n,(n) on a finite, locally free R 0 -module W ρ . We define a locally free sheaf E (m),ord N 2 ),ρ } is a system of locally free sheaves with G  N 2 ),ρ is canonically isomorphic to the pull-back of E ord U p (N 1 ,N 2 ),ρ from X ord n,U p (N 1 ,N 2 ) . In general W ρ has a filtration by R (m) n,(n) -invariant local direct-summands such that the action of R (m) n,(n) on each graded piece factors through L n,(n) . Thus E (m),ord U p (N 1 ,N 2 ),ρ has a G (m) n (A ∞ ) and GL m (O F,(p) ) invariant filtration by local direct summands such that each graded piece is the pull back of some E ord

The inverse system {E
depends only on U ′ and not on U. We will denote it . This isomorphism only depends on g ′ , U ′ 1 and U ′ 2 and not on g, U 1 and U 2 . This gives the system of sheaves {(R j π * Ω i n,γU gives a natural isomorphism which depends only on U ′ and not on U. This gives the system of sheaves is an isomorphism. These isomorphisms are equivariant for the actions of the groups G n,U ′ gives rise to an isomorphism and a canonical embeddding where the first map denotes the diagonal embedding. These maps do not depend on the choice of A univ . They are G This gives rise to canonical G

Moreover the composite maps
Hom ((π * Ω 1 Next we turn to the mixed characteristic case. If m ≥ m ′ and if U p is a neat open compact subgroup of G (m) N 1 ,N 2 ) depends only on (U p ) ′ and not on U p . We will denote it N 1 ,N 2 ) . N 22 ), then there is a natural map n (A ∞ ) ord,× then it is an isomorphism. Moreover this map only depends on g ′ , (U p 1 ) ′ (N 11 , N 12 ) and (U p 2 ) ′ (N 21 , N 22 ) and not on g, U p 1 (N 11 , N 12 ) and U p 2 (N 21 , N 22 ). This gives the system of sheaves n,γU p (N 1 ,N 2 ) gives a natural isomorphism which depends only on (U p ) ′ (N 1 , N 2 ) and not on U p (N 1 , N 2 ). This gives the system of sheaves N 2 ) is an isomorphism. These isomorphisms are G The natural maps N 2 ) ) are naturally identified with the sheaves Ω n,(U p ) ′ (N 1 ,N 2 ) (resp. Ξ n,(U p ) ′ (N 1 ,N 2 ) ). Moreover, under the identification N 1 ,N 2 ) . These identifications are equivariant for the actions of G n (A ∞ ) ord and Q m,(m−m ′ ) (O F,(p) ).

Generalized Shimura Varieties
We will introduce certain disjoint unions of mixed Shimura varieties, which are associated to L n,(i),lin and L n,(i) and P + n,(i) /Z(N n,(i) ) and P + n,(i) ; to L n,(i) ) and P (m),+ n,(i) ; and to P (m),+ n,(i) . The differences with the last section are purely book keeping. We then describe certain torus embeddings for these generalized Shimura varieties and discuss their completion along the boundary. These completions will serve as local models near the boundary of the toroidal compactifications of the X n,U and the A is an isomorphism, and GL m (F ) acts trivially on this space. (Use the fact that In the case m = 0 we drop it from the notation. Each Y The fibre over g ∈ L n,(i), . We define sheaves Ω + n,(i),U and Ξ + n,(i),U over X + n,(i),U as the quotients of We define the R n,(n),(i) /N(R n,(n),(i) )-action on E + (i),U by h(ξ 0 , ξ 1 ) = (ν(h) −1 ξ 0 , (•h −1 ) • ξ 1 ).
The inverse system {E + (i),U } has an action of L n,(i) (A ∞ ) and of L n,(i),lin (Q). Suppose that R 0 is an irreducible noetherian Q-algebra and that ρ is a representation of R n,(n),(i) on a finite, locally free R 0 -module W ρ . We define a locally free sheaf E + (i),U,ρ over X + n,(i),U × Spec R 0 by setting E + (i),U,ρ (W ) to be the set of (R (m) n,(n) /N(R n,(n),(i) ))(O W )-equivariant maps of Zariski sheaves of sets ,U,ρ } is a system of locally free sheaves with L n,(i) (A ∞ )-action and L n,(i),lin (Q)-action over the system of schemes . However the description of the actions of L n,(i) (A ∞ ) and L n,(i),lin (Q) involve ρ and not just ρ| L n−i,(n−i) . If g ∈ L n,(i) (A ∞ ) and γ ∈ L n,(i),lin (Q), then the natural maps We will also write Ω ♮ n,(i),U = L n,(i),lin (Q)\Ω + In the case m = 0 we drop it from the notation. Each X n,(i),lin (Z (p) )\X (m),ord,+ n,(i),U p (N 1 ,N 2 ) . The system of these spaces has a right action of L (m) n,(i) (A ∞ ) ord and a left action of N 2 ) . We define sheaves Ω ord,+ n,(i),U p (N 1 ,N 2 ) and Ξ ord,+ n,(i),U p (N 1 ,N 2 ) over X ord,+ n,(i),U p (N 1 ,N 2 ) as the quotients of n,(i),(U p ∩L n,(i),lin (A p,∞ ))(N 1 ) by U p . Then the systems of sheaves Ω ord,+ n,(i),U p (N 1 ,N 2 ) and Ξ ord,+ n,(i),U p (N 1 ,N 2 ) have commuting actions of L n,(i),lin (Z (p) ) and L n,(i) (A ∞ ) ord .
The inverse system {E + (i),U p (N 1 ,N 2 ) } has an action of L n,(i) (A ∞ ) ord and an action of L n,(i),lin (Z (p) ).
In the case m = 0 we will write simply A + n,(i),U . If g ∈ (P to be the coproduct of the maps as the coproduct of the maps This gives a left GL m (F )-action on the system of the A n,(i),lin (Q), so we don't do so.
We define a semi-abelian scheme G univ /A + n,(i),U by requiring that over the open and closed subscheme A it restricts to G univ . It is unique up to unique quasi-isomorphism. We also define a sheaf Ω + n,(i),U (resp. Ξ + n,(i),U ) over A + n,(i),U to be the unique sheaf which, for each h, restricts to Ω . Thus Ω + n,(i),U is the pull back by the identity section of Ω 1 G univ /A + n,(i),U . Then { Ω + n,(i),U } (resp. { Ξ + n,(i),U }) is a system of locally free sheaves on A + n,(i),U with a left (P + n,(i) /Z(N n,(i) ))(A ∞ )-action and a commuting right L n,(i),lin (Q)-action. There are equivariant exact sequences where π denotes the map A + n,(i),U → X + n,(i),U . Let E + (i),U denote the principal R n,(n),(i) -bundle on A + n,(i),U in the Zariski topology defined by setting, for W ⊂ A + n,(i),U a Zariski open, E + (i),U (W ) to be the set of pairs (ξ 0 , ξ 1 ), where We define the R n,(n),(i) -action on E + (i),U by h(ξ 0 , ξ 1 ) = (ν(h) −1 ξ 0 , (•h −1 ) • ξ 1 ).
The inverse system { E + (i),U } has an action of P + n,(i) (A ∞ ) and of L n,(i),lin (Q).
Suppose that R 0 is an irreducible noetherian Q-algebra and that ρ is a representation of R n,(n),(i) on a finite, locally free R 0 -module W ρ . We define a locally free sheaf E + (i),U,ρ over A + n,(i),U × Spec R 0 by setting E + (i),U,ρ (W ) to be the set of R ,U,ρ } is a system of locally free sheaves with P + n,(i) (A ∞ )-action and L n,(i),lin (Q)-action over the system of schemes .
Suppose that R 0 is an irreducible noetherian Q-algebra and that ρ is a representation of R n,(n),(i) on a finite, locally free R 0 -module W ρ . We define a locally free sheaf E ord,+ (i),U p (N 1 ,N 2 ),ρ over A ord,+ n,(i),U p (N 1 ,N 2 ) × Spec R 0 by setting E ord,+ (i),U p (N 1 ,N 2 ),ρ (W ) to be the set of R  N 2 ),ρ } is a system of locally free sheaves with P + n,(i) (A ∞ ) ord -action and L n,(i),lin (Z (p) )-action over the system of schemes {A ord,+ n,(i), can be identi- . However the description of the actions of P + n,(i) (A ∞ ) ord and L n,(i),lin (Z (p) ) involve ρ and not just ρ| R (i) . If g ∈ P + n,(i) (A ∞ ) ord,× and γ ∈ L n,(i),lin (Z (p) ), then the natural maps are isomorphisms. If ρ factors through R n,(n),(i) /N(R n,(n),(i) ) then E ord,+ (i),U p (N 1 ,N 2 ),ρ is canonically isomorphic to the pull-back of E ord,+ (i),U p (N 1 ,N 2 ),ρ from X ord,+ n,(i),U p (N 1 ,N 2 ) . In general W ρ has a filtration by R n,(n),(i) -invariant local direct-summands such that the action of R n,(n),(i) on each graded piece factors through R n,(n),(i) /N(R n,(n),(i) ). Thus E ord,+ (i),U p (N 1 ,N 2 ),ρ has a P + n,(i) (A ∞ ) ord and L n,(i),lin (Z (p) ) invariant filtration by local direct summands such that each graded piece is the pull back of some E ord,+ (i),U p (N 1 ,N 2 ),ρ ′ from X ord,+ n,(i),U p (N 1 ,N 2 ) . The next lemma follows from the discussion in section 3.4.
If g ∈ P (m),+ n,(i) (A ∞ ) and g −1 U g ⊂ U ′ , then we define to be the coproduct of the maps n,(i), U } a system of relative tori with right P to be the coproduct of the maps .

This gives a left action of L
to be the coproduct of the maps n,(i),lin (A ∞ ) ord and g ′ ∈ G is a subsheaf of X * ( S n,(i) )(R)). We will also write X * (S ).
Similarly we will write X * (S (m),ord,+ n,(i), U p (N ) ) ≥0 R (resp. X * (S (m),ord,+ n,(i), U p (N ) ) >0 R ) for the subsheaves (of monoids) of X * (S (m),ord,+ n,(i), U p (N ) ) R corresponding to C n,(i) )(R)). Again we will write X * (S (m),ord,+ n,(i), U p (N ) ) ≥0 R (resp. X * (S (m),ord,+ n,(i), U p (N ) ) ≥0 ) for the subsheaves (of monoids) of X * (S (m),ord,+ n,(i), U p (N ) ) R (resp. X * (S (m),ord,+ n,(i), U p (N ) )) consisting of sections that have non-negative pairing with each section of X * (S (m),ord,+ n,(i), U p (N ) ) >0 R . We will also write X * (S . to be the coproduct of the maps to be the coproduct of the maps   N 2 ) , which only depends on U p (and not U p ) and N 1 , N 2 , and which we will denote T (m),ord,+ n,(i),U p (N 1 ,N 2 ) . In the case m = 0 we will write simply T ord,+ n,(i),U p (N 1 ,N 2 ) . Note that T n,(i),lin (Z (p) ) and g ∈ P (m),+ n,(i) (A ∞ ) ord , then γ followed by δ equals δ followed by δγδ −1 , and g followed by δ equals δ followed by δgδ −1 . These actions are also all compatible with the actions on { S  n,(i), U . Suppose that R 0 is an irreducible, noetherian Q-algebra. Suppose also that U is a neat open compact subgroup of P + n,(i) (A ∞ ). If a is a global section of X * (S + n,(i),U ) >0 then L + U (a) is relatively ample for A + n,(i),U /X + n,(i),U . If π + denotes the map A + n,(i),U × Spec R 0 −→ X + n,(i),U × Spec R 0 , then we see that R i π + * L + U (a) = (0) for i > 0. (Because A + n,(i),U /X + n,(i),U is a torsor for an abelian scheme and L + U (a) is relatively ample for this morphism.) We will denote by (π A + /X + , * L) + U (a) the image π + * L + U (a). Suppose further that F is a locally free sheaf on X + n,(i),U ×Spec R 0 with L n,(i),lin (Q)-action. If a ♮ is a section of X * (S + n,(i),U ) >0,♮ we will define (π A + /X ♮ , * L ⊗ F ) + U (a ♮ ) as follows: Over a point y ♮ of Y ♮ n,(i),U we take the sheaf y,a (π A + /X + , * L) + U (a) y ⊗ F y over X ♮ n,(i),U,y ♮ × Spec R 0 , where y runs over points of Y + n,(i),U above y ♮ and a runs over sections of X * (S + n,(i),U ) y above a ♮ . It is a sheaf with an action of L n,(i),lin (Q). Lemma 4.2. Keep the notation and assumptions of the previous paragraph.

It is a S
(1) Proof: For the first part note that if y in Y + n,(i),U and if a ∈ X * (S + n,(i),U ) y then the stabilizer of a in {γ ∈ L n,(i) (Q) : γy = y} is finite, and that if U is neat then it is trivial. The second part follows from the observations of the previous paragraph together with proposition 0.13.3.1 of [EGA3].
Lemma 4.12. Suppose that U p is a neat open compact subgroup of P (m),+ n,(i) (A p,∞ ) and let (U p ) ′ denote the image of U p in P + n,(i) (A p,∞ ). Also suppose that N 2 ≥ N 1 ≥ 0 are integers. Let ∆ 0 be a smooth admissible cone decomposition for X * (S ord,+ n,(i),(U p ) ′ (N 1 ,N 2 ) ) and let Σ 0 be a compatible smooth admissible cone decomposition for X * (S (m),ord,+ n,(i),U p (N 1 ,N 2 ) ).
is an isomorphism.
We finish this section with an important vanishing result.
Proof: We will treat the case of T ♮,∧ n,(i),U,∆ 0 × Spf R 0 , the other case being exactly similar. We can immediately reduce to the case that the pull back to T +,∧ n,(i),U,∆ 0 × Spf R 0 of E is L n,(i),lin (Q)-equivariantly isomorphic to the pull back to T +,∧ n,(i),U,∆ 0 Spf R 0 of a locally free sheaf F with L n,(i),lin (Q)-action over X + n,(i),U × Spec R 0 .
Also suppose that R 0 is an irreducible noetherian Q-algebra (resp. Z (p) -algebra) with the discrete topology and that ρ is a representation of R n,(n),(i) on a finite locally free R 0 -module.

Compactification of Shimura Varieties.
We now turn to the compactification of the X n,U and the A : X n,U ֒→ X min n,U . These schemes are referred to as the minimal (or sometimes 'Baily-Borel') compactifications. (The introduction to [Pi] asserts that the scheme X min n,U is the minimal normal compactification of X n,U , although we won't need this fact.) For g ∈ G n (A ∞ ) and g −1 Ug ⊂ U ′ the maps There is a family of closed sub-schemes If g ∈ G n (A ∞ ) and if g −1 Ug ⊂ U ′ then the map is the coproduct of the maps where hg = g ′ h ′ with g ′ ∈ P + n,(i) (A ∞ ). We will write X min,∧ n,U,i for the completion of X min n,U along ∂ 0 i X min n,U . (See theorem 7.2.4.1 and proposition 7.2.5.1 of [La1].) There is also a canonically defined system of normal quasi-projective schemes with G n (A ∞ ) ord -action, {X ord,min n,U p (N 1 ,N 2 ) /Spec Z (p) }, together with compatible, dense open embeddings j min U p (N 1 ,N 2 ) : X ord n,U p (N 1 ,N 2 ) ֒→ X ord,min n,U p (N 1 ,N 2 ) , which are G n (A ∞ ) ord -equivariant. Suppose that g ∈ G n (A ∞ ) ord and that is quasi-finite. If p N 2 −N ′ 2 ν(g) ∈ Z × p and either N ′ 2 = N 2 or N ′ 2 > 0, then it is also finite. On F p fibres ς p acts as absolute Frobenius composed with the forgetful map. (See theorem 6.2.1.1, proposition 6.2.2.1 and corollary 6.2.2.9 of [La4].) Write ∂X ord,min n,U p (N 1 ,N 2 ) = X ord,min n,U p (N 1 ,N 2 ) − X ord n,U p (N 1 ,N 2 ) . There is a family of closed sub-schemes {∂ 0 i X ord,min n,U p (N 1 ,N 2 ) } are families of schemes with G n (A ∞ ) ord -action. We will write X ord,min,∧ n,U p (N 1 ,N 2 ),i for the completion of X ord,min n,U p (N 1 ,N 2 ) along ∂ 0 i X ord,min n,U p (N 1 ,N 2 ) . We have a decomposition ∂ 0 i X ord,min n,U p (N 1 , , where the second coproduct runs over h ∈ (P + n,(i) (A ∞ )\G n (A ∞ )/U p (N 1 , N 2 )) − (P + n,(i) (A ∞ ) ord,× \G n (A ∞ ) ord,× /U p (N 1 )). (Again see theorems 6.2.1.1 and proposition 6.2.2.1 of [La4].) [We explain why the map N 2 ) n , or even that (P + n,(i) ∩ P + n,(n) )(Z/p N 2 Z)\P + n,(n) (Z/p N 2 Z)/V ֒→ P + n,(i) (Z/p N 2 Z)\G n (Z/p N 2 Z)/V, where V = ker(P + n,(n) (Z/p N 2 Z) → L n,(n),lin (Z/p N 1 Z)). This is clear.] is the coproduct of the maps where hg = g ′ h ′ with g ′ ∈ P + n,(i) (A ∞ ) ord , and of the maps (Again see theorems 6.2.1.1 and proposition 6.2.2.1 of [La4].) If N ′ 2 ≥ N 2 ≥ N 1 then the natural map X ord,min is etale in a Zariski neighborhood of the F p -fibre, and the natural map is the coproduct of the maps In particular ς p acts as absolute Frobenius.
The schemes X ord,min n,U p (N 1 ,N 2 ) are not proper. There are proper integral models of the schemes X min n,U , but we have less control over them. More specifically suppose that U ⊂ G n (A p,∞ ×Z p ) is an open compact subgroup whose projection to G n (A p,∞ ) is neat. Then there is a normal, projective, flat Z (p) -scheme X min n,U with generic fibre X min n,U . If g ∈ G n (A p,∞ × Z p ) and if g −1 Ug ⊂ U ′ then there is a map g : X min n,U −→ X min n,U ′ extending the map g : X min n,U → X min n,U ′ . This gives the system {X min n,U } an action of G n (A p,∞ × Z p ). We set X min n,U = X min n,U × Z (p) F p . On X min n,U there is an ample line bundle ω U , and the system of line bundles {ω U } over {X min n,U } has an action of G n (A p,∞ × Z p ). The pull back of ω U to X n,U is G n (A p,∞ × Z p )-equivariantly identified with ∧ n[F :Q] Ω n,U .(See propositions 2.2.1.2 and 2.2.3.1 of [La4].) Moreover there are canonical sections such that g * Hasse U ′ = Hasse U whenever g ∈ G n (A p,∞ × Z p ) and U ′ ⊃ g −1 Ug. We will write X min,n-ord n,U for the zero locus in X min n,U of Hasse U . (See corollaries 6.3.1.7 and 6.3.1.8 of [La4].) Then X min n,U − X min,n-ord n,U is relatively affine over X min n,U associated to the sheaf of algebras for any a ∈ Z >0 . It is also affine over F p associated to the algebra The induced map on F p -fibres is an open and closed embedding X ord,min n,U p (N 1 ,N 2 ) ֒→ X min n,U p (N 1 ,N 2 ) − X min,n-ord n,U p (N 1 ,N 2 ) . (See proposition 6.3.2.2 of [La4].) In the case N 1 = N 2 = 0 this is in fact an isomorphism. (See lemmas 6.3.2.7 and 6.3.2.9 of [La4].) We remark that for N 2 > 0 this map is not an isomorphism. the definition of X ord n,U p (N 1 ,N 2 ) requires not only that the universal abelian variety is ordinary, the condition that defines X n,U p (N 1 ,N 2 ) − X min,n-ord n,U p (N 1 ,N 2 ) , but also that the universal subgroup of A univ [p N 2 ] is connected.
Also the pull back of  ≻0 W for some isotropic subspace W ⊂ V n and some (g, δ) ∈ G (m) n (A ∞ ) × π 0 (G n (R)) and is the set of R ≥0 -linear combinations of a finite set of elements of Herm V /W ⊥ × W m ; (2) if σ ∈ Σ then any face of σ also lies in Σ; (3) if σ, σ ′ ∈ Σ then either σ ∩ σ ′ = ∅ or σ ∩ σ ′ is a face of σ and σ ′ ; (acting only on the first factor); We remark that different authors use the term 'U-admissible cone decomposition' in somewhat different ways.
We call Σ ′ a refinement of Σ if every element of Σ is a union of elements of Σ ′ . We define a partial order on the set of pairs (U, Σ), where U ⊂ G if and only if U ′ ⊂ U and Σ ′ is a refinement of Σ. If g ∈ G (m) n (A ∞ ) and Σ is a U-admissible cone decomposition of G (m) There is a natural projection and ∆ of G n (A ∞ ) × π 0 (G n (R)) × C compatible if the image of every σ ∈ Σ is contained in an element of ∆. If in addition Σ is U-admissible, ∆ is U ′ -admissible and U ′ contains the image of U in G n (A ∞ ) we will say that (U, Σ) and (U ′ , ∆) are compatible and write (U, Σ) ≥ (U ′ , ∆ ′ ).
be an open compact subgroup and let N ≥ 0 be an integer and consider U p (N) ⊂ G (m) n (A ∞ ) ord,× . By a U p (N)-admissible cone decomposition) Σ of (G (m) n (A ∞ ) × π 0 (G n (R)) × C (m) ) ord we shall mean a set of closed subsets σ ⊂ (G ≻0 W for some isotropic subspace W ⊂ V n and some (g, δ) ∈ G (m) n (A ∞ ) × π 0 (G n (R)) and is the set of R ≥0 -linear combinations of a finite set of elements of Herm V /W ⊥ × W m ; (2) if σ ∈ Σ then any face of σ also lies in Σ; (3) if σ, σ ′ ∈ Σ then either σ ∩ σ ′ = ∅ or σ ∩ σ ′ is a face of σ and σ ′ ; n (A ∞ ) × π 0 (G n (R)) × C (m) ) ord , then γσu ∈ Σ; (6) there is a finite subset of Σ such that any element of Σ has the form γσu with γ ∈ G (m) n (Q) and u ∈ U p (N, N) and σ in the given finite subset.
for some open compact subgroup U p and for some N.
If Σ is a U p (N 1 , N 2 )-admissible cone decomposition of G (m) We call Σ ′ a refinement of Σ if every element of Σ is a union of elements of Σ ′ . We define a partial order on the set of pairs (U p is an open compact subgroup, N ∈ Z ≥0 and Σ is a U p (N)-admissible cone decomposition of (G (m) n (A ∞ ) × π 0 (G n (R)) × C (m) ) ord , as follows: we set There is a natural projection We will call admissible cone decompositions Σ of (G (m) n (A ∞ )×π 0 (G n (R))×C (m) ) ord and ∆ of (G n (A ∞ ) × π 0 (G n (R)) × C) ord compatible if the image of every σ ∈ Σ is contained in an element of ∆. If in addition Σ is U p (N)-admissible, ∆ is (U p ) ′ (N ′ )-admissible and (U p ) ′ (N ′ ) contains the image of U p (N) in G n (A ∞ ) ord we will say that (U p (N), Σ) and ((U p ) ′ (N ′ ), ∆) are compatible and write (U p  ) ≻0 R as follows: The cones in Σ(h) 0 over an element This does not depend on the representative h ′ we choose for y. It also only depends on h ∈ P Similarly if Σ is a U p (N)-admissible cone decomposition of n (A ∞ ) ord then we define an admissible cone decomposition Σ(h) 0 for ≻0 R as follows: The cones in Σ(h) 0 over an element y given as ) ≻0

R,y
, then there is a simplicial complex S(U, Σ) whose simplices are in bijection with the cones in G (m) n (Q)\Σ/U which have dimension bigger than 0, and have the same face relations. We will write S(U, Σ) ≤i for the subcomplex of S(U, Σ) consisting of simplices associated to the orbits of cones (g, δ) × σ ∈ Σ with σ ⊂ C (m),>0 (W ) for some W with dim F W ≤ i. We will also set an open subset of |S(U, Σ) ≤i |. Then one sees that If (U p (N), Σ) ∈ J (m),tor,ord n then there is a simplicial complex S(U p (N), Σ) ord whose simplices are in bijection with equivalence classes of cones of dimension greater than 0 in Σ, where σ and σ ′ are considered equivalent if σ ′ = γσu for some γ ∈ G (m) n (Q) and some u ∈ U p (N, N). We will write S(U p (N), Σ) ord ≤i for the subcomplex of S(U p (N), Σ) ord consisting of simplices associated to the orbits of cones (g, δ) × σ ∈ Σ with σ ⊂ C (m),>0 (W ) for some W with dim F W ≤ i. We will also set  [La4] for the assertions of the last three paragraphs.) Any of the (canonically quasi-isogenous) universal abelian varieties A univ /X n,U extend uniquely to semi-abelian varieties A univ ∆ /X n,U,∆ . The quasi-isogenies between the A univ extend uniquely to quasi-isogenies between the A univ ∆ . If g ∈ G n (A ∞ ) and (U, ∆) ≥ (U ′ , ∆ ′ )g then g * A univ ∆ ′ is one of the A univ ∆ . (See remarks 1.1.2.1 and 1.3.1.4 of [La4].) We will write ∂ i A (m) n,U,Σ for the pre-image under π A (m),tor /X min of ∂ i X min U . We also set n,U,Σ .
We will also write A Suppose that g −1 Ug ⊂ U ′ and that Σg is a refinement of Σ ′ . Suppose also that ֒→ X min,∧ n,U ′ ,i , which is commutative as a diagram of topological spaces (but not as a diagram of locally ringed spaces). The top square is commutative as a diagram of formal schemes and is compatible with the log structures. (Again see theorem 1.3.3.15 of [La4].) The pull back of A univ We will write |S(∂A . Moreover this is compatible with the log structures defined on each of the four formal schemes. (See theorem 7.1.4.1 of [La4].) If [σ] ∈ S(U p (N 1 , N 2 ), Σ) we will write (We remind the reader that the first superscript ord associates the 'ordinary' cone decomposition Σ ord to the cone decomposition Σ, while the second superscript ord is the notation we are using for the simplicial complex associated to an 'ordinary' cone decomposition.) We will write This only depends on Σ ord .
If (U p ) ′ is a neat subgroup of G n (A p,∞ ) containing the image of U p , and if ((U p ) ′ (N 1 , N 2 ), ∆) ∈ J tor n , and if Σ and ∆ are compatible; then for all h ∈ P (m),+ n,(i) (A ∞ ) ord with image h ′ ∈ P + n,(i) (A ∞ ) ord the cone decompositions Σ ord (h) 0 and ∆ ord (h ′ ) 0 are compatible and we have a diagram which is commutative as a diagram of topological spaces (but not as a diagram of locally ringed spaces). The top square is commutative as a diagram of formal schemes and is compatible with the log structures. (See theorem 7.1.4.1 of [La4].) The pull back of A univ is canonically quasi-isomorphic to the pull back of G univ from All this is compatible with passage to the generic fibre and our previous discussion. (Again see theorem 7.1.4.1 of [La4].) U,Σ ). Lemma 5.1. Suppose that R 0 is an irreducible, noetherian Q-algebra.
The third part follows from lemma 4.8.
We next deduce our first main observation.
Theorem 5.4. If i > 0 and U is neat then R i π X tor /X min , * E sub U,∆,ρ = (0). Similarly if i > 0 and U p is neat then R i π X ord,tor /X ord,min , * E ord,sub U p (N 1 ,N 2 ),∆,ρ = (0). Proof: The argument is the same in both cases, so we explain the argument only in the first case. Write X ∧ n,U,∆,i,h (resp. X min,∧ n,U,h,∂ 0 i X min n,U ) for the open and closed subset of X ∧ n,U,∆,i (resp. X min,∧ n,U,∂ 0 i X min n,U ) corresponding to T ♮,∧ n,(i),hU h −1 ∩P + n,(i) (A ∞ ),∆(h) (resp. X ♮ n,(i),hU h −1 ∩P + n,(i) (A ∞ ) ). (Recall that X ∧ n,U,∆,i is the completion of a smooth toroidal compactification of the Shimura variety X n,U along the locally closed subspace of the boundary corresponding to the parabolic subgroup P + n,(i) ⊂ G n . The formal scheme X min,∧ n,U,∂ 0 i X min n,U is the completion of the minimal (Baily-Borel) compactification of the same Shimura variety along the locally closed subspace of the boundary corresponding to the same parabolic. Each of these formal schemes is a disjoint union of sub-formal schemes indexed by certain elements h ∈ G n (A ∞ ).) n,U,Σ /X n,U ′ ,∆ (log ∞)) as shorthand for n,U,Σ /X n,U ′ ,∆ (log M Σ /M ∆ )). Then the collection n,U,Σ /X n,U ′ ,∆ (log ∞)}) is a system of locally free sheaves (for the Zariski topology) with G (m) n (A ∞ )-action. There are natural differentials n,U,Σ /X n,U ′ ,∆ (log ∞)) a complex. The tensor prod- Lemma 5.6. ( n,U ′ ,Σ ′ /X (log ∞).
for some U ′ and Σ 0 .
Proof: This follows from the properties of log differentials for log smooth maps (see section 2.2). For part 4 we also use lemma 5.1. For part 6 we also use the discussion of section 3.4 and a density argument.
The next lemma follows from lemma 4.10.
(1) The natural maps (2) The natural maps is a flat coherent O X n,U,∆ -module, and hence locally free of finite rank.
Next we record some results of one of us (K.-W.L).
. Then there are representations ρ i,j m,r of L n,(n) and a spectral sequence of sheaves on X min n,U ′ with first page This spectral sequence is G n (A ∞ )-equivariant.
Proof: Using part 2 of corollary 5.6 and parts 1 and 2 of lemma 5.1, we may reduce to the case that there is a cone decomposition ∆ compatible with Σ. By the preceding theorem it suffices to find ρ i,j m,r such that there is a spectral sequence of sheaves on X n,U ′ ,∆ with first page Moreover by lemma 5.7 we have that The result follows on combining this with parts 1, 2, 4 and 5 of lemma 5.8.
, then there is a continuous representation which is de Rham above p and has the following property: Suppose that v |p is a prime of F which is • either split over F + , • or inert but unramified over F + and Π is unramified at v; then , where q is the rational prime below v.
Proof: By the lemma ıΠ is the finite part of a cohomological, square integrable, automorphic representation of G n (A). The result now follows from corollary 1.3. is surjective. This follows using the long exact sequence in cohomology associated to the short exact sequence and the vanishing Let S denote the set of rational primes consisting of p and the primes where F ramifies. Also choose a neat open compact subgroup . Suppose that v is a place of F above a rational prime q ∈ S and let i ∈ Z. There is a unique element t v acts as 0 on any irreducible smooth representation of G n (Q q ) over C which is not a subquotient of an unramified principal series; • on an unramified representation Π q of G n (Q q ) the eigenvalue of t v by the characteristic function of G n (Z q ) we obtain a unique element T (i) v ∈ C[G n (Z q )\G n (Q q )/G n (Z q )] such that if Π q is an unramified representation of G n (Q q ) and if T . Suppose that q ∈ S is a rational prime. Let u 1 , ..., u r denote the primes of F + above Q which split u i = w i c w i in F , and let v 1 , ..., v s denote the primes of F + above q which do not split in F . Then under the identification where a i ∈ GL n (F w i ) is the diagonal matrix diag(1, ..., 1, ̟ w i ), and we may take d (1) w i = 1. We will call a topological Z p [G n ( Z S )\G n (A S )/G n ( Z S )]-algebra T of Galois type if for every there is a continuous pseudo-representation (see [T]) , which is also the image in the endomorphism algebra End (H 0 (X min N 2 ),ρt ⊗ Q p which is unramified outside S and satisfies

It suffices to show that there is a continuous pseudo-representation
for all v|q ∈ S and all i ∈ Z. (Because T will then automatically be valued in T S U p (N 1 ,N 2 ),ρt , by the Cebotarev density theorem. Note that if v is a prime of F split over F + and lying above a rational prime q ∈ S, then N 2 ),ρt .) We may then reduce to the case that ρ ⊗ Q p is irreducible. Let Suppose that t satisfies the inequality −2n ≥ (b τ,1 − t(p − 1)) + (b τ c,1 − t(p − 1)).
By lemma 5.11, where the sum runs over irreducible admissible representations of G n (A ∞ ) with Π U p (N 1 ,N 2 ) = (0) which occur in H 0 (X min ×Spec Q p , E sub ρt ). Further, from corollary 5.12, we deduce that there is a continuous representation N 2 ),ρt ) such that if v|q ∈ S then r is unramified at v and for all i ∈ Z. Taking T = tr r completes the proof of the lemma.
is a finitely generated Z p -submodule invariant under the action of the algebra Z p [G n ( Z S )\G n (A S )/G n ( Z S )], then let T ord,S U p (N ),ρ (W ) (resp. T ord,S U p (N 1 ,N 2 ),ρ (W )) denote the image of Z p [G n ( Z S )\G n (A S )/G n ( Z S )] in End Zp (W ). The next corollary follows from lemmas 6.1 and 6.2.
is a finitely generated Z p -submodule invariant under the action of the algebra Z p [G n ( Z S )\G n (A S )/G n ( Z S )], then T ord,S U p (N 1 ,N 2 ),ρ (W ) is of Galois type. We deduce from this the next corollary. is a finitely generated Z p -submodule invariant under the action of the algebra Z p [G n ( Z S )\G n (A S )/G n ( Z S )], then T ord,S U p (N ),ρ (W ) is of Galois type. Finally we deduce the following proposition.
Proposition 6.5. Suppose that ρ is a representation of L n,(n) over Z (p) . Suppose also that Π is an irreducible quotient of an admissible G n (A ∞ ) ord,× -sub-module Π ′ of H 0 (X ord,min , E ord,sub ρ ) Q p . Then there is a continuous semi-simple representation with the following property: Suppose that q = p is a rational prime above which F and Π are unramified, and suppose that v|q is a prime of F . Then where q is the rational prime below v.
Proof: Let S denote the set of rational primes consisting of p and the primes where F or Π ramifies. Also choose a neat open compact subgroup U p = G n ( Z S ) × U p S and integer N such that Π U p (N ) = (0).
As (Π ′ ) U p (N ) is a finite dimensional, and hence closed, subspace of the topological vector space H 0 (X ord,min , E ord,sub ρ ) Q p preserved by Z p [G n ( Z S )\G n (A S )/G n ( Z S )] and, as there is a Z p [G n ( Z S )\G n (A S )/G n ( Z S )]-equivariant map (Π ′ ) U p (N ) → → Π U p (N ) , there is a continuous homomorphism θ : T ord,S U p (N ),ρ ((Π ′ ) U p (N ) ) −→ Q p which for v|q ∈ S sends T (i) v to its eigenvalue on Π Gn(Zq) . Proposition 6.5 now follows from the above corollary and the main theorem on pseudo-representations (see [T]).
We remark that we don't know how to prove this proposition for a general irreducible subquotient of H 0 (X ord,min , E ord,sub ρ ) Q p (or indeed whether the corresponding statement remains true).
6.2. Interlude concerning linear algebra. Suppose that K is an algebraic extension of Q p . For a ∈ Q, we say that a polynomial P (X) ∈ K(X) has slopes ≤ a if P (X) = 0 and every root of P (X) in K has p-adic valuation ≤ a. (We normalize the p-adic valuation so that p has valuation 1.) If V is a K-vector space and T is an endomorphism of V , then we say that V admits slope decompositions for T , if for each a ∈ Q there is a decomposition V = V ≤a ⊕ V >a with the following properties: • T preserves V ≤a and V >a ; • V ≤a is finite dimensional; • if P (X) ∈ K[X] has slopes ≤ a then the endomorphism P (T ) restricts to an automorphism of V >a ; • there is a non-zero polynomial P (X) ∈ K[X] with slopes ≤ a such that the endomorphism P (T ) restricts to 0 on V ≤a . In this case V ≤a and V >a are unique, and we refer to them as the slope a decomposition of V with respect to T .
(1) If V is finite dimensional then it always admits slope decompositions.
(2) If K is a finite extension of Q p , if V is a K-Banach space, and if T is a completely continuous (see [Se]) endomorphism of V then V admits slope decompositions for T .
(3) Suppose that L/K is an algebraic extension and that V is a K vector space which admits slope decompositions with respect to an endomorphism T . Then V ⊗ K L also admits slope decompositions with respect to T . (4) Suppose that V 1 admits slope decompositions with respect to T 1 ; that V 2 admits a slope decomposition with respect to T 2 and that d : V 1 → V 2 is a linear map such that d • T 1 = T 2 • d.
Then for all a ∈ Q we have dV 1,≤a ⊂ V 2,≤a and dV 1,>a ⊂ V 2,>a .
Moreover ker d admits slope decompositions for T 1 , while Im d and coker d admit slope decompositions for T 2 . More specifically (coker d) >a = V 2,>a /(Im d) >a .
Suppose also that T is an endomorphism of V ∞ such that for all i > 1 If for each i the space V i admits slope decompositions for i, then V ∞ admits slope decompositions for T . (6) Suppose that is an exact sequence of K vector spaces and that T is an endomorphism of V that preserves V 1 . If V 1 and V 2 both admit slope decompositions with respect to T , then so does V . Moreover we have short exact sequences Proof: The first and third and fourth parts are straightforward. The second part follows from [Se].
For the fifth part one checks that V i,≤a is independent of i. If we set V ∞,≤a = V i,≤a for any i, and then these provide the slope a decomposition of V ∞ with respect to T . Finally we turn to the sixth part. Choose non-zero polynomials P i (X) ∈ K[X] with slopes ≤ a such that P i (T )V i,≤a = (0), for i = 1, 2. Set P (X) = P 1 (X)P 2 (X). Also set V ≤a = ker P (T ) and V >a = Im P (T ). We have complexes (0) −→ V 1,>a −→ V >a −→ V 2,>a −→ (0) and (0) −→ V 1,≤a −→ V ≤a −→ V 2,≤a −→ (0).
It suffices to show that these complexes are both short exact sequences. For then we see that, if Q(X) ∈ K[X] has slopes ≤ a, then the restriction of Q(T ) to V >a is an automorphism of V >a . Applying this to P (T ), we see that V ≤a ∩ V >a = (0). Moreover V ≤a + V >a contains V 1 and maps onto V 2 , so that V = V ≤a + V >a .
To show the first complex is short exact we need only check that V 1,>a = V >a ∩ V 1 , i.e. that V 1,≤a ∩ V >a = (0). So suppose that v ∈ V 1,≤a ∩ V >a then v = P (T )v ′ and P 1 (T )v = 0. Thus P 1 (T ) 2 P 2 (T )v ′ = 0 so the image of v ′ in V 2 lies in V 2,≤a and so P 2 (T )v ′ ∈ V 1 , and in fact P 2 (T )v ′ ∈ V 1,≤a . Finally we see that v = P 1 (T )P 2 (T )v ′ = 0, as desired.
To show the second complex is short exact we have only to show that V ≤a → V 2,≤a is surjective. So suppose that v ∈ V 2,≤a and suppose that v ∈ V lifts v.
We warn the reader that to the best of our knowledge it is not in general true that if V 1 ⊂ V is T -invariant then either V 1 or V /V 1 admits slope decompositions for T .
6.3. The ordinary locus of a toroidal compactification as a dagger space. We first review some general facts about dagger spaces. We refer to [GK] for the basic facts.
Suppose that K/Q p is a finite extension with ring of integers O K and residue field k. Suppose also that Y/O K is quasi-projective. Let Y denote the generic fibre Y × Spec K, let Y denote the special fibre Y × Spec k and let Y ∧ denote the formal completion of Y along Y . Let Y an (resp. Y † ) denote the rigid analytic (resp. dagger) space associated to Y . (For the latter see section 3.3 of [GK].) Thus Y an and Y † share the same underlying G-topological space, and in fact the completion (Y † ) ′ (see theorem 2.19 of [GK]) of Y † equals Y an . Let Y ∧ η denote the rigid analytic space associated to Y ∧ , its 'generic fibre'. Then Y ∧ η is identified with an admissible open subset ]Y [⊂ Y an . We will denote by Y † the admissible open dagger subspace of Y † with the same underlying topological space as ]Y [. Lemma 6.7. If Y and Y ′ are two quasi-projective O K -schemes as described in the previous paragraph and if f : Y → Y ′ is a morphism, then there is an induced Proof: The first part of the lemma is clear. For the second part, let Y ֒→ P M O K and Y ′ ֒→ P M ′ O K be closed embeddings. Let P ′ denote the closure of Y ′ in P M ′ O K . Also let P denote the closure of Y in P M O K × P M ′ O K . Then f extends to a map P → P ′ . The second part of the lemma follows from theorem 1.3.5 of [Be1] applied to Y ⊂ P and Y ′ ⊂ P ′ .
We will let H i rig (Y ) denote the rigid cohomology of Y in the sense of Berthelot -see for instance [LeS].
(1) If Y/O K is a smooth and quasi-projective scheme, then there is a canonical isomorphism (2) If f : Y → Z is a morphism of smooth quasi-projective schemes over O K then the following diagram is commutative: . Proof: For the first part apply theorem 5.1 of [GK] to the closure of Y in some projective space over O K . For the second part choose embeddings i : Y ֒→ P M O K and i ′ : Z ֒→ P M ′ O K . Let P ′ denote the closure of Z in P M ′ O K and P the closure of Y in P M O K × P ′ , so that f extends to a map P → P ′ . The desired result again follows from theorem 5.1 of [GK], because the isomorphisms of theorem 5.1 of [GK] are functorial under morphisms of the set up in that theorem.
[It is unclear to us whether this functoriality is supposed to be implied by the word 'canonical' in the statement of theorem 5.1 of [GK]. For safety's sake we sketch the argument for this functoriality. More precisely if f : X 1 → X 2 is a morphism of proper admissible formal Spf R-schemes which takes Y 1 ⊂ X 1,s to Y 2 ⊂ X 2,s , then we will show that the isomorphisms of theorem 5.1 of [GK] are compatible with the maps in cohomology induced by f . For part (a) we also suppose that we are given a map f * : f * F 2 → F 1 .
Using the notation of part (a) of theorem 5.1 of [GK], it suffices to show that the diagram (The functoriality of parts (b) and (c) follow easily from the functoriality of part (a).) The vertical morphisms arise from maps L • k → K • k of resolutions of the sheaves Ri * F k,X k and j † k F ′ k respectively. To define these resolutions one needs to choose affine covers {Y k,i } of Y k . We may suppose these are chosen so that f carries Y 1,i to Y 2,i for all i. Then L • k and K • k are the Cech complexes with The maps L • k → K • k arise from maps Here V runs over strict neighbourhoods of U∩]Y k,J [ X k in ]Y k [ X k and V ′ runs over strict neighbourhoods of ]Y k,J [ X k in ]Y k [ X k . The first isomorphism is justified in section 2.23 of [GK]. The second morphism arises because, for every V , we can find a V ′ so that V ′ ∩ U ⊂ V.
It suffices to show that if f U 1 ⊂ U 2 , then the diagrams commutative. But this is now clear.] Lemma 6.9. Suppose that f : X → Y is a proper morphism between Q p -schemes of finite type and that F /X is a coherent sheaf. Denote by f † : X † → Y † the corresponding map of dagger spaces and by F † the coherent sheaf on X † corresponding to F /X. Suppose also that V is an admissible open subset of Y † and that U is its pre-image in X † . Then where (R i f * F ) † denotes the coherent sheaf on Y † corresponding to (R i f * F )/Y . Proof: It suffices to check this in the case V = Y † . There is a chain of isomorphisms ( The first arrow is the transitivity of dagger and rigid analytification. The second arrow is Theorem 6.5 of [Kö]. The third arrow is Theorem 3.5 of [GK]. Since Y † is partially proper, Theorem 2.26 of [GK] implies that there is a unique isomorphism (R i f * F ) † ∼ = R i f † * F † which recovers the above map after passage to rigid spaces. We also obtain an element tr F ∈ End (H 0 (X ord, † , E sub ρ )) which commutes with the G n (A ∞ ) ord,× -action and satisfies tr F • ς p = p n 2 [F + :Q] . H 0 (X ord, † U p (N ),∆ , E sub, † U p (N ),∆,ρ ) ∼ −→ H 0 (X ord, † , E sub ρ ) U p (N ) . Proof: Use lemmas 5.1, 5.6, 5.7, 5.3 and 6.9.
6.4. The ordinary locus of the minimal compactification as a dagger space. Suppose that U p is a neat open compact subgroup of G n (A ∞,p ) and that N 2 ≥ N 1 ≥ 0. We will write X min,ord, † n,U p (N 1 ,N 2 ) for the dagger space associated to X min,ord n,U p (N 1 ,N 2 ) as described in the paragraph before lemma 6.7. Then the system of dagger spaces {X min,ord, † n,U p (N 1 ,N 2 ) } has an action of G n (A ∞ ) ord .
Let X min n,U p (N 1 ,N 2 ) [1/e U p (N 1 ,N 2 ) ] denote the open subscheme of X min n,U p (N 1 ,N 2 ) , where e U p (N 1 ,N 2 ) = 0. As ω ⊗(p−1)a is ample, X min n,U p (N 1 ,N 2 ) [1/e U p (N 1 ,N 2 ) ] is affine and so has the form Spec Z (p) [T 1 , ..., T s ]/I for some s and I. It is normal and flat over Z (p) .
For r ∈ p Q ≥0 let || || r denote the norm on Z (p) [T 1 , ..., T s ] defined by where i runs over Z s ≥0 and |(i 1 , ..., i s )| = i 1 +...+i s . We will write Z p T 1 , ..., T s r for the completion of Z (p) [T 1 , ..., T s ] with respect to || || r . Thus Z p T 1 , ..., T s 1 is the p-adic completion of Z (p) [T 1 , ..., T s ] and also the p-adic completion of Z p T 1 , ..., T s r for any r ≥ 1. Set Q p T 1 , ..., T s r = Z p T 1 , ..., T s r [1/p], the completion of Q[T 1 , ..., T s ] with respect to || || r . In the case r = 1 we will drop it from the notation. We will write Z p T 1 /r, ..., T s /r 1 for the || || r unit-ball in Q p T 1 , ..., T s r , i.e. for the set of power series i∈Z s ≥0 a i T i where a i ∈ Q p , and |a i | p ≤ r −| i| for all i, and |a i | p r | i| → 0 as | i| → ∞. We will also write Q p T 1 , ..., T s † = r>1 Q p T 1 , ..., T s r .
Let I r denote the ideal of Z p T 1 , ..., T s r generated by I and let I ′ r denote the intersection of I 1 with Z p T 1 , ..., T s r . Then I 1 is the p-adic completion of I. Moreover Z p T 1 , ..., T s 1 / I 1 is normal and flat over Z p , and X min,ord U p (N 1 ) = Spf Z p T 1 , ..., T s 1 / I 1 . Note that Z (p) [T 1 , ..., T s ]/(I, p) ∼ −→ Z p T 1 , ..., T s r /( I r , p) for all r ≥ 1. Thus ( I r , p) = ( I ′ r , p). We will also write I r,Qp (resp. I ′ r,Qp ) for the Q p span of I r (resp. I ′ r ) in Q p T 1 , ..., T s r . Then Sp Q p T 1 , ..., T s 1 / I 1,Qp ⊂ Sp Q p T 1 , ..., T s r / I ′ r,Qp ⊂ Sp Q p T 1 , ..., T s r / I r,Qp are all affinoid subdomians of X min,an U p (N,N 2 ) , the rigid analytic space associated to X min U p (N,N 2 ) × Spec Q p . Thus they are normal. Moreover Sp Q p T 1 , ..., T s r / I ′ r,Qp and Sp Q p T 1 , ..., T s r / I r,Qp −Sp Q p T 1 , ..., T s r / I ′ r,Qp forms an admissible open cover of Sp Q p T 1 , ..., T s r / I r,Qp . (Sp Q p T 1 , ..., T s r / I ′ r,Qp is the union of the connected components of Sp Q p T 1 , ..., T s r / I r,Qp , which contain a component of Then the map E r → E r ′ , which we will also denote i ′ r,r ′ , is completely continuous. The map tr F extends to a continuous Q p T 1 , ..., T s r p / I ′ r p ,Qp linear map t r : E r −→ E r p for r ∈ [1, r 2 ] ∩ p Q . We set so that E † = H 0 (X min,ord, † U p (N 1 ) , E sub, † U p (N 1 ),ρ ). We have that tr F | Er = t r . As t r is continuous and i ′ r p ,r is completely continuous we see that tr F : E r −→ E r and that this map is completely continuous. Thus each E r admits slope decompositions for tr F and hence by lemma 6.6 so does E † and E † ⊗ Q p . If a ∈ Q we thus have a well defined, finite dimensional subspace H 0 (X min,ord, † U p (N ) , E sub, † U p (N ),ρ ) Q p ,≤a ⊂ H 0 (X min,ord, † U p (N ) , E sub, † U p (N ),ρ ) ⊗ Qp Q p . (Defined with respect to tr F .) We set H 0 (X min,ord, † , E sub ρ ) Q p ,≤a = lim , E sub, † U p (N ),ρ ) Q p ,≤a , so that there are G n (A ∞ ) ord,× -equivariant embeddings H 0 (X min,ord, † , E sub ρ ) Q p ,≤a ⊂ H 0 (X min,ord, † , E sub ρ ) Q p ֒→ H 0 (X ord,min , E sub ρ ) Q p . We have proved the following lemma.
Lemma 6.12. H 0 (X min,ord, † , E sub ρ ) Q p ,≤a is an admissible G n (A ∞ ) ord,× -module. Combining this with corollary 6.5 we obtain the following result.
Corollary 6.13. Suppose that ρ is a representation of L n,(n) over Z (p) , that a ∈ Q and that Π is an irreducible G n (A ∞ ) ord,× -subquotient of H 0 (X min,ord, † , E sub ρ ) Q p ,≤a . Then there is a continuous semi-simple representation with the following property: Suppose that q = p is a rational prime above which F and Π are unramified, and suppose that v|q is a prime of F . Then where q is the rational prime below v.
We will next explain the consequences of these results for sheaves of differentials on A (m),ord, † n,U p (N 1 ,N 2 ),Σ . But we first need to record a piece of commutative algebra. ) ≤a .
The next corollary now follows from the proposition and lemma 6.6.
Corollary 6.18. Suppose that Π is an irreducible G n (A ∞ ) ord,× -subquotient of H i (A (m),ord, † , Ω s A (m),ord, † (log ∞) ⊗ I ∂A (m),ord, † ) ≤a ⊗ Qp Q p for some a ∈ Q. Then there is a continuous semi-simple representation with the following property: Suppose that q = p is a rational prime above which F and Π are unramified, and suppose that v|q is a prime of F . Then where q is the rational prime below v.

Galois representations.
In order to improve upon corollary 6.27 it is necessary to apply some simple group theory. To this end, let Γ be a topological group and let F be a dense set of elements of Γ. Let k be an algebraically closed, topological field of characteristic 0 and let d ∈ Z >0 . Let µ : Γ −→ k × be a continuous homomorphism such that µ(f ) has infinite order for all f ∈ F. For f ∈ F let E 1 f and E 2 f be two d-element multisets of elements of k × . Let M be an infinite subset of Z. For m ∈ M suppose that ρ m : Γ −→ GL 2d (k) be a continuous semi-simple representation such that for every f ∈ F the multi-set of roots of the characteristic polynomial of ρ m (f ) equals It is a, possibly disconnected, reductive group. There is a natural continuous homomorphism Note that ρ M ′ F is Zariski dense in G M ′ . We will use µ for the character of G M ′ which is projection to G m . For m ∈ M ′ we will let R m : G M ′ −→ GL 2d denote the projection to the factor indexed by m.
Lemma 7.1. For every g ∈ G M ′ (k) there are two d-element multisets Σ 1 g and Σ 2 g of elements of k × such that for every m ∈ M ′ the multiset of roots of the characteristic polynomial of R m (g) equals Σ 1 g ∐ Σ 2 g µ(g) m . Proof: It suffices to show that the subset of k × ×GL M ′ 2d (k) consisting of elements (t, (g m ) m∈M ′ ) such that there are d-element multisets Σ 1 and Σ 2 of elements of k × so that for all m ∈ M ′ the multiset of roots of the characteristic polynomial of g m equals Σ 1 ∐ Σ 2 t m , is Zariski closed. Let Pol 2d denote the space of monic polynomials of degree 2d. It even suffices to show that the subset X of k × × Pol M ′ 2d (k) consisting of elements (t, (P m ) m∈M ′ ) such that there are d-element multisets Σ 1 and Σ 2 of elements of k so that for all m ∈ M ′ the multiset of roots of P m equals Σ 1 ∐ Σ 2 t m , is Zariski closed.
There is a natural finite map , where S 2d denotes the symmetric group on 2d letters, define V (σm) to be the set of (t, (a m,i )) ∈ G m × (Aff 2d ) M ′ such that, for all m, m ′ ∈ M ′ we have (1 × π M ′ )V (σm) .
The lemma now follows from the finiteness of 1 × π M ′ . Then for all m ∈ M ′′ the only eigenvalue of R m (g) is 1. Thus g must be unipotent. However ker(G M ′′ → → G M ′ ) is reductive and so must be trivial.
Thus we can write G for G M ′ without danger of confusion.
Corollary 7.3. For every g ∈ G(k) there are two d-element multisets Σ 1 g and Σ 2 g of elements of k × such that for every m ∈ M the multiset of roots of the characteristic polynomial of R m (g) equals Σ 1 g ∐ Σ 2 g µ(g) m . Moreover if µ(g) has infinite order then the multisets Σ 1 g and Σ 2 g are unique. Proof: Choose non-empty finite subsets For each i we can find two d-element multisets Σ 1 g,i and Σ 2 g,i of elements of k × such that for every m ∈ M ′ i the multiset of roots of the characteristic polynomial of R m (g) equals Σ 1 g,i ∐ Σ 2 g,i µ(g) m .
Let m 1 ∈ M ′ 1 and let Σ denote the set of eigenvalues of R m 1 (g). Then, for every i, the multiset Σ 1 g,i consists of elements of Σ and the multiset Σ 2 g,i consists of elements of Σµ(g) −m 1 . Thus there are only finitely many possibilities for the pair of multisets (Σ 1 g,i , Σ 2 g,i ) as i varies. Hence some such pair (Σ 1 g , Σ 2 g ) occurs infinitely often. This pair satisfies the requirements of the lemma.
Proof: If µ were trivial on Z(G 0 ) 0 then it would be trivial on G 0 (because G 0 /Z(G 0 ) 0 is semi-simple), and so µ would have finite order, a contradiction. Thus µ| Z(G 0 ) 0 is non-trivial.
The space is a representation of the finite group G/G 0 and we can decompose is non-trivial, and so µ| Z(G) 0 is non-trivial.
For m ∈ M let X m denote the 2d-element multiset of characters of Z(G) 0 which occur in R m (taken with their multiplicity). If g ∈ G then we will write Y(g) m for the 2d-element multiset of pairs (χ, a), where χ is a character of Z(G) 0 and a is a root of the characteristic polynomial of g acting on the χ eigenspace of Z(G) 0 in R m . (The pair (χ, a) occurs with the same multiplicity as a has as a root of the characteristic polynomial of g acting on the χ-eigenspace of R m .) If Y ⊂ Y(g) m and if ψ ∈ X * (G) then we will set Yψ = {(χψ, aψ(g)) : (χ, a) ∈ Y}.
We warn the reader that this depends on g and not just on the set Y.
Lemma 7.5. Suppose that T /k is a torus and that X is a finite set of non-trivial characters of T . Let A be a finite subset of k × . Then we can find t ∈ T (k) such that χ(t) = a for all χ ∈ X and a ∈ A.
We can find ν ∈ X * (T ) such that (χ, ν) = 0 for all χ ∈ X. Thus we are reduced to the case T = G m , in which case we may take t to be any element of k × that does not lie in the divisible hull of the subgroup H of k × generated by A. (For example we can take t to be a rational prime such that all elements of a finite set of generators of H ∩ Q × are units at t.) Corollary 7.6. Suppose that T /k is a torus and that X is a finite set of characters of T . Then we can find t ∈ T (k) such that if χ = χ ′ lie in X then Lemma 7.7. If m, m ′ , m ′′ ∈ M, then we can decompose If µ m−m ′ = χ/χ ′ for all χ, χ ′ ∈ X m then the equation Y(g) m ′ = Y(g) 1 m,m ′ ,m ′′ ∐ Y(g) 2 m,m ′ ,m ′′ µ m ′ −m uniquely determines this decomposition.
Corollary 7.14. Suppose that E is a totally real or CM field and that π is a cuspidal automorphic representation such that π ∞ has the same infinitesimal as an algebraic representation of RS E Q GL n . Then there is a continuous semi-simple representation r p,ı : G E −→ GL n (Q p ) such that, if q = p is a prime above which π and E are unramified and if v|q is a prime of E, then r p,ı (π) is unramified at v and r p,ı (π)| ss W Ev = ı −1 rec Ev (π v | det | (1−n)/2 v ).
Proof: This can be deduced from theorem 7.13 by using lemma 1 of [So]. (This is the same argument used in the proof of theorem VII.1.9 of [HT].)