The adic, cuspidal, Hilbert eigenvarieties

2 The weight space 6 2.1 The Iwasawa algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 A blow up of the formal weight space . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The adic weight space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Properties of the universal character . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.1 Congruence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.2 A key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.3 Analyticity of the universal character . . . . . . . . . . . . . . . . . . . . 9

schemes, Coleman's theory was Q p -rigid analytic. Nevertheless Coleman observed that the characteristic series of the U p -operator acting on finite slope p-adic families of overconvergent modular forms had coefficients in the Iwasawa algebra (i.e. they were integral) and conjectured that there should exist an integral or positive characteristic theory of overconvergent modular forms. Following Coleman's intuition, we obtained such a theory for elliptic modular forms in [AIPC]. The present paper is an extension of [AIPC] to the case of Hilbert modular forms.
More precisely, in the present paper we accomplish the following. Let us first fix a totally real number field F of degree g over Q. Then let us recall (see for example [AIPH] or chapter §8 of the present paper) that there are two relevant algebraic groups attached to F , denoted by G := Res F/Q GL 2 and G * := G × Res F/Q Gm G m .
From the point of view of automorphic forms it is useful to work with modular forms on G, but the Shimura variety associated to G is not a moduli space of abelian varieties. Instead, the Shimura variety associated to G * is a moduli space of abelian varieties and so we first construct our modular sheaves for modular forms on G * , as in [AIPH], and then we descend these sheaves to the relevant varieties associated to G. a) Modular sheaves associated to G * .
This construction is accomplished in chapters 1 to 7, where we work with toroid al compactifications of the moduli spaces of abelian schemes with O F -multiplication and we denote by G the semi-abelian scheme which extends the universal abelian scheme. To fix ideas, let p denote a positive prime integer, T the torus Res F/Q G m and let Λ F := Z p [[T(Z p )]] be the associated Iwasawa algebra. We denote by W F the analytic adic space (called the weight space for modular forms on G * ) associated to the formal scheme W F := Spf Λ F and κ un : T(Z p ) −→ Λ × F the universal (weight) character. In particular, we have a natural decomposition of the adic weight space W F = W rig F ∪ W F,∞ , where W rig F is the adic space associated to the rigid analytic generic fiber of Spf Λ F (so this is the "old, p-adic weight space") and W F,∞ = {x ∈ W F | |p| x = 0}, sometimes called the "boundary of the weight space" and consisting in points with values in characteristic p-rings.
Let now N ≥ 4 be an integer relatively prime to p and let M(µ N , c) be the formal scheme associated to a projective toroidal compactification of the Shimura variety for G * of level (µ N , c).
Here c is a fractional ideal of F (see section §3 for more details.) Our main result is the construction of an integral family of sheaves of overconvergent modular forms, parametrized by the formal spectrum of the Iwasawa algebra Λ F . This overconvergent family extends the family of p-adic modular forms defined by Katz in [K] and used by Hida in [Hi2]. More precisely let us denote by W 0 F = Spf Λ 0 F the free" component of W F , where Λ 0 F is a complete regular local ring of dimension g + 1 with maximal ideal m and let Z := M(µ N , c) × W 0 F . We consider, for each r ≥ 0 the formal scheme Z r , which should be thought of as a "formal neighborhood of the ordinary locus in Z" and which is defined as the formal scheme which represents the functor associating to every m-adically complete Λ 0 F -algebra R the set of equivalence classes of tuples (h, η p , η 1 , η 2 , . . . , η g ), where h : SpfR −→ M(µ N , c) is a morphism of formal schemes and η p , η i ∈ H 0 SpfR, h * det ω (1−p)p r+1 A See section §6.3 for the definition of the equivalence relation between such tuples. Here A is the universal semi-abelian scheme over M(µ N , c), denoted G in the main body of the article, and T 1 , T 2 , . . . , T g are chosen elements of m, which together with p generate it (see section §2.1 for more details.) Let M r be the base-change, as formal schemes, of Z r to W F . We construct, for each r ≥ 0, a coherent sheaf w κ un r on M r . Let M r denote the adic analytic space associated to M r and ω un r the associated analytic coherent sheaf. Then ω κ un r is invertible and it satisfies the following properties.
1. The restriction of ω un r to the rigid analytic space M r × Spa(Zp,Zp) Spa(Q p , Z p ) is the sheaf defined in [AIPH], Definition 3.6.
2. for all classical weights k · χ : T(Z p ) −→ O * Cp , where k is an algebraic weight and χ a finite order character, the specialization of ω un r to k · χ is the restriction to M r of the sheaf ω k A (χ) of classical modular forms of weight k and nebentypus χ.

The family of sheaves {ω un
r } r≥0 is Frobenius compatible. See §6.4. b. The modular sheaves associated to G This construction is done in chapter 8. Let W G F , κ un G denote the formal weight space and respectively the universal character associated to the Iwasawa algebra Λ G F for the group G and let W G F −→ W F be the natural morphism of formal schemes. Let now M(µ N , c) G denote the formal Shimura variety for the group G (i.e. the formal completion along the special fiber of a projective, toroidal compactification of the Shimura variety for G). The toroidal compactifications for the Shimura varieties for G * and G can be chosen in such a way that we have a natural morphism of formal schemes α : M(µ N , c) −→ M(µ N , c) G . Moreover if ∆ denotes the quotient of the group of totally real units of O F by the square of the units congruent to 1 modulo N , this finite group acts naturally on M(µ N , c) by multiplication on the polarizations, such that: 1. The morphism α is finite,étale and Galois with Galois group ∆. It follows that M(µ N , c) G ∼ = (M(µ N , c))/∆.
2. For every r ≥ 0, we have a natural action of ∆ on M r × W F W G F lifting to an action on w κ un G r , which is the pull-back of w κ un r to M r × W F W G F . By finite,étale descent we obtain a coherent sheaf, still denoted w κ un G r , on M r,G := M r × W F W G F /∆. 3. If we denote by M r,G the analytic adic space associated to the formal scheme M r,G and by ω κ un G r the associated coherent sheaf, then ω κ un G r is invertible and the overconvergent modular forms for G are overconvergent sections of specializations of this modular sheaf. As in [AIPH], one can show by a cohomological argument that specialization is surjective on cuspidal forms.
The spectral theory of the operator U p on adic families of overconvergent modular forms allows us to construct an adic eigenvariety sitting over the analytic adic space associated to the Iwasawa algebra Λ G F . See §8.6.
Finally, this article generalizes and is crucially based on both [AIPC] and [AIPH]. In particular for many arguments we refer to loc. cit. Let us point out what is really new here: 1. The boundary of the weight space, both for G and G * are analytic spaces of dimension g − 1. Therefore the boundary overconvergent Hilbert modular forms (i.e. the overconvergent Hilbert modular forms in characteristic p) are parameterized by positive dimensional analytic spaces if g > 1, i.e. live in true analytic families.
2. In [AIPC], the universal integral modular sheaf w un r was a sheaf parameterized by the formal blow-up of the formal scheme Spf Λ with respect to the ideal m. Therefore the descent to the Iwasawa algebra in this paper improves [AIPC].
3. If p is ramified in O F the descent of the perfect sheaves of overconvergent Hilbert modular forms to finite levels by the use of Tate traces involves new problems due to the non smoothness of the associated Hilbert modular varieties in characteristic p.
Remark 1.1. In [AIPH] and also in this paper we work with toroidal compactifications M (µ N , c) of the integral models of the Shimura varieties associated to G * defined by P. Deligne and G. Pappas [DP], completing previous work of M. Rapoport [Ra]. These models are singular at primes dividing the discriminant of F . One could use one of the splitting models Pappas and M. Rapoport in [PR]. Such models depend on some auxiliary choices, namely an ordering of the embeddings of F in an algebraic closure, but they have the advantage of being smooth. The given map is an isomorphism over an open dense subscheme (the Rapoport locus, see §3); in particular the two moduli spaces differ only at primes p dividing the discriminant of F . Recall that the complement of the Rapoport locus is characterized by the fact that the sheaf ω G of invariant differentials of the versal semiabelian scheme G over M (µ N , c) is not locally free as O F ⊗ O M (µ N ,c) -module. On the other hand on a splitting model, ω G admits a filtration by invertible O M (µ N ,c) -modules, stable for the action of O F . These invertible sheaves allow to define Hilbert modular forms of non-parallel weight over the whole of M (µ N , c).
For our purposes, i.e. the construction of modular sheaves, it makes no difference which model we choose and we prefer to work with the minimal (and more canonical) one, the Deligne-Pappas model. The main reason is the fact that the key ingredient in the construction of the modular sheaves is the introduction of a different integral structure F of ω G which is locally free as O F ⊗ O IG n,r,I -module, e.g. even on the complement of the Rapoport locus in the formal scheme IG n,r,I (see [AIPH], Proposition 4.1, or section §4.1 of the present article).
for the universal character. Fix an isomorphism of topological groups It is a complete, regular, local ring with maximal ideal m. Furthermore Actually, there is also a canonical projection map Λ F → Λ 0 F obtained by sending all h ∈ H to 1. We let κ : T(Z p ) → (Λ 0 F ) be the composition of κ un and the above projection. We let χ : H → Λ * F be the composition of the inclusion H → T(Z p ) and the universal character.
We denote by W F , resp. W 0 F the m-adic formal scheme defined by Λ F resp. Λ 0 F . Then we have a natural map W F → W 0 F which is finite and flat.
Remark 2.1. In [AIPH] the weight space has been defined over the ring of integers of a finite extension K of Q p splitting F . The reason is that the classical weights are defined over K.
Here we prefer to work over Z p . As a consequence it will turn out that the characteristic series of the U p operator will have coefficients in the Iwasawa algebra Λ G F defined in Theorem 8.4, with no need to extend scalars.

A blow up of the formal weight space
Consider the blow up Spec Λ F of Spec Λ F with respect to the ideal m and let t : W F → W F be the associated m-adic formal scheme.
We describe in more detail the formal scheme W F . Notice that by the universal property of the blow up, the ideal sheaf I : open affine formal subscheme where I is generated by α (W α is empty unless α ∈ m \ m 2 ). In particular the m-adic topology on B α coincides with the α-adic topology.
One has variants W 0 F → W 0 F of the spaces introduced above and associated to the subalgebra Λ 0 F of Λ F . We also have a natural finite and flat morphism Choosing generators (p, T 1 , . . . , T g ) of m then W F is covered by the affinoids We let W F be the analytic adic space associated to W F . Namely, W F consists of the analytic points Spa(Λ F , Λ F ) an ⊂ Spa(Λ F , Λ F ). We denote by t : W F → W F the morphism of analytic adic spaces associated to t : Proof. For all α ∈ m the subset {x ∈ W F , 0 = |α| x ≥ |β| x , ∀β ∈ m} of W F equals W α by definition. Moreover, W F is covered by the W α . The conclusion follows.
Remark 2.3. Let us denote by W Berk F the subset of rank 1 points of W F . Then there is a map: with image included in [0, 1[ g+1 . This map may be helpful in order to understand W F . Let us denote by (x 0 , . . . , x g ) the coordinates on P g (R). Then Θ −1 ({x 0 = 0}) is the set of rank one points on the usual (adic) weight space over Spa(Q p , Z p ) associated to Λ F . Let us denote by W 0 F the analytic adic space attached to W 0 F . For every element α ∈ m we denote by W 0 α the analytic adic space associated to W 0 α . Finally, let us remark that the classical weights are points of the subspace of W F where p is invertible.

Properties
(2) Given (a 1 , . . . , a g ) ∈ Z g p , we have κ ρ(a 1 , . . . , Proof. (1) The group H is finite and its prime to p part maps isomorphically onto , and hence to Z g p , via the logarithm. In particular it injects into L via the quotient map and the subgroup . This proves the first claim.
Lemma 2.5. For every n ∈ Z ≥1 we have that κ ρ(p n−1 Z g p ) − 1 ⊂ m n . In particular we have for all n ∈ Z ≥1 Proof. Note that κ ρ(p n−1 a 1 , . . . , (using the logarithm). In particular 1+qp n−1 O F ⊗Z p is contained in ρ p n−1 O F ⊗ Z p via the identification above. The second claim follows.

A key lemma
We introduce a formalism inspired by Sen's theory that will be repeatedly used in the paper. Let n ∈ Z ≥1 and A 0 → A 1 · · · → A n be a tower of Λ 0 F -algebras which are domains. We assume that the group (O F /p n O F ) acts on A n by automorphisms of Λ 0 F -algebras and that A s is the sub-ring of A n fixed by the kernel H s of the map Let h ∈ A 0 and let p 0 = 0 ≤ p 1 ≤ · · · ≤ p n be a sequence of integers. Let c n ∈ h −pn A n be an element. Set c s = σ∈Hs σ.c n . We assume that: • c s ∈ h −ps A s for all s ≥ 0, ps A s for s ≥ 1 and b 0 = 1. Hereσ ∈ T(Z p ) is a lift of σ so that b s depends on c s and on the choices of lifts.
Lemma 2.6. 1. Another system of choices of liftsσ for the σ's would give an element b s and we have 2. We have the following congruence relations: Proof. The first point follows from lemma 2.5. To prove the second point, assume that s ≥ 1 and notice that One concludes by applying lemma 2.5 and also using the first point.

Analyticity of the universal character
We now study the analytic properties of the universal character. The degree of analyticity depends on the p-adic valuation of T 1 , · · · , T g . This motivates the following definition. For r s ∈ Q ≥1 we define the following rational open subsets of W 0 F : α is a rank one point, then α is a pseudo-uniformizer of the residue field k(x). Let us denote by v α : k(x) → R ∪ {∞} the valuation on k(x) normalized by v α (α) = 1. Notice that the norm p −vα(.) represents the equivalence class of |.| x . Then x ∈ W 0 α,I if and only if v α (p) ∈ I.
We now construct formal models. Take an element α ∈ m. We define B 0 α,I = H 0 (W α,I , O + W α,I ). Set W 0 α,I = Spf B 0 α,I . The analytic fiber of W 0 α,I is W 0 α,I . For various α's, the W 0 α,I glue to a formal scheme W 0 F,I with analytic fiber W 0 F,I is a p-adic formal scheme (the m-adic topology is the p-adic one). In the lemma below, G m , G a are considered as functors on the category of p-adic formal schemes equipped with a structural morphism to W 0 F . Let = 1 if p = 2 and = 3 if p = 2. The group Proposition 2.8. Let n ≥ 0 be an integer. Suppose that I ⊂ [0, p n ]. The character κ extends to a pairing Proof. Easy and left to the reader.
3 Hilbert modular varieties and the Igusa tower

Hilbert modular varieties
We recall the definition of Hilbert modular varieties following [Ra] and [DP]. Fix an integer N ≥ 4 and a prime p not dividing N . Let c be a fractional ideal of F and let c + be the cone of totally positive elements. Denote by D F the different ideal of O F . Let M (µ N , c) be the Hilbert modular scheme over Z p classifying triples (A, ι, Ψ, λ) consisting of: (1) abelian schemes A → S of relative dimension g over S, (2) an embedding ι : is the sheaf for theétale topology on S of symmetric O F -linear homomorphisms from A to the dual abelian scheme A ∨ and if P + ⊂ P is the subset of polarizations, then λ is an isomorphism ofétale sheaves λ : (P, P + ) ∼ = (c, c + ), as invertible O F -modules with a notion of positivity. The triple is subject to the condition that the map A ⊗ O F c → A ∨ is an isomorphism of abelian schemes (the so called Deligne-Pappas condition).
We write M (µ N , c) and M * (µ N , c) for a projective toroidal compactification, respectively the minimal or Satake compactification of M (µ N , c) (see [Ra]). Let M(µ N , c) (resp. M * (µ N , c)) be the associated formal schemes. They are endowed with a semi-abelian scheme G with O F -action.
There exist maximal open subscheme, respectively formal subscheme The complement is empty if p does not divide the discriminant of F and, in general, it is of codimension 2 in the characteristic p special fiber of M (µ N , c); see [DP]. We denote by Ha ∈ H 0 (M * (µ N , c) Fp , det ω p−1 G ) the Hasse invariant. We let Hdg ⊂ O M(µ N ,c) be the Hodge ideal defined by the Hasse invariant (see [AIPC,§A.1] for a precise definition: locally on M(µ N , c) it is the ideal generated by p and a (any) lift of a local generator of Ha det ω 1−p G ).

Canonical subgroups
Let A 0 be a Z p -algebra and α ∈ A 0 a non-zero element. We assume that A 0 satisfies the following: ( * ) A 0 is an integral domain, it is the α-adic completion of a Z p -algebra of finite type and p ∈ αA 0 . Let M (µ N , c) × Spec Zp Spec A 0 be the base change of the toroidal compactification via Spec A 0 → Spec Z p and let Y be the associated formal scheme over Spf A 0 .
Definition 3.1. For every integer r ∈ N denote by Y r → Y the formal scheme over Y representing the functor which to any α-adically complete A 0 -algebra R without α-torsion associates the equivalence classes of pairs h : Two pairs (h, η) et (h , η ) are declared equivalent if h = h and η = η (1 + p 2 α u) for some u ∈ R. We also denote by Y R r ⊂ Y r the open formal subscheme where the Rapoport condition holds (see §3).
Proposition 3.2. Assume that p ∈ α p k A 0 . Then for every integer 1 ≤ n ≤ r + k one has a canonical sub-group scheme H n of G[p n ] over Y r and H n modulo pHdg − p n −1 p−1 lifts the kernel of the n-th power of Frobenius. Moreover H n is finite flat and locally of rank p ng , it is stable under the action of O F , and the Cartier dual H D n isétale locally over Proposition 3.3. For every r ∈ Z ≥2 the isogeny given by dividing by the canonical subgroup H 1 of level 1 defines a finite morphism φ : Y r → Y r−1 . The restriction to the Rapoport locus φ : Y R r → Y R r−1 is finite and flat of degree p g . Proof. This is the content of [AIPC,Cor. A.2] which is written for general p-divisible groups. The last claim follows as relative Frobenius is finite and it is flat over the (smooth) Rapoport locus.

The partial Igusa tower 3.3.1 Construction
We use the notations of the previous section. Let A := A 0 [α −1 ]: it is a Tate ring in the sense of Huber [H1] with ring of definition here Y ad r , resp. Spa(A 0 , A 0 ) is the adic space associated to the formal scheme Y r , resp. Spf A 0 , and the fibre product is taken in the category of adic spaces.
Assume that p ∈ α p k A 0 and let r ∈ N and n ∈ N be an integer such that 1 ≤ n ≤ r + k. It follows from Proposition 3.2 that H D n over Y r isétale locally isomorphic to O F /p n O F . We let IG n,r → Y r be the Galois cover for the group (O F /p n O F ) * classifying the isomorphisms O F /p n O F → H D n , as group schemes, equivariant for the O F -action. We define IG n,r → Y r to be the formal scheme given by the normalization of Y r in IG n,r . See [AIPC,§3.2] for details. Such morphism is finite and is endowed with an action of (O F /p n O F ) * . One then gets a sequence of finite, The morphisms h : IG n,r → IG n−1,r are finite andétale over Y r . In particular there is a trace map Tr IG : h * O IGn,r → O IG n−1,r .

Ramification
Proposition 3.4. We have Proof. The claim for n ≥ 2 follows arguing as in [AIPC,Prop. 3.4]. We recall the argument. By normality the natural map IG n,r → H D n over Y r , associating to an isomorphism O F /p n O F → H D n the image of 1 ∈ O F /p n O F , extends to a morphism of formal schemes IG n,r → H D n over Y r . In particular we get a commutative diagram of formal schemes over Y r : which is cartesian over the analytic fiber Y r . In particular IG n,r → IG n−1,r is the normalization of the fppf (H n /H n−1 ) D -torsor over IG n−1,r obtained by the fibre product of the diagram above. One reduces to prove the claimed result for the trace of the morphism H D n → H D n−1 (over Y r ) and this follows from the relation between the different and the trace and a careful analysis of the different of (H n /H n−1 ) D given in [AIPC,Cor. A.2].
We are left to discuss the case n = 1. If p is unramified then the degree of IG 1,r → Y r is prime to p and the second claim of the proposition follows immediately. If p is ramified we let p be the product of all primes of O F over p. We introduce a variant of IG 1,r by setting IG 1,r to be the adic space over Y r classifying isomorphisms O F /pO F → H 1 p] D , as group schemes, equivariant for the O F -action. Here H 1 p is the kernel of multiplication by p on H 1 .
We have a natural map of adic spaces IG 1,r → IG 1,r → Y r . Taking normalizations we get morphisms of formal schemes We are left to estimate the image of the trace map associated to the morphism IG 1,r → IG 1,r . Arguing as at the beginning of the proof we get a commutative diagram of formal schemes over Y r , which is cartesian over Y r : Thus IG 1,r is the normalization of a torsor under ( It follows that Hdg is contained in the different of (H 1 /H 1 [p]) D over Y r and we conclude.
We immediately get the following Corollary 3.5. Let Spf R be an open of Y r such that the ideal sheaf Hdg is trivial and choose a generatorHa. For every 0 ≤ n ≤ r+k there exist elements c 0 = 1 and c n ∈Ha for n ≥ 1 such that Tr IG (c n ) = c n−1 for every n ≥ 1.

Frobenius
Recall from Proposition 3.3 that we have a Frobenius map φ : Y r → Y r−1 .
Proposition 3.6. There exists an Proof. As IG n,r is constructed by normalizing Y r in IG n,r , it suffices to construct a lift φ : IG n,r → IG n,r at the level of adic spaces.
Notice that H n+1 /H 1 is the canonical subgroup H n of level n of G = G/H 1 thanks to [AIPC,cor. A.2]. As multiplication by p on H n+1 defines an isomorphism H n+1 /H 1 ∼ = H n and hence an isomorphism

The basic constructions
Recall from §2.2 that we have introduced an m-adic formal scheme W 0 F , which is a formal model of the weight space. It is characterized by the property that the inverse image of the maximal α , one obtains a formal scheme X r,α over W 0 α . Set X r,α to be the associated analytic adic space over W α . For all choices of α, these formal schemes X r,α glue into a formal scheme X r → W 0 F . We let X r → W 0 F be the analytic adic space associated to X r . Let I = [p k , p k ] ⊂ [1, ∞] be an interval. We defined a formal scheme W 0 F,I → W 0 F and now we consider X r,I = X r × W 0 F W 0 F,I and X r,α,I = X r,α × W 0 α W 0 α,I . Let n ∈ N be an integer such that 1 ≤ n ≤ r + k. Applying the considerations of §3.3 we obtain anétale cover of adic spaces IG n,r,α,I → X r,α,I , n , as group schemes, equivariant for the O F -action. This is a morphism of adic spaces associated to a morphism of formal schemes IG n,r,α,I → X r,α,I .
For various α ∈ m these adic spaces and formal schemes glue and we obtain IG n,r,I → X r,I and IG n,r,I → X r,I .

Equations
In this subsection we give local equations for some of the spaces defined so far. We have: If α ∈ m \ m 2 , this is a regular ring. Otherwise this ring is 0. Consider an interval I = [p k , p h ] with k ≥ 0 an integer and h ≥ k an integer or h = ∞. Take α ∈ m \ m 2 . If 1 ∈ I and α = p, then B 0 α,I = B α . If α = p and 1 / ∈ I, B 0 α,I = 0. Assume now that α = p.
In the second case B 0 α,I is a regular ring and, in particular, it is normal. In the first case, one checks that B α,I is normal by verifying that it is Cohen-Macaulay and regular in codimension 1 (Serre's criterion). Let Lemma 3.7. The rings R and R are normal.
Moreover A is Cohen-Macaulay and flat over Z p , thus A/pA is also Cohen-Macaulay. As a result, , it is Cohen-Macaulay. Let us check that R is regular in codimension 1. Let P be a codimension 1 prime ideal of R. Then R P is easily seen to be regular if α ∈ P. Assume that α lies in P. Then P is a generic point of . Either Ha p r+1 ∈ P and in that case P maps to the generic point P of an irreducible component of A/(pA, Ha). By [AG] the ring (A/pA) P is a DVR so let t be a generator of its maximal ideal. Ift denotes a lift of t in R thent is a generator of the maximal ideal of R P and we are done. Otherwise, w ∈ P and in that case P maps to a generic point of A/pA, and w is a generator of the maximal ideal of R P . The normality of the ring R follows along similar lines.
Corollary 3.8. The formal schemes X r,[p k ,p h ] are normal.

Overconvergent modular sheaves in characteristic 0
In this section we will construct sheaves of overconvergent Hilbert modular forms over the adic space W F \ {|p| = 0}. This was already accomplished in [AIPH] but our goal now is to provide canonical integral models for the modular sheaves constructed in [AIPH].

A modified integral structure on ω G
Fix an interval I = p k , p k with k and k integers such that k ≥ k ≥ 0. Let r ∈ Z ≥1 and fix a positive integer n with n ≤ r + k. Let G be the semi-abelian scheme over X r,I . It follows from Proposition 3.2 that there exists a canonical subgroup H n ⊂ G[p n ].
Let g n : IG n,r,I → X r,I be the partial Igusa tower defined in §3.4. Let ω G be the sheaf of invariant differentials of G. It follows from [AIPC,Cor. A.2] that the kernel of the map ω G /p n ω G → ω Hn is annihilated by Hdg p n −1 p−1 ω G . We deduce that the projection map ω G → ω G /p n Hdg − p n −1 p−1 ω G factors via ω Hn . One then has a commutative diagram of f ppf sheaves of abelian groups over X r,I : where all vertical arrows are surjective and the horizontal arrow is the Hodge-Tate map. Over IG n,r,I we have a universal section P ∈ H D n which is the image of 1 via the universal morphism Proposition 4.1. Let F be the inverse image in ω G of the O IG n,r,I -submodule of ω G /p n Hdg − p n −1 p−1 ω G spanned by HT(P ). Then F is a locally free O F ⊗ O IG n,r,I -module of rank 1, the cokernel of F ⊂ ω G is annihilated by Hdg 1 p−1 and the map HT • ψ n defines an isomorphism of O F ⊗ O IG n,r,Imodules: Proof. This is a variant of [AIPH], Prop. 3.4. Let U := Spf R ⊂ IG n,r,I be an open formal affine subscheme such that ω G | U is free of rank g as an R-module. Write M ∈ M n×n R/p n Hdg − p n −1 p−1 R for the matrix of the linearization of the map HT over U . Thanks to [AIPC,prop. A.3] it has determinant ideal equal to Hdg 1 p−1 . In particular Hdg Let M ∈ M n×n (R) be any lift of M . Its determinant δ is Hdg 1 p−1 (up to unit). Let S ⊂ ω G | U be the submodule spanned by the columns of M . Then δω G ⊂ S. Since p n Hdg − p n −1 p−1 ω G = p n Hdg − p n p−1 · δω G ⊂ S one deduces that F| U coincides with the S. In particular it is a free R-module of rank g. By definition it is stable for the action of O F .
For every x ∈ O F /p n O F the image y ∈ HT(ψ n (x)) lies by construction in the image of S in ω G /p n Hdg − p n −1 p−1 ω G . As Hdg − p n p−1 = Hdg − p n −1 p−1 · Hdg 1 p−1 and as ω G /F is annihilated by Hdg 1 p−1 it follows that any two lifts y and y in S differ by an element lying in Hdg − p n −1 p−1 · Hdg 1 p−1 ω G = Hdg − p n −1 p−1 · S. We then get a well defined map O F /p n O F → F/p n Hdg − p n p−1 F inducing HT • ψ n when composed with the projection to ω G /p n Hdg − p n p−1 Fω G . This provides the HT . By construction its restriction to U is a surjective map of free R/p n Hdg − p n p−1 R-modules of rank g and hence it is an isomorphism. It follows that S = F| U is a free O F ⊗ R-module of rank 1 concluding the proof of the Proposition.
We denote by f n : F n,r,I → IG n,r,I the torsor for the group 1 + p n Hdg − p n One has an action of (O F ⊗ Z p ) * on F n,r,I , lifting the action of (O F /p n O F ) * on IG n,r,I , given by λ · (ω, 1) = (λω, λ). We then get a well defined action of the group

The sheaves of overconvergent forms
Fix an interval I = p k , p k with k and k integers such that k ≥ k ≥ 0. Let r, n ∈ Z ≥0 . We assume that r Proof. The claim is local on IG n,r,I . Let α ∈ m \ m 2 . We prove the claim over an open U = Spf R ⊂ IG n,r,α,I over the open W 0 α,I = Spf B 0 α,I of W 0 F . By construction we have p/α p k ∈ B 0 α,I and αHdg −p r+1 ⊂ R. Hence pHdg −p r+k+1 ⊂ R. In particular pHdg −p n ⊂ R and the second claim follows.
Proposition 2.8 implies that the character κ extends to a character as the subsheaf of f n, * O F n,r,I of sections transforming according to the character κ −1 under the action of 1 + p n Hdg − p n p−1 Res O F /Z G a . It is an invertible sheaf over IG n,r,I . Define w n,r,I ⊂ g n, * w 1 n,r,I as the subsheaf of (g n Proposition 4.3. The sheaf w n,r,I is an invertible O X r,I -module of rank 1.
The rest of this section is devoted to the proof of Proposition 4.3. We follow closely [AIPC,§5] by starting with the following: Lemma 4.4. Let (O X r,I ) 00 be the ideal of topologically nilpotent elements of O X r,I . Suppose that r ≥ 1 (resp. r ≥ 2 if p = 2). Then κ((O F ⊗ Z p ) * ) − 1 ⊂ Hdg O X r,I 00 and for every integer such that 2 ≤ ≤ r + k we have Proof. We deal with the case p = 2 leaving to the reader the case p = 2. The claim is local on X r,I . We restrict ourselves to an open formal affine subscheme U = Spf R mapping to the open W 0 α of W 0 F defined by an element α ∈ m \ m 2 . By construction, κ((O F ⊗ Z p ) * ) − 1 ⊂ αB 0 α and since αHdg −1 ⊂ O X r,I 00 , we can conclude that the first point holds. Using lemma 2.5 we see that for ≥ 2, we have that κ(1 Arguing as in lemma 4.2 we deduce from the assumption that ≤ r + k that pHdg −p ∈ O X r,I 00 . On the other hand as r ≥ 1, then α ∈ Hdg p 2 O X r,I so that α p −2 ∈ Hdg p O X r,I . As p −1 p−1 < p , it follows that We also have the following: Lemma 4.5. The inclusion O IG n,r,I → f n, * O F n,r,I defines an isomorphism O IG n,r,I /qp n O IG n,r,I → w 1 n,r,I /qp n w 1 n,r,I .

Proof.
Consider an open formal affine subscheme U = Spf R ⊂ IG n,r,I mapping to the open formal subscheme W 0 α of W 0 F defined by some α ∈ m \ m 2 . Assume that ω G | U is free. The choice of an elements ∈ F| U lifting s := HT (1) defines a section of the morphism F n,r,I | U ∼ = IG n,r,I | U and hence an isomorphism fs : w 1 n,r,I | U → O IG n,r,I | U given by evaluating the functions ats. Two different liftss ands differ by an element of 1+p n Hdg − p n p−1 O F ⊗R thanks to proposition 4.1. Proposition 4.2 implies that 1 + p n Hdg − p n . As I = [p k , p k ] we conclude that κ(1 + p k +1+n O F ⊗ R) ⊂ 1 + p n +1 R (and similarly κ(1 + p k +3+n O F ⊗ R) ⊂ 1 + qp n R for p = 2). Thus fs ≡ fs modulo qp n . This provides the inverse to the isomorphism in the lemma.
Let U = Spf R be an open affine formal subscheme of X r,I . Suppose that ω G is free over U . Thanks to Corollary 3.5 for every non-negative integer n such that 0 ≤ n ≤ r + k there exist elements c 0 = 1 and c n ∈Ha − p n −1 p−1 O IGn,r (Spf R) for n ≥ 1 such that Tr IG (c n ) = c n−1 for every n ≥ 1. If n satisfies r + k ≥ n ≥ k + 3 (resp. n ≥ k + 4 if p = 2) we define a projector: e cn : g n, * w 1 n,r,I (R) →Ha The following lemma proves Proposition 4.3: Lemma 4.6. Let s ∈ g n, * w 1 n,r,I (R) be an element such that s ≡ 1 mod p (in the sense of Lemma 4.5). Then e cn (s) ∈ w n,r,I (R) and w n,r,I (R) is the free R-module generated by e cn (s).
Proof. The proof is entirely analogous to the proof of [AIPC,Lemme 5.4]. Write s = 1 + ph for a section h ∈ F n,r,I (R). We get In this formula,σ is an arbitrary lift of σ to T(Z p ). SinceHa p r+k+1 | p and p n −1 p−1 < p r+k+1 , it follows that p σ∈(O F /p n O F ) * κ(σ)σ(c n h) ∈ R 00 F n,r,I (R) where R 00 is the ideal of topologically nilpotent elements in R.
We need to show that This follows from lemmas 2.6 and 4.4. As a consequence, e cn (s) belongs to w n,r,I (R) and one checks easily that it is a generator using the normality of R as in [AIPC,Lemme 5.4].
4.3 Properties of w n,r,I

Functoriality
Fix intervals I ⊂ I, r and r such that r ≥ r and integers n ≥ n so that (I , r , n ) and (I, r, n) satisfy the assumptions given at the beginning of §4.2 . We have the following commutative diagram: Proposition 4.7. The morphism above is an isomorphism.
Proof. Consider O X r ,I → ι * w −1 n,r,I ⊗ w n ,r ,I . This last sheaf is the subsheaf of (g n • f n ) * O F n ,r ,I consisting of sections on which (O F ⊗Z p ) * · 1+p n Hdg − p n p−1 O F ⊗Z p ) acts trivially. This coincides with the sheaf O X r ,I by the normality of X r ,I . The composite map O X r ,I → ι * w −1 n,r,I ⊗ w n ,r ,I → O X r ,I is the identity. This proves the claim.
We simplify the notations and write w I instead of w n,r,I .

Frobenius
Propositions 3.3 and 3.6 provide compatible morphisms φ : X r,I → X r−1,I and IG n+1,r,I → IG n,r−1,I obtained by composing the projection IG n+1,r,I → IG n,r,I and the Frobenius map φ : IG n,r,I → IG n,r−1,I . Let us recall the description of the morphism IG n+1,r,I → IG n,r−1,I . Let F : G → G/H 1 = G be the canonical isogeny between the semi-abelian schemes G over X r,I and G over X r−1,I . This morphism induces a surjective morphism of canonical subgroups H n+1 → H n+1 /H 1 ∼ = H n of G and G respectively. Dualizing we get an injective morphism F D : H D n → H D n+1 . The map φ : IG n+1,r,I → IG n,r ,I associates to a morphism ψ : making the following diagram commute: We then get the commutative diagram: where the morphism F n+1,r,I → F n,r−1,I is given by mapping a differential w ∈ F to pw ∈ F ⊂ ω G . One checks that this is well defined by using the following commutative diagram: Proposition 4.8. The morphism φ * w I → w I is an isomorphism.
Proof. The proof is analogous to the proof of Proposition 4.7.

Perfect overconvergent modular forms
In this section we define a sheaf of perfect overconvergent Hilbert modular forms over the weight space W 0 F and in the next we will show that one can undo the perfectisation. Taking the limits we get formal schemes IG n,∞,I → X ∞,I over W 0 F . Varying n we get a tower of formal schemes · · · → IG n+2,∞,I → IG n+1,∞,I → IG n,∞,I . Let IG ∞,∞,I be the projective limit. As the index r varies now, we denote by G r → X r,I the semi-abelian scheme and by Hdg r ⊂ O X r,I the Hodge ideal defined by G r .

The anti-canonical tower
Recall from §3.3 that associated to the finite morphism IG n,r,I → IG n−1,r,I we have a trace map Tr IG : O IG n,r,I → O IG n−1,r,I . These are compatible for varying r and define a trace map Tr IG : O IG n,∞,I → O IG n−1,∞,I .

Tate traces
Let α ∈ m \ m 2 . Denote by IG ∞,∞,α,I → X ∞,α,I the base change of the formal schemes above to W 0 α,I → W 0 F . Let h r : X ∞,α,I → X r,α,I be the projection map onto the r-th factor.
Proposition 5.2. One has Tate traces: as soon as p r (p − 1) > 2g + 1. For every non-negative integer k ≥ r + 1 define B 0 α,I,p −k := B 0 α,I α p −k . One proves as in §3.4.1 that it is a normal ring. Let W 0 α,I,p −k be the associated α-adic formal scheme and let W 0 α,I,p −k := Spa B 0 α,I,p −k α −1 , B 0 α,I,p −k be the associated analytic adic space. Define X r,α,I,p −k to be the fiber product X r,α,I × W 0 α,I W 0 α,I,p −k . Define X r,α,I,p −(r+1) to be the normalization of X r,α,I in X r,α,I,p −(r+1) (see §3.3). For general k ≥ r + 1 let X r,α,I,p −k be the base-change of X r,α,I,p −(r+1) via the map W 0 α,I,p −k → W 0 α,I,p −(r+1) . The associated analytic adic space is X r,α,I,p −k such that we have morphisms 5.2.1 An explicit description of diagram 1 in §5.2 be an open formal affine so that the sheaf ω G is trivial. We will describe the fiber of the above chain of morphisms over U . Choose a lift of the Hasse invariant viewed as a scalarHa. The fiber of U in X r,α,I,p −(r+1) is the formal spectrum of Here the variable u and the equation uv − α p k −k are missing in case k = ∞. Arguing as in the proof of Lemma 3.7 it follows that R is a normal ring.
It is finite and free as an R-module with basis α a/p k for 0 ≤ a ≤ p k+1−r − 1. The associated formal scheme Spf R k is the open of X r,α,I,p −k over the open U ⊂ X.
Then the restriction of the diagram (1) to U is given by the ring homomorphisms:

The unramified case
We first assume that p is unramified in F . This implies that A is formally smooth over Z p . It follows from [AIPC, Lemme 6.1] applied to the extension finite and flat extension and that Tr(R k ) ⊂ p g α − (2g+1) p r S k . This implies that for all r ≥ r and k ≥ r + 1, In particular, defining we deduce that, if p r (p − 1) > 2g + 1, the image of h r, * O X r+s,α,I is contained in α −1 O X r,α,I,p −k ∩ O X r,α,I α −1 which is α −1 · O X r,α,I since X r,α,I,p −k → X r,α,I is a finite and dominant morphism and X r,α,I is normal. The Proposition follows from this.

The general case
We now drop the assumption that p is unramified in F . In this situation M(µ N , c) is not formally smooth. Nevertheless the Rapoport locus M(µ N , c) R ⊂ M(µ N , c) is the smooth locus and its complement is of codimension at least 2. We let X R r,α,I,p −k ⊂ X r,α,I,p −k be the open formal subscheme where the Rapoport condition holds.
Arguing as in the unramified case, we obtain a map Tr r := 1 p sg Tr φ s : h r, * O X R r+s,α,I → α −1 O X R r,α,I . Lemma 5.3. The formal scheme X R r,α,I is Zariski dense in X r,α,I .
Proof. This follows easily from the explicit equations. Note that we crucially use here that the complement of the Rapoport locus is of codimension 1 in the non-ordinary locus of the special fiber of M(µ N , c).
Consider the following commutative diagram: where Q runs over all codimension 1 prime ideals in R. Thus we are left to check that αTr r (f ) ∈ R Q whenever p ∈ Q.
If α ∈ Q as wHa p r+1 − α = 0, we deduce that the image of Q in M(µ N , c) lies in the ordinary locus and in particular in the Rapoport locus. Thus αTr r (f ) ∈ R Q . If α ∈ Q, then Q is a generic point of the special fiber of R and since the Rapoport locus is Zariski dense, Q lies in the Rapoport locus. Thus αTr r (f ) ∈ R Q . we have f = f s + h with f s ∈ R s [1/α] and h ∈ R so that we may assume that f ∈ R s [1/α]. Let f n + f n−1 a n−1 + · · · + a 0 = 0 with a 0 , . . ., a n−1 ∈ R be an integral relation.

The sheaf of perfect, overconvergent Hilbert modular forms
Applying Tr s one gets f n + f n−1 Tr h (a n−1 ) + · · · + Tr h (a 0 ) = 0 and as a i ∈ R for s large enough we have Tr s (a i ) ∈ R ∩ α −1 R s . For h ≥ 0 the morphism R s → R s+h is a finite dominant morphism of normal rings. Hence R s /α → R s+h /α is injective so that R s /α → R/α is injective as well. We deduce that αTr s (a i ) ∈ αR s and hence that Tr s (a i ) ∈ R s . Thus f is integral over R s and, hence, f ∈ R s proving the first claim of the lemma.
(2) The inverse image of Spf R in IG n,∞,α,I is equal to Spf R n with R n integral over R and R α −1 ⊂ R n α −1 finite andétale. In particular contains R and is integral over R and, hence by the first claim, it must be equal to R. Let R ∞ be the inverse image of Spf R in IG ∞,∞,α,I . Consider an element x ∈ R ∞ fixed by (O F ⊗ Z p ) * . There exists n large enough and x n ∈ R n such that x − x n = αx for some x ∈ R ∞ . In particular x is fixed by 1 + p n O F ⊗ Z p . Thanks to Proposition 5.1 for every n ≥ n there exists an element c n ∈ R n such that Tr R n /Rn (c n ) = α. In particular the higher cohomology groups of (1 + p n O F /p n O F ) acting on R n are annihilated by α.
For every s there exists n(s) ≥ n such that x ∈ (R n(s) /α s ) 1+p n O F ⊗Zp and hence there exists y s ∈ R n such that y s ≡ αx modulo α s . We deduce that y s converges to an element y for s → ∞ such that αx = y. Hence x ∈ R Proof. We prove the first claim. Let U := Spf R be an open formal subscheme of X ∞,α,I . Suppose that Hdg 1 is a principal ideal over Spf R with generatorHa 1 . Let Spf R n (resp. Spf R ∞ ) be the inverse image of U in IG n,∞,α,I (resp. in IG ∞,∞,α,I ). Due to Corollary 5.4 it suffices to exhibit an invertible element x ∈ R ∞ such that σ(x) = κ −1 (σ)x for every σ ∈ (O F ⊗ Z p ) * . Proposition 5.1 implies that there exist elements c n ∈Ha −1 1 R n such that Tr Rn/R n−1 (c n ) = c n−1 and c 0 = 1. Define b n : The last claim can be proved as in §4.3.

Descent
In this section we prove that the sheaf w perf I defined in §5.3 can be descended to some finite level.

Comparison with the sheaf w I
Consider an interval I = [p k , p k ] with k and k non negative integers. Thanks to proposition 4.3 we have an invertible sheaf w I over X r,I . Recall that we have a projection map h r : X ∞,I → X r,I . Then: Proposition 6.1. There exists a canonical isomorphism w perf I h * r w I .
Proof. Over X ∞,α,I we have a chain of isogenies where G s is the versal semi-abelian scheme over X s,I . Denote by C n,r → G r [p n ] the kernel of (F n ) D : G r [p n ] D → G r+n [p n ] D ). Clearly C n,r = H n (G r+n ) D . The isogeny F : G r+1 → G r induces a morphism C n,r+1 → C n,r which is generically an isomorphism. Over IG n,∞,I we have a universal morphism We then get an injective homomorphism h * r w I → w perf I . Moreover Proof. We prove the first statement. The second follows arguing as in [AIPC,Lemme 6.6] using the Tate traces constructed in Proposition 5.2. Let U := Spf A be a formal open affine subscheme of M(µ N , c) over which ω G is trivial. Let Ha be a lift of Ha. Arguing as in the proof of Lemma 3.7 one deduces that the inverse image of U in X r,α,I is Spf R with R := A⊗ Zp B α u, w /(wHa r −α, αu−p) and that for integers 1 ≤ h < k [AIPC,Lemme 6.4] shows that the map R → lim k R 1,k is an isomorphism. This proves the claim.
Theorem 6.4. The sheaf w perf I descends to an invertible sheaf w I over X r,α,I for r ≥ sup{4, 1 + log p 2g + 1} if p ≥ 3 and r ≥ sup{6, 1 + log p 2g + 1} if p = 2. More precisely w I is the subsheaf of O X r,α,I -modules of w perf I characterized by the fact that for every interval J = [p k , p k ] with k ≥ k ≥ 0 integers and denoting ι J,I : X r,α,J → X r,α,I the natural morphism, then ι * J,I w I is the sheaf w J of Proposition 4.3 compatibly with the identification of Proposition 6.1.
Moreover w I is free of rank 1 over every formal affine subscheme U ⊂ M(µ N , c) such that ω G | U is trivial.
Proof. The proof is analogous to the proof of [AIPC,Thm. 6.4]. We set where the limit is taken over integers k, k . Let U := Spf A be a formal open affine subscheme of M(µ N , c) where ω G is trivial. Let W := Spf B be the inverse image of U in X r,α,I . We prove that w I | W is a free O W -module of rank 1 and it descends w perf I | W . We prove the claim for the minimal r possible, i.e., r = 4 if p ≥ 3 and r = 6 for if p = 2. LetHa r be a lift of the Hasse invariant over U . Thanks to Corollary 3.5 and Proposition 5.1 we can find elements c n ∈Ha Using lemma 2.6, we deduce that: • The sequence b n converges to an element b ∞ , Consider an interval J = [p k , p k+1 ] and the commutative diagram: O IG n,r,α,J (T ) such that c 0 = 1, Tr IG (c n ) = c n−1 and c n = c n if n ≤ r. Take a section s ∈ O F k+r,r,α,J (T ) which is 1 mod p 2 and which generates w r+k,r,J (T ) (see lemma 4.5, and note that 1 = k + r − (k + 1) − 2 if p = 2 and 1 = k + r − (k + 1) − 4 if p = 2). . Let f : and it follows from 2.6 that Over the interval [p k , p k+1 ], we have p ∈ α p k B 0 α,I , and it follows that m n B 0 α,I ⊂ (α p n−1 , α p k +p n−2 , · · · , α (n−1)p k +1 ).
When p = 2, one proves similarly that It follows that . Taking the Tate trace Tr r : h r, * O X ∞,α,J → α −1 O X r,α,J constructed in Proposition 5.2 we conclude that Tr r (b ∞ ) = f (1 + α 2 Tr r (u) +Ha − p r+k −1 p−1 p 2 Tr r (v)). As Tr r (u), Tr r (v) ∈ α −1 O X r,α,J (T ) by Proposition 5.2, we deduce that Tr r (b ∞ ) is a generator of w J | T . Since the construction of Tr r (b ∞ ) is functorial in J we get that w I (W ) = Tr r (b ∞ ) · lim k+1≥k ≥k≥0 O X r,α,[p k ,p k ] (W ) = T r r (b ∞ )B thanks to Lemma 6.3. The theorem follows.
Proposition 6.5. The sheaves w I over each X r,α,I glue to a sheaf still denoted w I over X r,I . Let φ : X r+1,I → X r,I be the Frobenius. We have an isomorphism Proof. Due to Theorem 6.4 the sheaf w I over X r,α,I is canonically determined by the sheaves w perf I and the sheaves w J for J = [p k , p k ]. These glue for varying α and have compatible Frobenius morphisms by §4.3.2. The claim follows.
There is a cartesian diagram of formal schemes: (2) The sheaf g w I (here I = [1, ∞]) is an invertible sheaf over Z r and g g w I = w I .
(1) The map g is the base-change of the morphism W 0 F → W 0 F which is the m-adic completion of a blow-up, therefore it is proper. Thus C := g O Xr (W ) is a finite B-module. The map B → C is an isomorphism over the ordinary locus of U and, hence, it is generically an isomorphism. As X r is also normal it follows that B = C as claimed.
(2) Let Z := g −1 W . Thanks to (1) it suffices to prove that w I | Z is a free O Z -module of rank 1. Let Z perf be the inverse image of Z in X ∞ . We will actually prove that w perf I | Z perf is a free O Z perf -module of rank 1 and find a generator b ∞ whose trace will be a generator of w I | Z .
We apply the construction of §3.3 with A 0 = Z p . We thus obtain formal schemes Y s together with partial Igusa towers IGY n,s → Y s for n ≤ s.
Passing to the limit over Frobenius, we obtain IGY n,∞ → Y ∞ . There is an obvious commutative diagram: for integers n ≤ r and elements c n ∈Ha −p r r O IGY n,∞ (W ) for general r ≤ n so that c 0 = 1 and Tr IG (c n ) = c n−1 . We call c n the pull back of c n in O IGn,∞ (Z). We can now repeat the proof of Theorem 6.4 using these elements c n to obtain a trivialization b ∞ of w perf I | Z perf whose trace gives the trivialization of w I | Z .
We set w κ = g w I .

The main theorem
Let H be the torsion subgroup of T(Z p ). Let χ : H → Λ F be the restriction of the universal character to H. Due to Lemma 2.4 it is a quotient of (O F /p 2 O F ) * and we view χ as a character of (O Let w χ be the subsheaf of (h 2 ) O IGM 2,r where (O F /p 2 O F ) * acts via the character χ −1 : H → Λ * F composed with the projection (O F /p 2 O F ) * → H. This is a coherent sheaf, invertible over the ordinary locus and over the analytic fiber M r .
We define w κ un := (s w κ ) ⊗ w χ where s : M r → Z r is the projection. We let ω κ un be the associated sheaf over M r .
Theorem 6.7. The sheaf ω κ un over M r enjoys the following properties: 1. the restriction of ω κ un to the classical analytic space M r × Spa(Zp,Zp) Spa(Q p , Z p ) is the sheaf defined in [AIPH,Def. 3.6]; 2. for all locally algebraic weight kχ : T(Z p ) → O * Cp where k is an algebraic weight and χ is a finite character, then ω κ un | kχ = ω k (χ) is the sheaf of weight k modular forms and nebentypus χ.
3. If i : M r+1 → M r is the inclusion and φ : M r+1 → M r is the Frobenius, then we have a canonical isomorphism i ω κ un φ ω κ un .

Overconvergent forms in characteristic p
Specializing the sheaf ω κ un of Theorem 6.7 to characteristic p points of W F we obtain sheaves of overconvergent Hilbert modular forms in characteristic p. The goal of this section is to describe them via a construction purely in characteristic p.

Formal schemes attached to the Hilbert modular variety in characteristic p
Let m be the maximal ideal (T 1 , · · · , T g ) of Λ 0 F,{∞} is the blow-up of W 0 F,{∞} along m. In section 6.3 we defined an m-adic formal scheme Z r → W 0 F . We set Z r, In §3.4 we defined a formal scheme X r,{∞} over W 0 F,{∞} and it follows from the definitions that: Let IGZ n,r,{∞} = IG n × Spec Fp Z r,{∞} be the partial Igusa tower of level n over Z r,{∞} . Passing to the limit over n, we get an m-adic formal scheme h : IGZ ∞,r,{∞} → Z r,{∞} which carries an action of T(Z p ).
Proof. Easy and left to the reader.
We have the following: Proposition 7.5. For any normal, m-adically complete torsion free Λ 0 F /pΛ 0 F -algebra, the Rvalued points IGZ n,r,∞ (R) classify tuples x, η T 1 , · · · , η Tg , ψ n where n is a O F -linear morphism of group schemes which is an isomorphism over (Spec R) ord .
One then concludes that {b n } n is a Cauchy sequence of elements of R ∞ converging to a unit b ∞ := lim n b n of R ∞ having the property that σ( 7.5 Comparison with the sheaf w [1,∞] In this section we work over X r,{∞} and prove that the specialization at {∞} of the sheaf w [1,∞] of Theorem 6.4 equals the pull back to X r,{∞} of the sheaf w {∞} defined in §7.4. Fix now an interval [p k , p k+1 ] with k ≥ k 0 . Let Spf C be the inverse image of Spf B in X r,α,[p k ,p k+1 ] . Fix elements c n ∈Ha − p n −1 p−1 O IG n,r,α,[p k ,p k+1 ] (Spf C) for r + k 0 + 1 ≤ n ≤ r + k satisfying Tr IG (c n ) = c n−1 for n ≥ r + k 0 + 2 and Tr IG (c r+k 0 +1 ) = c r+k 0 . There is a generator f of the sheaf w I over Spf C such that Using lemma 2.6 one more time, we deduce that f = b r+k 0 mod α p k 0 −p k 0 −1 . As a consequence, As this relation holds on all intervals [p k , p k+1 ], it follows that Tr r ( Comparing the definition of b r+k 0 and the construction of the sheaf w {∞} we obtain that the restriction of w [1,∞] to X r,{∞} is w {∞} .

Analytic overconvergent Hilbert modular forms in characteristic p
We now let M r, Let us recall that we denoted by κ un : T(Z p ) → (Λ F /pΛ F ) * the universal mod p character.
Let ω κ un be the pull back of ω κ un to M r,{∞} . An r-overconvergent Hilbert modular form of weight κ un is a global section of ω κ un . Here is the desired,à la Katz, description.
Proposition 7.8. An r-overconvergent modular form f of weight κ un is a functorial rule which associates to a tuple (R, R + ), x : Spa(R, R + ) → M r,{∞} , ψ : O F ⊗ Z p lim n x * H D n an element in f (x, ψ) ∈ R, where: 1. (R, R + ) is a complete affinoid Tate algebra, 2. x * H D n is the pullback of the dual canonical subgroup of level n to Spa(R, R + ), 5. There exists a rational cover Spa(R, R + ) = ∪Spa(R i , R + i ) and for each i a bounded and open subring R i,0 ⊂ R + i such that x G| Spa(R i ,R + i ) comes from a semi-abelian scheme G 0 over Spf R i,0 and the isomorphism ψ| Spa(R i ,R + i ) comes from a group scheme morphism ψ 0 : O F ⊗ Z p → lim n (G 0 [F n ]) D defined over SpfR i,0 (where [F n ] means the kernel of the n-th power of the Frobenius isogeny).
Proof. Take (R, R + ) as in the proposition. Without loss of generality, we can assume that R has a noetherian ring of definition. Using 5), we observe that the rule f defines compatible sections of H 0 (Spf R i,0 , w {∞} ) ⊗ R i,0 R which glue, by the sheaf property, to a section of ω κ un on Spa(R, R + ).  [AIPH], intro., p. 6).
Lemma 8.1. The group ∆ acts freely on M (µ N , c). One can form the quotient Proof. Since M (µ N , c) is a projective scheme, it can be covered by open affine subschemes fixed by the action of ∆. Thus we can form the quotient M (µ N , c) G . We now show that ∆ acts freely on M (µ N , c). This can be proved over an algebraically closed field k where the freeness of the action amounts to proving the following: Consider an abelian variety with real multiplication (A, ι, Ψ, λ) over k as in §3 and a totally positive unit ∈ O +, * F . Let α : A → A be an automorphism commuting with the O F -action, the level N structure Ψ and such that λ • α = α ∨ • λ. Then α ∈ U N is a totally positive unit, congruent to 1 modulo N .
As α respects the level N structure, it suffices to prove that α is an endomorphism lying in O F . Suppose this is not the case. Then E := F [α] ⊂ End 0 (A) would be a commutative algebra of dimension at least 2g. It must be a field, else A would decompose as a product of at least two abelian varieties of dimension < g, with real multiplication by F which is impossible. As a maximal commutative subalgebra of End 0 (A) has dimension ≤ 2g, it follows that E is a CM field of degree 2 over F . Moreover the Rosati involution associated to any O F -invariant polarization induces complex conjugation on E. As the rank of the group of units in O E is equal to the rank of the group of units of O F by Dirichlet's unit theorem, it follows that there exists an integer n ≥ 2 such that α n ∈ O * F and α n−1 ∈ O * F . Hence ζ = (α/ᾱ) is a primitive n-root of unity in O E . It preserves every O F -equivariant polarization λ as λ −1 • ζ ∨ • λ • ζ = ζζ = 1. As N ≥ 4 it follows from Serre's lemma that ζ = 1 leading to a contradiction. We are left to show that ∆ acts freely on the boundary D := M (µ N , c) G \ M (µ N , c) G . Recall that the boundary is the union of its connected components parametrized by the cusps of the minimal compactification. Each connected component of the boundary is stratified. More precisely, for each connected component, there is a polyhedral decomposition Σ of the cone of totally positive elements M + inside a fractional ideal M ⊂ F determined by the cusp. To every cone σ ∈ Σ corresponds a stratum S σ ⊂ D. By construction of the toroidal compactification, if ∈ U N , then S 2 σ = S σ . We now claim that the action of ∆ on the set of all strata in D is free. This follows from the fact that the stabilizer of σ ∈ Σ is a finite subgroup of O +, * F , thus trivial. This concludes the proof.

Descending the sheaf w κ un
We follow closely [AIPH]; see especially the Introduction and §4. First of all the weight space associated to G is the formal scheme W G F defined by the Iwasawa algebra Λ G There is a natural map of formal schemes W G F → W F defined by the group homomorphism This induces a map of analytic adic spaces ι : W G F → W F . We denote by κ un Consider the formal scheme M r × W F W G F . Let w κ un G be the pull-back of the universal sheaf to M r × W F W G F . As a consequence of lemma 8.1 the group ∆ acts freely on the formal schemes M r × W F W G F . We denote by M r,G the quotients by the group ∆. We now claim that the action of ∆ on M r × W F W G F can be lifted to an action on the sheaf w κ un G . Since w κ un G is defined without any reference to polarization we have an isomorphism µ : [ ] * w κ un G → w κ un G for all ∈ O +, * F . We modify the action by multiplying µ by ν( ). One then verifies, see [AIPH,§4.1], that this action factors through the group ∆. By finiteétale descent we obtain a sheaf that we continue to denote w κ un G over M r,G . We let M r,G be the analytic fiber of M r,G and denote by ω κ un G the invertible sheaf on M r,G associated to w κ un G .

8.3
The cohomology of the sheaf ω κ un G (−D) There is an obvious map M r,G → M(µ N , c) G . Let D denote the boundary divisor in M(µ N , c) G . We also denote by D its inverse image in M r,G . Recall that M * (µ N , c) is the minimal compactification of M (µ N , c). Certainly, the construction of M r admits a variant where one uses M * (µ N , c) as a starting point instead of M (µ N , c). Let us denote by M r the resulting formal schemes.
We also define M r,G := M r × W F W G F /∆. Let h : M r,G → M * r,G be the canonical projection. The main result of this section is the following cohomology vanishing theorem: Theorem 8.2. We have R i h w κ un G (−D) = 0 for all i > 0.
Proof. This is a variant of [AIPH], Theorem 3.17. Recall that M (µ N , c) G is the quotient of M (µ N , c) by ∆. Let M * (µ N , c) be the minimal compactification and let M * (µ N , c) G be its quotient by ∆. We have a map h : M (µ N , c) G → M * (µ N , c) G . Let L be a torsion invertible sheaf on M (µ N , c) G . Then we claim that R i h L(−D) = 0 for i > 0. This follows from [AIPH], Prop. 6.4. Note that in that reference the proposition is stated for the trivial sheaf, but the proof works without any change for a torsion sheaf.
The map h : M r,G → M * r,G is an isomorphism away from the cusps and in particular away from the ordinary locus, so we are left to prove the statement for the map h ord : M ord,G → M * ord,G over the ordinary locus. In this case, the sheaf w κ un G (−D) is invertible. Recall that the ring Λ G F is semi-local and complete. Let n be a maximal ideal of Λ G F corresponding to a character η : (O F /pO F ) * × (F p ) × → F × q where F q is a finite extension of F p . We are left to prove the vanishing for the sheaf F := w κ un G (−D)/n over M ord,G . This is an invertible sheaf on its support M ord (µ N , c) G,Fq → M ord,G . Moreover F ⊗q−1 = O M ord (µ N ,c) G,Fq because the order of the character η divides q − 1, and we can conclude.
Corollary 8.3. Let g : M r,G → W G F be the projection to the weight space.
1. We have the following vanishing result: R i g ω κ un G (−D) = 0 for all i > 0.
2. For every point κ ∈ W G , is the space of r-overconvergent, cuspidal, arithmetic Hilbert modular forms of weight κ.
3. There exists a finite covering of the weight space W is a projective Banach R i -module, for every i.
Proof. We have a sequence of maps g : M r,G g 1 → M * r,G g 2 → W G F . The map g 2 is affine and therefore has no non zero higher cohomology. Moreover, Theorem 8.2 implies that R i (g 2 ) ω κ un G (−D) = 0. The second point follows easily. For the last point, fix some open Spa(R i , R + i ) ⊂ W G F . Since the sheaf ω κ un G (−D) is invertible, there is a finite covering ∪Spa(S i , S + j ) of M r,G | Spa(R i ,R + i ) such that the sheaf ω κ un G (−D) is trivial on every open of this covering. Arguing as in [AIPC,Prop. 6.9] one proves that each S i is a projective Banach R i -module. We can form the Chech complex associated to the covering ∪Spa(S i , S + j ), which, by (1), is a resolution of g ω κ un G (−D)(Spa(R i , R + i )). Moreover, each term appearing in the complex is a projective Banach module and thus g ω κ un G (−D)(Spa(R i , R + i )) is a projective Banach module. r,G of the adic space M r,G and we denote again by g o , resp. g the structural morphisms to W G F (see §8.2). The Koecher principle states that:
Proof. Since the complement of M (µ N ) in M (µ N ) is contained in the ordinary locus one may restrict to proving the two assertions on ordinary loci. This is classical: unravelling the constructions one is reduced to prove the claims for the structural sheaves on the adic Igusa tower. This can be proven at the level of formal schemes and then, by devissage, reducing to the Igusa tower IG The restriction of our construction to the ordinary locus gives back the theory of families of p-adic modular forms of Katz [K] and Hida [Hi2]; see [AIPS,§5.4] for details. In particular one gets q-expansions at the cusps.
g o * ω κ un → g o * p 1, * (p * 2 ω κ un ) π * ,−1 −→ g o * p 1, * (p * 1 ω κ un ) q −1 Trp 1 −→ g o * ω κ un Here Tr p 1 is the trace map of the finite flat morphism p 1 . One checks on q-expansions that dividing by the normalization factor q is a well defined operation. Using the Koecher principle of §8.4 we get a map g * ω κ un → g * ω κ un from the global sections of ω κ un over M r ( c) to the global sections over M r (c), which are denoted T , for not dividing p, and U , for = pO F or a prime ideal dividing p. This provides the definition of the Hecke operators for the overconvergent (cuspidal) forms defined in Theorem 6.7 for the group G * . Taking the quotient under the group ∆ we define such Hecke operators also for the arithmetic overconvergent cuspidal modular forms defined in §8.2 for the group G.

The adic eigenvariety for arithmetic Hilbert modular forms
In [AIPH], Theorem 5.1, we constructed a cuspidal eigenvariety over W G F \ {|p| = 0} for the p-adic, arithmetic Hilbert modular forms. We now extend it over W G F . Let Frac(F ) (p) be the group of fractional ideals prime to p. Let Princ(F ) +,(p) be the group of positive elements which are p-adic units. The quotient Frac(F ) (p) /Princ(F ) +,(p) = Cl + (F ) is the strict class group of F . For all c ∈ Frac(F ) (p) we have defined an adic space M r , that we now denote by M r (c) in order to mark its dependance on c and a sheaf ω κ un G over M r (c). Let g c : M r (c) → W G F be the projection. For all x ∈ Princ(F ) +,(p) we canonically identify (g c ) ω κ un G (−D) and (g xc ) ω κ un G (−D) as in [AIPH], Def. 4.6.
Taking the limit over r we thus obtain a sheaf of projective Banach modules c∈Cl + (F ) (g c ) ω κ un G (−D).
Thanks to §8.5 this sheaf carries an action of the Hecke algebra H p of level prime to p as well as an action of the U p operator and of the operators U P i (see [AIPH], §4.3). Moreover, having taken the limit over r implie the U p -operator is compact thanks to Proposition 3.3. Applying [AIPC], Appendice B, we obtain the following theorem.

The characteristic series
2. The spectral variety Z G = V (P G ) → W G F is locally finite, flat and partially proper over the weight space.
3. There is an eigenvariety E G → Z G , finite and torsion free over Z G , which parametrizes finite slope eigensystems of overconvergent, arithmetic Hilbert modular forms.