Rigid analytic tower associated to the parabolic subgroup

Chapter III: The Hodge-Newton filtration for Hodge-Newton reducible local

4.2 Rigid analytic tower associated to the parabolic subgroup

For our proof of Theorem 4.1.5, we construct an intermediate tower of rigid analytic spaces associated to the parabolic subgroupP.

4.2.1. For the rest of this chapter, we fix an EL realization(G,e[b],{µ})of our datum (G,[b],{µ})and takePeandeLas in Lemma 3.1.4. We continue to assume thatGeis of the form

Ge= Res_O|ZpGLn,

whereO is the ring of integers of some finite unramified extension ofQp. We also takeeLj,Lj,bj, µj for j =1,2,· · · ,r as in Theorem 3.2.2. Then Theorem 3.2.2 gives a Hodge-Newton decomposition of X

X = X₁×X₂× · · · ×X_r (4.2.1.1) and the induced Hodge-Newton filtration ofX

0⊂ X^(r⁾ ⊂ X^(r−¹⁾ ⊂ · · · ⊂ X⁽¹⁾ = X, (4.2.1.2) where each quotientX⁽^j)/X^(j⁺¹⁾ ' XjcarriesLj-structure that arises from the datum (Lj,[bj],{µj})with the choice bj ∈ [bj].

4.2.2. Following Mantovan in [Man08], Definition 9, we define a set-valued functor RZP,be on NilpZ˘p as follows: for any R ∈ NilpZ˘p, we set RZ

P,be (R) to be the set of isomorphism classes of triples(X,X^•, ι)where

• Xis a p-divisible group over Rwith an action ofO (see Example 2.4.5);

• X^•is a filtration ofp-divisible groups over R

0 ⊂ X^(r) ⊂ X^(r−1) ⊂ · · · ⊂ X⁽¹⁾ = X,

which is preserved by the action ofO such that the quotientsX⁽^j)/X⁽^j⁺¹⁾ are p-divisible groups (with the induced action ofO);

• ι: XR/p→ X_R/pis a quasi-isogeny which is compatible with the action ofO and induces quasi-isogeniesι⁽^j) : X_R/p⁽^j) −→ X⁽^j)

R/pfor j = 1,2,· · · ,r, such that for alla∈O and j =1,2,· · ·,r,

detR(a,Lie(X^(j)))=det(a,Fil⁰(X⁽^j))_˘

Qp). Mantovan in [Man08], Proposition 11 proved that the functor RZ

P,be is represented by a formal scheme which is formally smooth and locally formally of finite type over ˘Zp. We write RZ

P,be also for this representing formal scheme, and RZ^rig

P,be for its rigid analytic generic fiber. In addition, we write X

P,be and X^•

P,be respectively for the universal filteredp-divisible group over RZ

P,be and the associated “universal filtration”.

Remark. As in [Man08], Definition 10, we can also define a tower of étale covers RZ^∞

P,be = {RZ^K^f^p

X,Pe} over RZ^rig

P,be with a natural action of Pe(Qp) × Jb(Qp) and a Weil descent datum overE, whereKfp

0runs over open and compact subgroups ofPe(Zp).

4.2.3. By the functoriality of Rapoport-Zink spaces described in Proposition 2.6.6, the embeddingG,−→Geinduces a closed embedding

RZ_G,b,−→ RZ

G,be .

In addition, we have a natural map eπ₂ : RZ

P,be −→RZ

G,be

defined by(X,X^•, ι) 7→ (X, ι)on the points. We define RZ_P,b:=RZ

P,be ×_RZ

G,be RZ_G,b. Then we have the following Cartesian diagram:

RZ_P,b RZ_G,b

RZP,be RZ

G,be

π₂

eπ₂

Moreover, π₂ is a local isomorphism which gives an isomorphism on the rigid analytic generic fiber since eπ₂ has the same property (see [Man08], Theorem 36 and [Sh13], Proposition 6.3.).

We want to describe the universal property of the closed embedding RZP,b ,−→

RZP,be in an analogous way to the universal property of RZG,b ⊂ RZb described in 2.6.2. For this, we choose a decomposition ofΛ

Λ=Λ₁⊕Λ₂⊕ · · · ⊕Λr

corresponding to the decomposition ofeL in (3.2.1.1). We setΛ⁽^j) =Λ₁⊕ · · · ⊕Λj

for j =1,2,· · · ,r, and denote byΛ^•the filtration 0 ⊂ Λ⁽¹⁾ ⊂ · · · ⊂ Λ^(r) = Λ.

Then for anyZp-algebraRwe have

P(R)= {g ∈G(R):g(Λ^•_R)= Λ^•_R}. Now consider a morphism f : Spf(R) → RZ

P,be for someR∈Nilp_˘

Zp. Let(X,X^•)be ap-divisible group over Spec(R)with a filtration which pulls back to(f^∗X

P,be , f^∗X^•

P,be ) over Spf(R). We denote byD(X^•)the filtration of Dieudonné modules

0= D(X/X⁽¹⁾) ⊂ D(X/X⁽²⁾) ⊂ · · · ⊂D(X/X^(r)) ⊂ D(X)

induced by X^• via (contravariant) Dieudonne theory. We choose tensors (ˆti) on D(X)[1/p]as in 2.6.2. Then f factors through RZ_P,bif and only if eπ2◦ f factors through RZ_G,b ,−→RZ

G,be , which is equivalent to existence of a (unique) family of tensors(ti)onD(X)such that

(i) for some ideal of definitionJofRcontainingp, the pull-back of(ti)overR/J agrees with the pull-back of(ˆti)overR/J,

(ii) for a p-adic liftRof Rwhich is formally smooth over ˘Zp, theR-scheme P_R := IsomR

[D(X)_R,(ti)_R],[Λ^∗⊗_Z_p R,(si ⊗1)]

defined in 2.6.2 is aG-torsor, and consequently theR-scheme P_R⁰ :=IsomR

[D(X^•)_R,(ti)_R],[(Λ^•)^∗ ⊗_Z_pR,(si⊗1)]

is aP-torsor,

(iii) the Hodge filtration ofXis a {µ}-filtration with respect to(ti).

Here the scheme P_R⁰ in (ii) classifies the isomorphismsD(X)_R Λ^∗_R which map the tensors(ti)to(si ⊗1)and the filtrationD(X^•)_R to(Λ^•)^∗ ⊗_Z_pR.

We obtain the “universal p-divisible group" X_P,b over RZ_P,b with the associated

“universal filtration” X_P,b^• by taking the pull-back of X

P,be and X^•

P,be over RZ_P,b. We also obtain a family of “universal tensors" (t_i^univ,P)onD(X_P,b)by applying the universal property to an open affine covering of RZ_P,b. Moreover, this family has a “étale realization” (t^univ,P_i,ét ) on the Tate moduleTp(X_P,b)(see [Kim13], Theorem 7.1.6.).

4.2.4. The formal scheme RZ_P,bis formally smooth and locally formally of finite type over ˘Z^pby construction. Hence it admits a rigid analytic generic fiber, which we denote by RZ^rig_P,b. Moreover, sinceπ₂gives an isomorphism on the rigid analytic generic fiber, we have a Jb(Qp)-action and a Weil descent datum over E on RZ^rig_P,b induced by the corresponding structures on RZ^rig_G,b.

For any open compact subgroupKp0

ofP(Zp), we define the following rigid analytic étale cover of RZ^rig_P,b:

RZ^K^p

P,b :=Isom_RZrig P,b

[Λ^•,(si)],[Tp(X_P,b^• ),(t^univ,P_i,ét )] . Kp0.

TheJb(Qp)-action and the Weil descent datum overE on RZ^rig_P,bpull back to RZ^K^p

P,b. We denote by RZ^∞_P,b:={RZ^K^p

P,b}the tower of these covers with Galois groupP(Zp).

The Galois action on this tower gives rise to a naturalP(Qp)-action which commutes with theJb(Qp)-action and the Weil descent datum overE(cf. [Kim13], Proposition 7.4.8.). Hence the cohomology groups

Hⁱ(RZ^K^p

P,b)= H_cⁱ(RZ^K_P,b^p ⊗_˘

QpCp,Ql(dim RZ^K^p

P,b))

form a tower {Hⁱ(RZ^K^p

P,b)} for eachi, which are endowed with a natural action of P(Qp) × W_E × Jb(Qp). Moreover, for any admissible l-adic representation ρ of Jb(Qp), the groups

H^i,j(RZ^∞_P,b)_ρ:=lim

−−→

K_p⁰

Ext_J^j

b(Qp)(Hⁱ(RZ^K^p

P,b), ρ) satisfy the following properties (cf. 2.6.10):

(1) The groupsH^i,j(RZ^∞_P,b)_ρvanish for almost alli,j.

(2) There is a natural action of P(Qp) × W_E on eachH^i,j(RZ^∞_P,b)_ρ. (3) The representationsH^i,j(RZ^∞_P,b)_ρare admissible.

We can thus define a virtual representation ofP(Q^p) × W_E H^•(RZ^∞_P,b)_ρ := Õ

i,j≥0

(−1)ⁱ⁺^jH^i,j(RZ^∞_P,b)_ρ.

Remark. Alternatively, we can obtain the tower RZ^∞_P,bas the pull-back of the tower RZ^∞

P,be over RZ^rig_P,b.

Dalam dokumen On Hodge-Newton reducible local Shimura data of Hodge type (Halaman 46-50)