**of the Rapoport-Zink space for GSp(4)**

### Tetsushi Ito and Yoichi Mieda

Abstract. In this paper, we study the*ℓ-adic cohomology of*
the Rapoport-Zink tower for GSp(4). We prove that the smooth
representation of GSp_{4}(Q*p*) obtained as the*ith compactly sup-*
ported *ℓ-adic cohomology of the Rapoport-Zink tower has no*
quasi-cuspidal subquotient unless *i* = 2,3,4. Our proof is
purely local and does not require global automorphic methods.

**1** **Introduction**

In [RZ96], M. Rapoport and Th. Zink introduced certain moduli spaces of quasi-
isogenies of *p-divisible groups with additional structures called the* *Rapoport-Zink*
*spaces. They constructed systems of rigid analytic coverings of them which we*
call the *Rapoport-Zink towers, and established the* *p-adic uniformization theory of*
Shimura varieties generalizing classical ˇCerednik-Drinfeld uniformization. These
spaces uniformize the rigid spaces associated with the formal completion of certain
Shimura varieties along Newton strata.

Using the *ℓ-adic cohomology of the Rapoport-Zink tower, we can construct a*
representation of the product*G(*Q*p*)*×J(*Q*p*)*×W*(Q*p**/*Q*p*), where*G*is the reductive
group over Q*p* corresponding to the Shimura datum, *J* is an inner form of it, and
*W*(Q*p**/*Q*p*) is the Weil group of the *p-adic ﬁeld* Q*p*. It is widely believed that this
realizes the local Langlands and Jacquet-Langlands correspondences (cf. [Rap95]).

Classical examples of the Rapoport-Zink spaces are the Lubin-Tate space and the
Drinfeld upper half space; these spaces were extensively studied by many people
and many important results were obtained (cf. [Dri76], [Car90], [Har97], [HT01],
[Dat07], [Boy09] and references therein). However, very little was known about the
*ℓ-adic cohomology of other Rapoport-Zink spaces.*

The aim of this paper is to study cuspidal representations in the*ℓ-adic cohomol-*
ogy of the Rapoport-Zink tower for GSp_{4}(Q*p*). Let us denote the Rapoport-Zink
space for GSp_{4}(Q*p*) by *M*˘. It is a special formal scheme over Z*p** ^{∞}* =

*W*(F

*p*) in the sense of Berkovich [Ber96]. Let

*M*˘

^{rig}be the Raynaud generic ﬁber of

*M*˘, that is, the generic ﬁber of the adic space

*t( ˘M*) associated with

*M*˘. Using level structures

2010*Mathematics Subject Classification. Primary: 14G35; Secondary: 22E50, 11F70.*

at*p, we can construct the Rapoport-Zink tower*

*· · · −→M*˘*m+1*^{rig} *−→M*˘_{m}^{rig} *−→ · · · −→M*˘2^{rig} *−→M*˘1^{rig} *−→M*˘0^{rig}=*M*˘^{rig}*,*
where *M*˘*m*^{rig} *−→M*˘^{rig} is an ´etale Galois covering of rigid spaces with Galois group
GSp_{4}(Z*/p** ^{m}*Z). We take the compactly supported

*ℓ-adic cohomology (in the sense*of [Hub98]) and take the inductive limit of them. Then, on

*H*_{RZ}* ^{i}* := lim

*−→*

_{m}*H*

_{c}*( ˘*

^{i}*M*

*m*

^{rig}

*⊗*Q

*p∞*Q

*p*

^{∞}*,*Q

*ℓ*) (hereQ

*p*

*= FracZ*

^{∞}*p*

*), we have an action of a product*

^{∞}GSp_{4}(Q*p*)*×J(*Q*p*)*×W*(Q*p**/*Q*p*),
where*J* is an inner form of GSp_{4}.

The main theorem of this paper is as follows:

**Theorem 1.1 (Theorem 3.2)** *The* GSp_{4}(Q*p*)-representation *H*_{RZ}^{i}*⊗*Q*ℓ* Q*ℓ* *has no*
*quasi-cuspidal subquotient unlessi*= 2,3,4.

For the deﬁnition of quasi-cuspidal representations, see [Ber84, 1.20]. Note that
since *M*˘*m*^{rig} is 3-dimensional for every *m≥*0,*H*_{RZ}* ^{i}* = 0 unless 0

*≤i≤*6.

Our proof of this theorem is purely local. We do not use global automorphic
methods. The main strategy of the proof is similar to that of [Mie10a], in which
the analogous result for the Lubin-Tate tower is given; we construct the formal
model *M*˘*m* of *M*˘*m*^{rig} by using Drinfeld level structures and consider the geometry
of its special ﬁber. However, our situation is much more diﬃcult than the case of
the Lubin-Tate tower. In the Lubin-Tate case, the tower consists of aﬃne formal
schemes *{*Spf*A*_{m}*}**m**≥*0, and we can associate it with the tower of aﬃne schemes
*{*Spec*A*_{m}*}**m**≥*0. In [Mie10a], the second author deﬁned the stratiﬁcation on the
special ﬁber of Spec*A** _{m}* by using the kernel of the universal Drinfeld level structure,
and considered the local cohomology of the nearby cycle complex

*RψΛ along the*strata. On the other hand, our tower

*{M*˘

*m*

*}*

*m*

*≥*0 does not consist of aﬃne formal schemes and there is no canonical way to associate it with a tower of schemes. To overcome this problem, we take a sheaf-theoretic approach. For each direct summand

*I*of (Z

*/p*

*Z)*

^{m}^{4}, we will deﬁne the complex of sheaves

*F*

*m,I*on ( ˘

*M*

*m*)

_{red}so that the cohomology

*H*

*(( ˘*

^{i}*M*

*m*)red

*,F*

*m,I*) substitutes for the local cohomology of

*RψΛ along*the strata deﬁned by

*I*in the Lubin-Tate case. For the deﬁnition of

*F*

*m,I*, we use the

*p-adic uniformization theorem by Rapoport and Zink.*

There is another diﬃculty; since a connected component of *M*˘ is not quasi-
compact, the representation *H*_{RZ}* ^{i}* of GSp

_{4}(Q

*p*) is far from admissible. Therefore it is important to consider the action of

*J(*Q

*p*) on

*H*

_{RZ}

*, though it does not appear in our main theorem. However, the cohomology*

^{i}*H*

*(( ˘*

^{i}*M*

*m*)

_{red}

*,F*

*m,I*) has no apparent action of

*J(*Q

*p*), since

*J*(Q

*p*) does not act on the Shimura variety uniformized by

*M*˘. We use the variants of formal nearby cycle introduced by the second author in
[Mie10b] to endow it with an action of *J(*Q*p*). Furthermore, to ensure the smooth-
ness of this action, we use a property of ﬁnitely generated pro-p groups (Section
2). In fact, extensive use of the formalism developed in [Mie10b] make us possible
to work mainly on the Rapoport-Zink tower itself and avoid the theory of *p-adic*
uniformization except for proving that *M*˘*m* is locally algebraizable. However, for
the reader’s convenience, the authors decided to make this article as independent of
[Mie10b] as possible.

The authors expect that the converse of Theorem 1.1 also holds. Namely, we
expect that *H*_{RZ}^{i}*⊗*_{Q}*ℓ* Q*ℓ* has a quasi-cuspidal subquotient if *i*= 2,3,4. We hope to
investigate it in a future work.

The outline of this paper is as follows. In Section 2, we prepare a criterion for the
smoothness of representations over Q*ℓ*. It is elementary but very powerful for our
purpose. In Section 3, we give some basic deﬁnitions concerning with the Rapoport-
Zink space for GSp(4) and state the main theorem. Section 4 is devoted to introduce
certain Shimura varieties related to our Rapoport-Zink tower and recall the theory
of*p-adic uniformization. The proof of the main theorem is accomplished in Section*
5. The ﬁnal Section 6 is an appendix on cohomological correspondences. The results
in the section are used to deﬁne actions of GSp_{4}(Q*p*) on various cohomology groups.

**Acknowledgment** The second author would like to thank Noriyuki Abe and
Naoki Imai for the stimulating discussions.

**Notation** Let *p* be a prime number and take another prime *ℓ* with *ℓ* *̸*= *p. We*
denote the completion of the maximal unramiﬁed extension of Z*p* by Z*p** ^{∞}* and its
fraction ﬁeld by Q

*p*

*. Let*

^{∞}**Nilp**=

**Nilp**

_{Z}

*be the category of Z*

_{p∞}*p*

*-schemes on which*

^{∞}*p*is locally nilpotent. For an object

*S*of

**Nilp, we put**

*S*=

*S⊗*Z

*F*

^{p∞}*p*.

In this paper, we use the theory of adic spaces ([Hub94], [Hub96]) as a framework
of rigid geometry. A rigid space overQ*p** ^{∞}* is understood as an adic space locally of
ﬁnite type over Spa(Q

*p*

^{∞}*,*Z

*p*

*).*

^{∞}Every sheaf and cohomology are considered in the ´etale topology. Every smooth
representation is considered overQ*ℓ* or Q*ℓ*. For aQ*ℓ*-vector space *V*, we put *V*_{Q}

*ℓ* =
*V* *⊗*Q*ℓ*Q*ℓ*.

**2** **Preliminaries: smoothness of representations of** **profinite groups**

Let **G** be a linear algebraic group over a *p-adic ﬁeld* *F*. In this section, we give
a convenient criterion for the smoothness of a **G(F**)-representation over Q*ℓ*. The
following theorem is essential:

**Theorem 2.1** *LetKbe a closed subgroup of*GL* _{n}*(Z

*p*)

*and*(π, V)

*a ﬁnite-dimensional*

*representation over*Q

*ℓ*

*of*

*K*

*as an abstract group. Assume that there exists a*

*K-*

*stable*Z

*ℓ*

*-lattice*Λ

*of*

*V. Then this representation is automatically smooth.*

In order to prove this theorem, we require several facts on pro-p groups. Put
*K*_{1} =*K∩*(1 +*pM** _{n}*(Z

*p*)), which is a pro-p open subgroup of

*K*.

**Lemma 2.2** *The pro-p* *group* *K*_{1} *is (topologically) ﬁnitely generated.*

*Proof.* By [DdSMS99,*§*5.1], the proﬁnite group GL* _{n}*(Z

*p*) has ﬁnite rank. In partic- ular,

*K*

_{1}, a closed subgroup of GL

*(Z*

_{n}*p*), has ﬁnite topological generators.

**Lemma 2.3** *Every subgroup of ﬁnite index of* *K*_{1} *is open.*

*Proof.* In fact, this is true for every ﬁnitely generated pro-p group; this is due to
Serre [Ser94, 4.2, Exercices 6)]. See also [DdSMS99, Theorem 1.17], which gives a
complete proof.

**Remark 2.4** More generally, every subgroup of ﬁnite index of a ﬁnitely generated
proﬁnite group is open ([NS03], [NS07a], [NS07b]). It is a very deep theorem.

**Lemma 2.5** *Let* *G* *be a pro-ℓ* *group. Then every homomorphism* *f*: *K*_{1} *−→* *G* *is*
*trivial.*

*Proof.* Let*H* be an open normal subgroup of *G* and denote the composite *K*_{1} *−−→*^{f}*G−→G/H* by*f** _{H}*. By Lemma 2.3, Ker

*f*

*is an open normal subgroup of*

_{H}*K*

_{1}. Thus

*K*

_{1}

*/*Ker

*f*

*is a ﬁnite*

_{H}*p-group. On the other hand,*

*G/H*is a ﬁnite

*ℓ-group. Since we*have an injection

*K*

_{1}

*/*Ker

*f*

_{H}*,−→*

*G/H*, we have

*K*

_{1}

*/*Ker

*f*

*= 1, in other words,*

_{H}*f*

*= 1. Therefore the composite*

_{H}*K*

_{1}

*−−→*

^{f}*G−−→*

^{∼}^{=}lim

*←−*

^{H}*G/H*is trivial. Hence we have

*f*= 1, as desired.

*Proof of Theorem 2.1.* Since*K*_{1} is an open subgroup of*K, we may replaceK* by*K*_{1}.
Take a*K*_{1}-stableZ*ℓ*-lattice Λ of*V*. Then, Λ/ℓΛ is a ﬁnite abelian group. Therefore,
by Lemma 2.3, there exists an open subgroup*U* of*K*_{1} which acts trivially on Λ/ℓΛ.

In other words, the homomorphism *π*: *K*_{1} *−→* GL(Λ) *⊂* GL(V) maps *U* into the
subgroup 1+ℓEnd(Λ). Since*U* is a closed subgroup of 1+pM* _{n}*(Z

*p*) and 1+

*ℓ*End(Λ) is a pro-ℓ group, by Lemma 2.5, the homomorphism

*π|*

*U*:

*U*

*−→*1 +

*ℓ*End(Λ) is trivial. Namely,

*π|*

*U*is a trivial representation.

**Lemma 2.6** *Let* *F* *be a* *p-adic ﬁeld and* **G** *a linear algebraic group over* *F. Then*
*every compact subgroupK* *of***G(F**)*can be realized as a closed subgroup of*GL* _{n}*(Z

*p*)

*for somen.*

*Proof.* Take an embedding **G** *,−→* GL* _{m}* deﬁned over

*F*. Since

**G(F**) is a closed subgroup of GL

*(F),*

_{m}*K*is also a closed subgroup of GL

*(F). Therefore we have a faithful continuous action of*

_{m}*K*on

*F*

*. By taking a Q*

^{m}*p*-basis of

*F*, we have a faithful continuous action of

*K*on Q

^{n}*p*for some

*n. Since*

*K*is compact, it is well-known that there is a

*K-stable*Z

*p*-lattice in Q

^{n}*p*. Hence we have a continuous injection

*K ,−→*GL

*(Z*

_{n}*p*). Since

*K*is compact, it is isomorphic to a closed subgroup of GL

*n*(Z

*p*).

**Corollary 2.7** *Let* *F* *and* **G** *be as in the previous proposition. Let* *I* *be a ﬁltered*
*ordered set and* *{K*_{i}*}**i**∈**I* *be a system of compact open subgroups of* **G(F**) *indexed*
*byI.*

*Let*(π, V)*be a (not necessarily ﬁnite-dimensional)*Q*ℓ**-representation of***G(F**)*as*
*an abstract group. Assume that there exists an inductive system* *{V*_{i}*}**i**∈**I* *of ﬁnite-*
*dimensional*Q*ℓ**-vector spaces satisfying the following:*

**–** *For every* *i∈I,* *V*_{i}*is endowed with an action of* *K*_{i}*as an abstract group.*

**–** *For every* *i∈I,* *V*_{i}*has a* *K*_{i}*-stable* Z*ℓ**-lattice.*

**–** *There exists an isomorphism* lim*−→*^{i}^{∈}^{I}*V*_{i}*−−→*^{∼}^{=} *V* *as* Q*ℓ**-vector spaces such that the*
*composite* *V*_{i}*−→*lim*−→*^{i}^{∈}^{I}*V*_{i}*−−→*^{∼}^{=} *V* *is* *K*_{i}*-equivariant for every* *i∈I.*

*Then*(π, V) *is a smooth representation of* **G(F**).

*Proof.* Let us take *x* *∈* *V* and show that Stab_{G(F}_{)}(x), the stabilizer of *x* in **G(F**),
is open. There exists an element *i* *∈* *I* such that *x* lies in the image of *V*_{i}*−→* *V*.
Take*y∈V** _{i}* which is mapped to

*x. By Theorem 2.1 and Lemma 2.6,*

*V*

*is a smooth representation of*

_{i}*K*

*. Therefore Stab*

_{i}

_{K}*(y) is open in*

_{i}*K*

*, hence is open in*

_{i}**G(F**).

Since *V*_{i}*−→* *V* is *K** _{i}*-equivariant, we have Stab

_{K}*(y)*

_{i}*⊂*Stab

_{K}*(x)*

_{i}*⊂*Stab

_{G(F}_{)}(x).

Thus Stab** _{G(F)}**(x) is open in

**G(F**), as desired.

**Remark 2.8** Although we need the corollary above only for the case *F* = Q*p*, we
proved it for a general*p-adic ﬁeld* *F* for the completeness.

**3** **Rapoport-Zink space for** GSp(4)

**3.1** **The Rapoport-Zink space for** GSp(4) **and its rigid ana-** **lytic coverings**

In this subsection, we recall basic deﬁnitions concerning with Rapoport-Zink spaces.

General deﬁnitions are given in [RZ96], but here we restrict them to our special case.

LetX be a 2-dimensional isoclinic *p-divisible group over* F*p* with slope 1/2, and
*λ*_{0}: X *−−→*^{∼}^{=} X* ^{∨}* a (principal) polarization of X, namely, an isomorphism satisfying

*λ*

^{∨}_{0}=

*−λ*0. Consider the contravariant functor

*M*˘:

**Nilp**

*−→*

**Set**that associates

*S*with the set of isomorphism classes of pairs (X, ρ) consisting of

**–** a 2-dimensional *p-divisible group* *X* over *S,*

**–** and a quasi-isogeny (cf. [RZ96, Deﬁnition 2.8]) *ρ*: X*⊗*_{F}_{p}*S* *−→X⊗**S**S,*
such that there exists an isomorphism *λ*: *X* *−→* *X** ^{∨}* which makes the following

diagram commutative up to multiplication by Q^{×}*p*:
X*⊗*_{F}_{p}*S* ^{ρ}^{//}

*λ*0*⊗*id

*X⊗**S**S*

*λ**⊗*id

X^{∨}*⊗*_{F}_{p}*S* *X*^{∨}*⊗**S**S.*^{ρ}

oo *∨*

Note that such *λ* is uniquely determined by (X, ρ) up to multiplication by Z^{×}* _{p}* and
gives a polarization of

*X. It is proved by Rapoport-Zink that*

*M*˘is represented by a special formal scheme (cf. [Ber96]) over SpfZ

*p*

*. Moreover,*

^{∞}*M*˘is separated over SpfZ

*p*

*[Far04, Lemme 2.3.23]. However,*

^{∞}*M*˘is neither quasi-compact nor

*p-adic.*

We put *M*¯=*M*˘red, which is a scheme locally of ﬁnite type and separated over F*p*.
It is known that*M*¯is 1-dimensional (for example, see [Vie08]) and every irreducible
component of *M*¯is projective over F*p* [RZ96, Proposition 2.32]. In particular, *M*¯
has a locally ﬁnite quasi-compact open covering.

Let *D(*X)_{Q} = (N,Φ) be the rational Dieudonn´e module of X, which is a 4-
dimensional isocrystal over Q*p** ^{∞}*. The ﬁxed polarization

*λ*

_{0}gives the alternating pairing

*⟨*

*,*

*⟩*

*λ*0:

*N*

*×N*

*−→*Q

*p*

*(1). We deﬁne the algebraic group*

^{∞}*J*over Q

*p*as follows: for a Q

*p*-algebra

*R, the group*

*J(R) consists of elements*

*g*

*∈*GL(R

*⊗*

_{Q}

*p*

*N*) such that

**–** *g* commutes with Φ,

**–** and *g* preserves the pairing *⟨* *,* *⟩**λ*0 up to scalar multiplication, i.e., there exists
*c(g)∈R** ^{×}* such that

*⟨gx, gy⟩*

*λ*0 =

*c(g)⟨x, y⟩*

*λ*0 for every

*x, y*

*∈R⊗*Q

*p*

*N*.

It is an inner form of GSp(4), since *D(*X)_{Q} is the isocrystal associated with a basic
Frobenius conjugacy class of GSp(4).

In the sequel, we also denote*J*(Q*p*) by*J. Every elementg* *∈J* naturally induces
a quasi-isogeny *g*: X *−→* X and the following diagram is commutative up to Q^{×}*p*-
multiplication:

X ^{g}^{//}

*λ*0

X

*λ*0

X* ^{∨}* X

^{∨}*.*

^{g}oo *∨*

Therefore, we can deﬁne the left action of *J* on *M*˘ by *g*: *M*˘(S) *−→* *M*˘(S);

(X, ρ)*7−→*(X, ρ*◦g*^{−}^{1}).

We denote the Raynaud generic ﬁber of*M*˘by*M*˘^{rig}. It is deﬁned as*t( ˘M*)*\V*(p),
where*t( ˘M*) is the adic space associated with *M*˘(cf. [Hub94, Proposition 4.1]). As
*M*˘ is separated and special over Z*p** ^{∞}*,

*M*˘

^{rig}is separated and locally of ﬁnite type over Spa(Q

*p*

^{∞}*,*Z

*p*

*). Since*

^{∞}*M*˘has a locally ﬁnite quasi-compact open covering,

*M*˘

^{rig}is taut by [Mie10b, Lemma 4.14]. Moreover, by using the period morphism [RZ96, Chapter 5], we can see that

*M*˘

^{rig}is 3-dimensional and smooth over Spa(Q

*p*

^{∞}*,*Z

*p*

*) (cf. [RZ96, Proposition 5.17]).*

^{∞}Next we will consider level structures. Let *X*e be the universal *p-divisible group*
over *M*˘and *X*e^{rig} be the associated *p-divisible group over* *M*˘^{rig}. Note that *X*e^{rig} is
an ´etale *p-divisible group. Let us ﬁx a polarization* e*λ*: *X*e *−→* *X*e* ^{∨}* which is com-
patible with

*λ*

_{0}, i.e., satisﬁes the condition in the deﬁnition of

*M*˘. Let S be a connected rigid space over Q

*p*

*(i.e., a connected adic space locally of ﬁnite type over Spa(Q*

^{∞}*p*

^{∞}*,*Z

*p*

*)), S*

^{∞}*−→*

*M*˘

^{rig}a morphism over Q

*p*

*and*

^{∞}*X*e

_{S}

^{rig}the pull-back of

*X*e

^{rig}. Fix a geometric point

*x*of S and an isomorphism

*T*

*(µ*

_{p}

_{p}*∞*

*,S*)

*= Z*

_{x}*p*(1)

*∼*= Z

*p*. Then

*λ*e induces an alternating bilinear form

*ψ*

_{λ}_{e}on the

*π*

_{1}(S, x)-module (T

_{p}*X*e

^{rig})

*;*

_{x}*ψ*_{e}* _{λ}*: (T

_{p}*X*e

^{rig})

_{x}*×*(T

_{p}*X*e

^{rig})

_{x}*−→T*

*(µ*

_{p}

_{p}*∞*

*,S*)

_{x}*∼*=Z

*p*

*.*

Fix a freeZ*p*-module*L*of rank 4 and a perfect alternating bilinear form*ψ*_{0}: *L×L−→*

Z*p*. Put *K*_{0} = GSp(L, ψ_{0}), *V* =*L⊗*_{Z}*p*Q*p* and *G* = GSp(V, ψ_{0}). Let *T*(S, x) be the
set consisting of isomorphisms *η*: *L−−→*^{∼}^{=} (T*p**X*e^{rig})*x* which map *ψ*0 toZ^{×}*p*-multiples of
*ψ*_{e}* _{λ}*. It is independent of the choice of e

*λ*and

*T*

*(µ*

_{p}

_{p}*∞*

*,S*)

_{x}*∼*=Z

*p*, since they are unique up to Z

^{×}*-multiplication. Obviously, the groups*

_{p}*K*

_{0}and

*π*

_{1}(S, x) naturally act on

*T*(S, x).

For an open subgroup *K* of *K*_{0}, a *K-level structure of* *X*e_{S}^{rig} means an element
of (T(S, x)/K)^{π}^{1}^{(S,x)}. Note that, if we change a geometric point *x* to *x** ^{′}*, the sets
(T(S, x)/K)

^{π}^{1}

^{(S,x)}and (T(S, x

*)/K)*

^{′}

^{π}^{1}

^{(S,x}

^{′}^{)}are naturally isomorphic. Thus the notion of

*K-level structures is independent of the choice of*

*x. The functor that associates*S with the set of

*K-level structures of*

*X*e

_{S}

^{rig}is represented by a ﬁnite Galois ´etale covering

*M*˘

_{K}^{rig}

*−→M*˘

^{rig}, whose Galois group is

*K*0

*/K. Since*

*T*(S, x) is a

*K*0-torsor,

*M*˘

*K*

^{rig}0 coincides with

*M*˘

^{rig}. If

*K*

*is an open subgroup of*

^{′}*K, we have a natural*morphism

*p*

*KK*

*:*

^{′}*M*˘

_{K}^{rig}

*′*

*−→*

*M*˘

_{K}^{rig}. Therefore, we get the projective system of rigid spaces

*{M*˘

*K*

^{rig}

*}*

*K*indexed by the ﬁltered ordered set of open subgroups of

*K*

_{0}, which is called the

*Rapoport-Zink tower. Obviously, the group*

*J*acts on the projective system

*{M*˘

_{K}^{rig}

*}*

*K*.

Let*g* be an element of*G* and *K* an open subgroup of*K*_{0} which is enough small
so that *g*^{−}^{1}*Kg* *⊂* *K*_{0}. Then we have a natural morphism *M*˘*K*^{rig} *−→* *M*˘_{g}^{rig}*−*1*Kg* over
Q*p** ^{∞}*. If

*g*

*∈*

*K*

_{0}, then it is given by

*η*

*7−→η◦g; for other*

*g, it is more complicated*[RZ96, 5.34]. In any case, we get a right action of

*G*on the pro-object “lim

*←−*”

*M*˘

_{K}^{rig}.

**Definition 3.1**We put

*H*

_{RZ}

*= lim*

^{i}*−→*

^{K}*H*

_{c}*( ˘*

^{i}*M*

*K*

^{rig}

*⊗*Q

*p∞*Q

*p*

^{∞}*,*Q

*ℓ*).

Here *H*_{c}* ^{i}*( ˘

*M*

_{K}^{rig}

*⊗*

_{Q}

*p∞*Q

*p*

^{∞}*,*Q

*ℓ*) is the compactly supported

*ℓ-adic cohomology of*

*M*˘

_{K}^{rig}

*⊗*Q

*p∞*Q

*p*

*deﬁned in [Hub98]; note that*

^{∞}*M*˘

_{K}^{rig}is separated and taut. By the constructions above,

*G×J*acts on

*H*

_{RZ}

*on the left (the action of*

^{i}*j*

*∈*

*J*is given by (j

*)*

^{−1}*). Obviously the action of*

^{∗}*G*on

*H*

_{RZ}

*is smooth. On the other hand, it is known that the action of*

^{i}*J*on

*H*

_{RZ}

*is also smooth. This is due to Berkovich (see [Far04, Corollaire 4.4.7]); see also Remark 5.12, where we give another proof of the smoothness. Hence we get the smooth representation*

^{i}*H*

_{RZ}

*of*

^{i}*G×J*.

Our main theorem is the following:

**Theorem 3.2 (Non-cuspidality)** *The smooth representation* *H*^{i}

RZ,Q*ℓ* *of* *Ghas no*
*quasi-cuspidal subquotient unlessi*= 2,3,4.

For the deﬁnition of quasi-cuspidal representations, see [Ber84, 1.20].

Theorem 3.2 is proved in Section 5.

**3.2** **An integral model** *M* ˘

^{m}**of** *M* ˘

*K*

^{rig}

*m*

For an integer*m* *≥*1, let *K** _{m}* be the kernel of GSp(L, ψ

_{0})

*−→*GSp(L/p

^{m}*L, ψ*

_{0}). It is an open subgroup of

*K*

_{0}. We can describe the deﬁnition of

*K*

*-level structures more concretely. As in the previous subsection, we ﬁx a polarizatione*

_{m}*λ*of

*X*e

^{rig}which is compatible with

*λ*

_{0}. It induces the alternating bilinear morphism between ﬁnite

´etale group schemes *ψ*_{e}* _{λ}*:

*X*e

^{rig}[p

*]*

^{m}*×X*e

^{rig}[p

*]*

^{m}*−→*

*µ*

_{p}*. Let S*

^{m}*−→*

*M*˘

^{rig}be as in the previous subsection. Then a

*K*

*m*-level structure of

*X*e

_{S}

^{rig}naturally corresponds bijectively to an isomorphism

*η*:

*L/p*

^{m}*L*

*−−→*

^{∼}^{=}

*X*e

_{S}

^{rig}[p

*] between ﬁnite ´etale group schemes such that there exists an isomorphism Z*

^{m}*/p*

*Z*

^{m}*−−→*

^{∼}^{=}

*µ*

_{p}

^{m}*which makes the following diagram commutative:*

_{,S}*L/p*^{m}*L×L/p*^{m}*L* ^{ψ}^{0} ^{//}

*η**×**η* *∼*=

Z*/p** ^{m}*Z

*∼*=

e

*X*_{S}^{rig}[p* ^{m}*]

*×X*e

_{S}

^{rig}[p

*]*

^{m}

^{ψ}^{e}

^{λ}^{//}

*µ*

_{p}

^{m}

_{,S}*.*

For simplicity, we write *M*˘*m*^{rig} for *M*˘_{K}^{rig}* _{m}* and

*p*

*for*

_{mn}*p*

_{K}

_{m}

_{K}*. In this subsection, we construct a formal model*

_{n}*M*˘

*m*of

*M*˘

*m*

^{rig}by following [Man05,

*§*6]. Let

*S*be a formal scheme of ﬁnite type over

*M*˘

^{rig}and denote by

*X*e

*the pull-back of*

_{S}*X*e to

*S*. A Drinfeld

*m-level structure ofX*e

*is a morphism*

_{S}*η*:

*L/p*

^{m}*L−→X*e

*[p*

_{S}*] satisfying the following conditions:*

^{m}**–** the image of *η* gives a full set of sections of *X*e* _{S}*[p

*],*

^{m}**–** and there exists a morphism Z*/p** ^{m}*Z

*−→*

*µ*

_{p}

^{m}*which makes the following dia- gram commutative:*

_{,S}*L/p*^{m}*L×L/p*^{m}*L* ^{ψ}^{0} ^{//}

*η**×**η*

Z*/p** ^{m}*Z

e

*X** _{S}*[p

*]*

^{m}*×X*e

*[p*

_{S}*]*

^{m}

^{ψ}^{e}

^{λ}^{//}

*µ*

_{p}

^{m}

_{,}

_{S}*.*

It is known that the functor that associates *S* with the set of Drinfeld *m-level*
structures of *X*e* _{S}* is represented by the formal scheme

*M*˘

*m*which is ﬁnite over

*M*˘ (cf. [Man05, Proposition 15]). Note that, unlike the case of Lubin-Tate tower,

*M*˘

*m*

is not necessarily ﬂat over *M*˘. It is easy to show that *M*˘*m* gives a formal model of

*M*˘*m*^{rig}, namely, the Raynaud generic ﬁber of *M*˘*m* coincides with *M*˘*m*^{rig}. We denote
( ˘*M**m*)_{red} by*M*¯*m*, which is a 1-dimensional scheme over F*p*.

There is a natural left action of*J* on*M*˘*m*which is compatible with that on*M*˘_{m}^{rig}.
On the other hand, the natural action *K*0 on *L/p*^{m}*L* induces a right action of *K*0

on*M*˘*m*, which is compatible with that on *M*˘_{K}^{rig}* _{m}*.

We can also describe *M*˘*m* as a functor from **Nilp** to **Set; for an object** *S* of
**Nilp, the set** *M*˘*m*(S) consists of isomorphism classes of triples (X, ρ, η), where
(X, ρ)*∈M*˘*m*(S) and *η*: *L/p*^{m}*L−→X[p** ^{m}*] is a Drinfeld

*m-level structure of*

*X. By*this description, the action of

*j*

*∈J*on

*M*˘

*m*is given by (X, ρ, η)

*7−→*(X, ρ

*◦j*

^{−}^{1}

*, η).*

On the other hand, the action of*g* *∈K*_{0} on*M*˘*m* is given by (X, ρ, η)*7−→*(X, ρ, η*◦g).*

By [Man04, Lemma 7.2], *{M*˘*m**}**m≥0* forms a projective system of formal schemes
equipped with the commuting action of *J* and *K*_{0}.

**3.3** **Compactly supported cohomology of** *M* ¯

*m*

For*m* *≥*0, we denote the set of quasi-compact open subsets of *M*¯*m* by *Q**m*. It has
a natural ﬁltered order by inclusion.

**Definition 3.3** For an object *F* of *D** ^{b}*( ¯

*M*

*m*

*,*Z

*ℓ*) or

*D*

*( ¯*

^{b}*M*

*m*

*,*Q

*ℓ*), we put

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*) = lim

_{U}*−→*

_{∈Q}*m*

*H*_{c}* ^{i}*(U,

*F|*

*U*).

Assume that *F* has a *J*-equivariant structure, namely, for every *g* *∈* *J* an isomor-
phism*φ** _{g}*:

*g*

^{∗}*F*

*−−→ F*

^{∼}^{=}is given such that

*φ*

_{gg}*′*=

*φ*

_{g}*′*

*◦g*

^{′∗}*φ*

*for every*

_{g}*g, g*

^{′}*∈J*. Then

*J*naturally acts on

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*) on the right. Therefore we get a left action of

*J*on

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*) by taking the inverse

*J*

*−→J;*

*g*

*7−→g*

^{−}^{1}.

**Theorem 3.4** *LetF*^{◦}*be an object ofD*_{c}* ^{b}*( ¯

*M*

*m*

*,*Z

*ℓ*)

*andF*

*the object ofD*

^{b}*( ¯*

_{c}*M*

*m*

*,*Q

*ℓ*)

*associated withF*

^{◦}*. Assume that we are given a*

*J-equivariant structure ofF*

^{◦}*(thus*

*F*

*also has a*

*J-equivariant structure). Then*

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*)

*is a ﬁnitely generated*

*smooth*

*J-representation.*

*Proof.* Let *U* be an element of *Q**m*. By [Far04, Proposition 2.3.11], there exists a
compact open subgroup *K**U* of *J* which stabilizes *U*. Then *H*_{c}* ^{i}*(U,

*F|*

*U*) is a ﬁnite- dimensionalQ

*ℓ*-vector space endowed with the action of

*K*

*and has the*

_{U}*K*

*-stable Z*

_{U}*ℓ*-lattice Im(H

_{c}*(U,*

^{i}*F*

^{◦}*|*

*U*)

*−→*

*H*

_{c}*(U,*

^{i}*F|*

*U*)). Therefore

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*) is a smooth

*J-*representation by Corollary 2.7.

To prove that *H*_{c}* ^{i}*( ¯

*M*

*m*

*,F*) is ﬁnitely generated, we may assume

*m*= 0, for

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*) =

*H*

_{c}*( ¯*

^{i}*M*0

*, p*

_{0m}

_{∗}*F*). In this case, we can use the similar method as in [Far04, Proposition 4.4.13]. Let us explain the argument brieﬂy. By [Far04, Th´eor`eme 2.4.13], there exists

*W*

*∈ Q*0 such that ∪

*g**∈**J**gW* = *M*¯0. We put *K* =
*{g* *∈* *J* *|* *gW* = *W}* and Ω = *{g* *∈* *J* *|* *gW* *∩* *W* *̸*= ∅}. As in the proof of
[Far04, Proposition 4.4.13],*K* is a compact open subgroup of *J* and Ω is a compact

subset of *J. For* *α* = ([g1], . . . ,[g*n*]) *∈*(J/K)* ^{n}*, we put

*W*

*α*=

*g*1

*W*

*∩ · · · ∩g*

*n*

*W*and

*K*

*= ∩*

_{α}

_{n}*j=1**g*_{j}*Kg*^{−}_{j}^{1}. For an open covering *{gW}**g**∈**J/K*, we can associate the ˇCech
spectral sequence

*E*_{1}* ^{r,s}*= ⊕

*α**∈*(J/K)^{−}^{r+1}

*H*_{c}* ^{s}*(W

_{α}*,F|*

*W*

*α*) =

*⇒H*

_{c}*( ¯*

^{r+s}*M*0

*,F*).

Consider the diagonal action of*J* on (J/K)^{−}* ^{r+1}*. The coset

*J\{α∈*(J/K)

^{−}

^{r+1}*|W*

_{α}*̸*=∅}

is ﬁnite; indeed, if *W*_{α}*̸*= ∅ for *α* = ([g_{1}], . . . ,[g_{−}* _{r+1}*])

*∈*(J/K)

*, then*

^{−r+1}*g*

_{1}

^{−}^{1}

*α*

*∈*

*{*1

*} ×*Ω/K

*× · · · ×*Ω/K, which is a ﬁnite set.

Take a system of representatives *α*_{1}*, . . . , α** _{n}* of the coset above. Then there is
a natural isomorphism ⊕

*α**∈**J α**j**H*_{c}* ^{s}*(W

_{α}*,F|*

*W*

*α*)

*∼*= c-Ind

^{J}

_{K}

_{αj}*H*

_{c}*(W*

^{s}

_{α}

_{j}*,F|*

*W*

*). Hence*

_{αj}*E*

_{1}

^{r,s}*∼*= ⊕

*n*

*j=1*c-Ind^{J}_{K}

*αj**H*_{c}* ^{s}*(W

_{α}

_{j}*,F|*

*W*

*) is a ﬁnitely generated*

_{αj}*J-module, since the*cohomology

*H*

_{c}*(W*

^{s}

_{α}

_{j}*,F|*

*W*

*) is ﬁnite-dimensional for each*

_{αj}*j. By this and the fact*that a ﬁnitely generated smooth

*J-module is noetherian [Ber84, Remarque 3.12],*we conclude that

*H*

_{c}*( ¯*

^{i}*M*0

*,F*) is ﬁnitely generated.

**Lemma 3.5** *Let* *F* *be an object of* *D*^{b}* _{c}*( ¯

*M*

*m*

*,*Q

*ℓ*)

*with a*

*K*

_{0}

*/K*

_{m}*-equivariant struc-*

*ture. Let*

*n*

*be an integer with*0

*≤*

*n*

*≤*

*m*

*and put*

*G*= (p

_{nm}

_{∗}*F*)

^{K}

^{n}

^{/K}

^{m}*. Then we*

*haveH*

_{c}*( ¯*

^{i}*M*

*m*

*,F*)

^{K}

^{n}

^{/K}*=*

^{m}*H*

_{c}*( ¯*

^{i}*M*

*n*

*,G*).

*Proof.* Since the cardinality of*K*_{n}*/K** _{m}* is prime to

*ℓ, (−*)

^{K}

^{n}

^{/K}*commutes with*

^{m}*H*

_{c}*. Therefore, we have*

^{i}*H*_{c}* ^{i}*( ¯

*M*

*m*

*,F*)

^{K}

^{n}

^{/K}*= lim*

^{m}

_{U}*−→*

_{∈Q}*m*

*H*_{c}* ^{i}*(U,

*F|*

*U*)

^{K}

^{n}

^{/K}*= lim*

^{m}

_{V}*−→*

_{∈Q}*n*

*H*_{c}* ^{i}*(

*p*^{−}_{nm}^{1}(V),*F|*_{p}^{−1}_{nm}_{(V}_{)})*K**n**/K**m*

= lim_{V}*−→*_{∈Q}

*n*

*H*_{c}* ^{i}*(

*V, p*_{nm}* _{∗}*(

*F|*

*p*

^{−}*nm*

^{1}(V)))

*K*

*n*

*/K*

*m*

= lim_{V}*−→*_{∈Q}

*n*

*H*_{c}* ^{i}*
(

*V,*(

*p*_{nm}* _{∗}*(

*F|*

*p*

^{−}*nm*

^{1}(V)))

*K*

*n*

*/K*

*m*)

= lim_{V}*−→*_{∈Q}

*n*

*H*_{c}* ^{i}*(V,

*G|*

*V*) =

*H*

_{c}*( ¯*

^{i}*M*

*n*

*,G*).

**Definition 3.6** A *system of coeﬃcients* over the tower *{M*¯*m**}**m**≥*0 is the data *F* =
*{F**m**}**m**≥*0 where*F**m* is an object of*D*^{b}* _{c}*( ¯

*M*

*m*

*,*Q

*ℓ*) with a

*K*

_{0}

*/K*

*-equivariant structure such that (p*

_{m}

_{nm}

_{∗}*F*)

^{K}

^{n}

^{/K}*=*

^{m}*F*

*n*for every integers

*m,*

*n*with 0

*≤*

*n*

*≤*

*m. Then, by*Lemma 3.5, we have

*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*

*m*)

^{K}

^{n}

^{/K}*=*

^{m}*H*

_{c}*( ¯*

^{i}*M*

*n*

*,F*

*n*). We put

*H*

_{c}*( ¯*

^{i}*M*

*∞*

*,F*) = lim

*−→*

^{m}*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*

*m*).

If each *F**m* is endowed with a *J-equivariant structure which commutes with the*
given *K*0*/K**m*-equivariant structure, and for every 0 *≤* *n* *≤* *m* the *J-equivariant*
structures on*F**m* and *F**n* are compatible under the identiﬁcation (p_{nm}_{∗}*F**m*)^{K}^{n}^{/K}* ^{m}* =

*F*

*n*, then we say that we have a

*J*-equivariant structure on

*F*. Such a structure naturally induces the action of

*J*on

*H*

_{c}*( ¯*

^{i}*M*

*∞*

*,F*).

By replacing “D^{b}* _{c}*( ¯

*M*

*m*

*,*Q

*ℓ*)” with “D

_{c}*( ¯*

^{b}*M*

*m*

*,*Z

*ℓ*)”, we may also deﬁne a

*system*

*of integral coeﬃcients*

*F*

*over*

^{◦}*{M*¯

*m*

*}*

*m*

*≥*0, the cohomology

*H*

_{c}*( ¯*

^{i}*M*

_{∞}*,F*

*) and a*

^{◦}*J-*equivariant structure on

*F*

*.*

^{◦}**Corollary 3.7** *Let* *F*^{◦}*be a system of integral coeﬃcients over* *{M*¯*m**}**m**≥*0 *with a*
*J-equivariant structure and* *F* *the system of coeﬃcients associated with* *F*^{◦}*. Then*
*H*_{c}* ^{i}*( ¯

*M*

_{∞}*,F*)

*is a smooth*

*K*

_{0}

*×J-representation andH*

_{c}*( ¯*

^{i}*M*

_{∞}*,F*)

^{K}

^{m}*is a ﬁnitely gen-*

*erated smoothJ-representation for every integerm*

*≥*0.

*Proof.* The smoothness is clear from Theorem 3.4 and the deﬁnition of*H*_{c}* ^{i}*( ¯

*M*

*∞*

*,F*).

Since *H*_{c}* ^{i}*( ¯

*M*

*∞*

*,F*)

^{K}*=*

^{m}*H*

_{c}*( ¯*

^{i}*M*

*m*

*,F*

*m*), the second assertion also follows from Theo- rem 3.4.

**4** **Shimura variety and** *p-adic uniformization*

In this section, we introduce certain Shimura varieties (Siegel threefolds) related
to our Rapoport-Zink tower. Let us ﬁx a 4-dimensional Q-vector space *V** ^{′}* and an
alternating perfect pairing

*ψ*

*:*

^{′}*V*

^{′}*×V*

^{′}*−→*Q. For an integer

*m*

*≥*0 and a compact open subgroup

*K*

^{p}*⊂*GSp(V

_{A}

^{′}

_{∞}*,p*) = GSp(V

_{A}

^{′}

_{∞}*,p*

*, ψ*

^{′}_{A}

_{∞}*,p*), consider the functor Sh

_{m,K}*from the category of locally noetherian Z*

^{p}*p*

*-schemes to the category of sets that associates*

^{∞}*S*with the set of isomorphism classes of quadruples (A, λ, η

^{p}*, η*

*) where*

_{p}**–** *A* is a projective abelian surface over *S* up to prime-to-pisogeny,
**–** *λ*: *A−→A** ^{∨}* is a prime-to-p polarization,

**–** *η** ^{p}* is a

*K*

*-level structure of*

^{p}*A,*

**–** and *η** _{p}*:

*L/p*

^{m}*L−→A[p*

*] is a Drinfeld*

^{m}*m-level structure*

(for the detail, see [Kot92,*§*5]). Two quadruples (A, λ, η^{p}*, η** _{p}*) and (A

^{′}*, λ*

^{′}*, η*

^{′}

^{p}*, η*

_{p}*) are said to be isomorphic if there exists a prime-to-pisogeny from*

^{′}*A*to

*A*

*which carries*

^{′}*λ*to aZ

^{×}_{(p)}-multiple of

*λ*

*,*

^{′}*η*

*to*

^{p}*η*

^{′}*and*

^{p}*η*

*p*to

*η*

^{′}*. We put Sh*

_{p}*K*

*= Sh0,K*

^{p}*. It is known that if*

^{p}*K*

*is suﬃciently small, Sh*

^{p}

_{m,K}*is represented by a quasi-projective scheme over Z*

^{p}*p*

*with smooth generic ﬁber. In the sequel, we always assume that*

^{∞}*K*

*is enough small so that Sh*

^{p}*m,K*

*is representable. We denote the special ﬁber of Sh*

^{p}*m,K*

^{p}(resp. Sh_{K}* ^{p}*) by Sh

_{m,K}*(resp. Sh*

^{p}

_{K}*).*

^{p}For a compact open subgroup *K*^{′}* ^{p}* contained in

*K*

*and an integer*

^{p}*m*

^{′}*≥*

*m, we*have the natural morphism Sh

*m*

^{′}*,K*

^{′}

^{p}*−→*Sh

*m,K*

*. This is a ﬁnite morphism and is moreover ´etale if*

^{p}*m*

*=*

^{′}*m.*

Next we recall the*p-adic uniformization theorem, which gives a relation between*
*M*˘ and Sh*K** ^{p}*. Let us ﬁx a polarized abelian surface (A0

*, λ*

*A*0) over F

*p*such that

*A*

_{0}[p

*] is an isoclinic*

^{∞}*p-divisible group with slope 1/2. Note that such (A*

_{0}

*, λ*

_{A}_{0}) exists; for example, we can take (A

_{0}

*, λ*

_{A}_{0}) = (E

^{2}

*, λ*

^{2}

*), where*

_{E}*E*is a supersingular elliptic curve over F

*p*and

*λ*

*E*is a polarization of

*E. By deﬁnition, the rational*Dieudonn´e module

*D(A*

_{0}[p

*])*

^{∞}_{Q}is isomorphic to

*D(*X)

_{Q}. Thus, by the subsequent lemma, there is an isomorphism of isocrystals

*D(A*

_{0}[p

*])*

^{∞}_{Q}

*∼*=

*D(*X)

_{Q}which preserves the natural polarizations.

**Lemma 4.1** *We use the notation in [RR96,§1]. Let* *d≥*1 *be an integer.*

i) *Let* *b* *be an element ofB*(GSp_{2d}) *and* *b*^{′}*the image of* *b* *under the natural map*
*B*(GSp_{2d})*−→B(GL*2d). Then *b* *is basic if and only if* *b*^{′}*is basic.*

ii) *The map* *B(GSp*_{2d})_{basic} *−→B(GL*_{2d})_{basic} *induced from i) is an injection.*

*Proof.* Note that the center of GSp_{2d} coincides with that of GL_{2d}. Thus i) is clear,
since *b* (resp. *b** ^{′}*) is basic if and only if the slope morphism

*ν*

*: D*

_{b}*−→*GSp

_{2d}(resp.

*ν*_{b}*′*: D*−−→*^{ν}* ^{b}* GSp

_{2d}

*,−→*GL

_{2d}) factors through the center of GSp

_{2d}(resp. GL

_{2d}).

We prove ii). By [RR96, Theorem 1.15], it suﬃces to show that the natural
map *π*_{1}(GSp_{2d}) *−→* *π*_{1}(GL_{2d}) is injective. Take a maximal torus *T* (resp. *T** ^{′}*) of
GSp

_{2d}(resp. GL

_{2d}) such that

*T*

*⊂*

*T*

*. Then, since Sp*

^{′}_{2d}(resp. SL

_{2d}) is simply connected,

*π*

_{1}(GSp

_{2d}) (resp.

*π*

_{1}(GL

_{2d})) can be identiﬁed with the quotient of

*X*

*(T) (resp.*

_{∗}*X*

*(T*

_{∗}*)) induced by*

^{′}*c:*

*T*

*−*G

*m*(resp. det :

*T*

^{′}*−*G

*m*), where

*c*denotes the similitude character of GSp

_{2d}. In particular, both

*π*

_{1}(GSp

_{2d}) and

*π*

_{1}(GL

_{2d}) are isomorphic to Z.

The commutative diagram

GSp_{2d} ^{c}^{// //}

G*m*
*z**7→**z*^{d}

GL_{2d} ^{det} ^{// //}G*m*

induces the commutative diagram

*X** _{∗}*(T)

^{// //}

*X** _{∗}*(G

*m*)

*×**d*

*π*_{1}(GSp_{2d})

*X** _{∗}*(T

*)*

^{′}^{// //}

*X*

*(G*

_{∗}*m*)

*π*1(GL2d).

In particular, the natural map*π*_{1}(GSp_{2d})*−→π*_{1}(GL_{2d}) is injective.

Therefore, there is a quasi-isogeny X *−→* *A[p** ^{∞}*] preserving polarizations. If
we replace (X

*, λ*0) by the polarized

*p-divisible group (A*0[p

*], λ*

^{∞}*A*0) associated with (A

_{0}

*, λ*

_{A}_{0}), the

*G-representation*

*H*

_{RZ}

*remains unchanged. Thus, in order to prove Theorem 3.2, we may assume that (X*

^{i}*, λ*

_{0}) = (A

_{0}[p

*], λ*

^{∞}

_{A}_{0}). In the remaining part of this article, we always assume it. Moreover, we ﬁx an isomorphism

*H*1(A0

*,*A

^{∞}*)*

^{,p}*∼*=

*V*

_{A}

^{′}*∞,p*preserving alternating pairings.

Denote the isogeny class of (A_{0}*, λ*_{A}_{0}) by*ϕ*and put*I** ^{ϕ}*= Aut(ϕ). We have natural
group homomorphisms

*I*

^{ϕ}*,−→*

*J*and

*I*

^{ϕ}*,−→*Aut(H

_{1}(A

_{0}

*,*A

^{∞}*)) = GSp(V*

^{,p}_{A}

^{′}*∞*

*,p*).

These are injective.

Let*Y*_{K}* ^{p}* be the reduced closed subscheme of Sh

_{K}*such that*

^{p}*Y*

_{K}*(F*

^{p}*p*) consists of triples (A, λ, η

*) where the*

^{p}*p-divisible group associated with (A, λ) is isogenous to*(X

*, λ*

_{0}). It is the basic (or supersingular) stratum in the Newton stratiﬁcation of Sh

_{K}*. Note that (A, λ, η*

^{p}*)*

^{p}*∈*Sh

_{K}*(F*

^{p}*p*) belongs to

*Y*

_{K}*(F*

^{p}*p*) if and only if (A, λ)

*∈ϕ*([Far04, Proposition 3.1.8], [Kot92,

*§*7]). We denote the formal completion of Sh

_{K}*along*

^{p}*Y*

_{K}*by (Sh*

^{p}

_{K}*)*

^{p}

^{∧}

_{/Y}*Kp*.

Now we can state the *p-adic uniformization theorem:*