mark kisin harvard - clay mathematics institute · harvard 1. main theorem let (g,x) be a shimura...

Integral models of Shimura varieties II

Mark Kisin

Harvard

1

Main Theorem

Let (G,X) be a Shimura datum of abelian type.

Kp = GZp(Zp) ⊂ G(Qp) hyperspecial, Kp ⊂ G(Apf) compact open

K = KpKp ⊂ G(Af).

ShK(G,X) = G(Q)\X ×G(Af)/K

is a variety over the reflex field E = E(G,X).

2

Main Theorem



K = KpKp ⊂ G(Af).



Theorem. Let p > 2, and λ|p be a prime of E. There is a smooth OEλ-scheme S K(G,X) extending ShK(G,X) such that the G(Ap

f)-action on

ShKp(G,X) = lim← Kp

ShKpKp(G,X)

extends to the OEλ-scheme

S Kp(G,X) = lim← Kp

S KpKp(G,X).

2

Main Theorem



K = KpKp ⊂ G(Af).



Theorem. Let p > 2, and λ|p be a prime of E. There is a smooth OEλ-scheme S K(G,X) extending ShK(G,X) such that the G(Ap

f)-action on

ShKp(G,X) = lim← Kp

ShKpKp(G,X)

extends to the OEλ-scheme


S KpKp(G,X).

If R is a regular, formally smooth OEλ-algebra, then

S Kp(G,X)(R) = S Kp(G,X)(R[1/p]).

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Theorem. Let p > 2, and λ|p be a prime of E. There is a smoothOEλ-scheme S K(G,X) extending ShK(G,X) such that the


S KpKp(G,X)

is equivariant and satisfies the extension property.

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S KpKp(G,X)


Suppose that (G,X) is of Hodge type.

(G,X) ↪→ (GSp(V, ψ), S±),

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S KpKp(G,X)



(G,X) ↪→ (GSp(V, ψ), S±),

S K(G,X) the normalization of the closure of

θ : ShK(G,X) ↪→ ShK′(GSp, S±) ↪→ S K′(GSp, S±)

where S K′(GSp, S±) is an integral model of ShK′(GSp, S±) obtained fromits interpretation as a moduli space of polarized abelian varieties.

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S KpKp(G,X)



(G,X) ↪→ (GSp(V, ψ), S±),




Here K′ = K′pK′p,

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S KpKp(G,X)



(G,X) ↪→ (GSp(V, ψ), S±),




Here K′ = K′pK′p,

K′p ⊂ GSp(VQp), the stabilizer of a lattice VZp ⊂ VQp, and

GZp ↪→ GL(VZp) (Prasad-Yu).

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ThenS Kp(G,X) := lim

← KpS KpKp(G,X)

is G(Apf)-equivariant.

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← KpS KpKp(G,X)


S Kp(G,X) satisfies the extension property because S K′p(GSp, S±) does.

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← KpS KpKp(G,X)



Have to show: S K(G,X) is a smooth OEλ-scheme.

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← KpS KpKp(G,X)



Have to show: S K(G,X) is a smooth OEλ-scheme.

Idea: Describe the local structure of S Kp(G,X) in terms of moduli of“p-adic Hodge structures”, or more precisely p-divisible groups.

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Hodge cycles.

Let E(G,X) ⊂ K ⊂ C be a field, and

x ∈ ShK(G,X)(K) ↪→ ShK′(GSp, S±)(K)

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Hodge cycles.



Thenx ; A = Ax polarized abelian var./K

5

Hodge cycles.




G ↪→ GL(V ) is defined by a collection of tensors {sα} ∈ V ⊗.

5

Hodge cycles.




G ↪→ GL(V ) is defined by a collection of tensors {sα} ∈ V ⊗.We saw these give rise to Hodge cycles

sα,x ∈ H1(AC,Q)⊗ = H1(AC,Q)⊗

5

Hodge cycles.





sα,x ∈ H1(AC,Q)⊗ = H1(AC,Q)⊗

and hence using the comparison between etale and singular cohomology to

sα,x,` ∈ H1et(AK,Q`)

⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.

for any prime `.

5

Hodge cycles.





sα,x ∈ H1(AC,Q)⊗ = H1(AC,Q)⊗

and hence using the comparison between etale and singular cohomology to


⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.

for any prime `.

The Q`-vector space nH1et(AK,Q`) is equipped with an action of Gal(K/K).

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⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.

The Q`-vector spaceH1et(AK,Q`) is equipped with an action of Gal(K/K).

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⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.


Theorem. (Deligne: “Hodge =⇒ Absolute Hodge”)


⊗ are fixed by Gal(K/K).

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⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.





This is motivated by the Hodge conjecture which predicts that the sα,x,`are the classes of algebraic cycles, so would at least be fixed by an opensubgroup Gal(K/K).

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⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.






Now let VZ(p)= VZp ∩ V ⊂ VQp, a Z(p)-lattice in V.

Choose {sα} ⊂ V ⊗Zp such that GZp ⊂ GL(VZp) is the stabilizer of {sα}.

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⊗ ∼−→ H1(AC,Q)⊗ ⊗Q Q`.






Now let VZ(p)= VZp ∩ V ⊂ VQp, a Z(p)-lattice in V.

Choose {sα} ⊂ V ⊗Zp such that GZp ⊂ GL(VZp) is the stabilizer of {sα}.

The tensors sα,x,p ∈ H1et(AK,Zp)

⊗ are Gal(K/K) -invariant

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Now suppose K/Eλ is finite, with residue field of k.

Suppose Ax has good reduction:

x ; x ∈ ShK(G,X)(k)

Attached to Ax we have

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H1et(Ax,K,Zp) Axoo o/ o/ o/ o/ o/ o/ o/ o/ o/ o/ ///o/o/o/o/o/o/o/o/o/o

�� O�O�O�O�O�O�O�O

H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ

Gx = lim−→Ax[pn]

fff& f& f& f& f& f& f& f& f& f& f& f& f& f&

888x8x8x8x8x8x8x8x8x8x8x8x8x8x8x8x

Here Ax is the abelian scheme over OK extending Ax; i.e the Neron model.

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ





One has Fontaine’s p-adic comparison isomorphism

H1et(Ax,K,Zp)⊗Zp Bcris

∼−→ H1cris(Ax/W (k))⊗W (k) Bcris

which is the analogue of the de Rham isomorphism.

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ









Bcris is a K0 = W (k)[1/p]-algebra with an action of Gal(K/K) and ϕ.

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ









Bcris is a K0 = W (k)[1/p]-algebra with an action of Gal(K/K) and ϕ.

This package is the p-adic analogue of a Hodge structure of weight 1.

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ






sα,x,p ∈ H1et(Ax,K,Zp)

⊗ maps to sα,x,0 ∈ H1cris(Ax/W (k)) ⊗ Qp, which is

invariant by ϕ, and in Fil0.

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ









To equip our integral p-adic Hodge structure with a “G-structure” andconsider its deformations one needs the following

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ










Lemma. (Key lemma) We have {sα,x,0} ⊂ H1cris(Ax/W (k))⊗,

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H1cris(Ax/W (k))

Gal(K/K)

Abs. Frobenius ϕ











and the group fixing the sα,x,0 is a reductive subgroup

GW (k) ⊂ GL(H1cris(Ax/W (k))).

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The Lemma lets us do two things:

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1) The sα extend to integral sections of the de Rham cohomology of theuniversal family

A → S K(G,X)

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A → S K(G,X)

2) One can define a deformation space of “p-divisible groups with G-structure”, and show it is smooth (Faltings),

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A → S K(G,X)

2) One can define a deformation space of “p-divisible groups with G-structure”, and show it is smooth (Faltings),

Then using 1), one can identify this deformation space with the completionof S K(G,X) at x.

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p-adic Hodge theory.

For the proof of the lemma we need some techniques from p-adic Hodgetheory.

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Lemma. We have {sα,x,0} ⊂ H1cris(Ax/W (k))⊗, and the group fixing

the sα,x,0 is a reductive subgroup


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Definition. A crystalline Zp-representation is a finite free Zp-moduleL equipped with an action of Gal(K/K). such that

dimK0(L⊗Bcris)Gal(K/K) = rkZpL.

(one always has ≤).

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E.g L = H1et(Ax,K,Zp) - actually all the L’s we work with will arise from

this one by tensor operations.

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E.g L = H1et(Ax,K,Zp) - actually all the L’s we work with will arise from

this one by tensor operations.

We denote by RepcrisZp the category of crystalline Zp-representations.

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Let W = W (k), fix a uniformiser π ∈ K, and let E(u) ∈ W [u] be theEisenstein polynomial for π.

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Set S = W [[u]] equipped with a Frobenius ϕ which acts as the usualFrobenius on W and sends u to up.

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Set S = W [[u]] equipped with a Frobenius ϕ which acts as the usualFrobenius on W and sends u to up.

Let Modϕ/S denote the category of finite free S-modules M equipped witha Frobenius semi-linear isomorphism

1⊗ ϕ : ϕ∗(M)[1/E(u)]∼−→M[1/E(u)].

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We have the following ’Black (magic) Box’ theorem

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Theorem. There exists a fully faithful tensor functor

M : RepcrisZp → Modϕ/S

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Let M = M(L).

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Let M = M(L).

1. There is a canonical isomorphism

Dcris(L) := (Bcris ⊗ L)Gal(K/K) ∼−→M/uM[1/p]

compatible with Frobenius.

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Let M = M(L).





OEur ⊗Zp L∼−→ OEur ⊗S M,

where OEur is a faithfully flat, and formally etale S(p)-algebra.

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Let M = M(L).







3. If L = H1et(Ax,K,Zp), then

ϕ∗(M/uM)∼−→ H1

cris(Ax/W )

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Let M = M(L).







3. If L = H1et(Ax,K,Zp), then

ϕ∗(M/uM)∼−→ H1

cris(Ax/W )

The functor M is still somewhat mysterious. It is constructed using p-adicHodge theory. But see Scholze’s final talk (?).

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Proof of the Key lemma:

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We apply the above theory with L = H1et(Ax,K,Zp).

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Recall that GZp ⊂ GL(L) is the reductive group defined by {sα,x,p}.

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We may view the sα,x,p as morphisms sα : 1→ L⊗ in RepcrisZp .

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Applying the functor M we obtain morphisms sα : 1→M⊗ in Modϕ/S.

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Applying the functor M we obtain morphisms sα : 1→M⊗ in Modϕ/S.

Specializing the (sα) at u = 0 gives

sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗

which gives the first part of the lemma.

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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗


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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗


Have to show GS ⊂ GL(M) defined by the (sα) is reductive.

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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗



Let M′ = L⊗S andP ⊂ HomS(M′,M)

the subscheme of isomorphisms between M′ and M taking sα,x,p to sα.

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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗





The fibres of P are either empty or a torsor under GZp.

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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗






We know the sα,x,p define a reductive subgroup in GL(L) by assumption,so it suffices to show that P is a GZp-torsor: All such torsors are trivial(Lang’s lemma).

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sα,x,0 ∈ ϕ∗(M/uM)⊗∼−→ H1

cris(Ax)⊗






We know the sα,x,p define a reductive subgroup in GL(L) by assumption,so it suffices to show that P is a GZp-torsor: All such torsors are trivial(Lang’s lemma).

So we have to show that P is flat over S with non-empty fibres.

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Claim: P = HomS,sα(M′,M) ⊂ HomS(M′,M) is a GZp-torsor.

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For R a S-algebra, we set PR = P ×SpecS SpecR.

Step 1: PS(p)is a GZp-torsor:

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AsOEur ⊗Zp L

∼−→ OEur ⊗S M,

POEuris a trivial GZp-torsor.

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AsOEur ⊗Zp L



Since OEur is faithfully flat over S(p) and S(p), PS(p)is a GZp-torsor.

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AsOEur ⊗Zp L




Step 2: PS[1/pu] is a GZp-torsor:

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AsOEur ⊗Zp L





Let U ⊂ SpecS[1/up] the maximal open subset over which P is flat withnon-empty fibres.

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AsOEur ⊗Zp L






U contains the generic point by Step 1, so SpecS[1/p]\U is finite.

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AsOEur ⊗Zp L







Since the sα are Frobenius invariant, and ϕ∗(M)[1/E(u)]∼−→M[1/E(u)]

we havePS[1/E(u)]

∼−→ ϕ∗(PS[1/E(u)]).

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Since the sα are ϕ-invariant, and ϕ∗(M)[1/E(u)]∼−→M[1/E(u)] we have

PS[1/E(u)]∼−→ ϕ∗(PS[1/E(u)]).

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PS[1/E(u)]∼−→ ϕ∗(PS[1/E(u)]).

Let y ∈ U.

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PS[1/E(u)]∼−→ ϕ∗(PS[1/E(u)]).

Let y ∈ U.If ϕn(y) 6= (E(u)) for n = 0, 1, 2, . . . then ϕn(y) /∈ U by descent.

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PS[1/E(u)]∼−→ ϕ∗(PS[1/E(u)]).


If (E(u)) * ϕ−n(y) n = 1, 2, . . . then ϕ−n(y) /∈ U.

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PS[1/E(u)]∼−→ ϕ∗(PS[1/E(u)]).


If (E(u)) * ϕ−n(y) n = 1, 2, . . . then ϕ−n(y) /∈ U.One of these conditions always hold, which contradicts finiteness.

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Step 3: PK0 is a GZp-torsor, where S→ K0 := W [1/p] via u 7→ 0.

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Since Dcris(L[1/p]) = M/uM[1/p],

Bcris ⊗ L∼−→ Bcris ⊗M/uM.

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Step 4: PK0[[u]] is a GZp-torsor: There is a canonical ϕ-equivariant isomor-phism

M⊗S K0[[u]]∼−→ K0[[u]]⊗K0 M/uM[1/p]

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Step 4: PK0[[u]] is a GZp-torsor: There is a canonical ϕ-equivariant isomor-phism

M⊗S K0[[u]]∼−→ K0[[u]]⊗K0 M/uM[1/p]

PK0[[u]]∼−→ PK0 ⊗K0 K0[[u]], which is a GZp-torsor by Step 3.

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Step 5: P is a GZp-torsor:

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So far we have proved that P is a torsor over

SpecS(p) SpecS[1/pu] SpecK0[[u]]

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This covers U = SpecS - closed point. So P |U is a GZp-torsor.

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Consider

(GL(VZp)/GZp)(U)→ H1(GZp, U)→ H1(GL(VZp), U) = 0

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Consider


AsGZp is reductive GL(VZp)/GZp is affine, and an U -point of GL(VZp)/GZpextends to SpecS.

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Consider



Hence any GZp-torsor over U extends to SpecS, and hence is trivial.

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Consider




Hence P |U is trivial, and there is a M|U∼−→M′|U taking sα to sα.

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Consider




Hence P |U is trivial, and there is a M|U∼−→M′|U taking sα to sα.

Since any vector bundle over U has a canonical extension to S, this impliesthat P is the trivial GZp-torsor.

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Now suppose that Kp is parahoric.

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Example: Suppose GZp is a reductive model of G, as before, let B ⊂GZp ⊗ Fp be a Borel subgroup, and take

Kp = {g ∈ GZp(Zp) : g 7→ g ∈ B(Fp) ⊂ GZp(Fp)}

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E.g Γ0(p)-structure for elliptic curves.

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Theorem. (MK-Pappas) Suppose p > 2, and GQp splits over a tamelyramified extension. If λ|p is a prime of E, there exists an OEλ-schemeS Kp(G,X) extending ShKp(G,X) such that

1.S Kp(G,X) = lim

← KpS KpKp(G,X)

is G(Apf) equivariant, and satisfies the extension property, as before.

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1.S Kp(G,X) = lim

← KpS KpKp(G,X)


2. ShKp(G,X)/E proper =⇒ S KpKp(G,X)/OEλ proper.

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1.S Kp(G,X) = lim

← KpS KpKp(G,X)



3. The strictly henselian local rings on S K(G,X) have normal, Cohen-Macaulay irreducible components.

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1.S Kp(G,X) = lim

← KpS KpKp(G,X)



3. The strictly henselian local rings on S K(G,X) have normal, Cohen-Macaulay irreducible components.

4. There is an explicit description of the local structure in terms oforbit closures of G acting on a certain Grassmannian; “local mod-els”.

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1.S Kp(G,X) = lim

← KpS KpKp(G,X)



3. The strictly henselian local rings on S K(G,X) have normal, Cohen-MacCaulay irreducible components.

4. There is an explicit description of the local structure in terms oforbit closures of G acting on a certain Grassmannian; “local mod-els”.

This was conjectured by Rapoport and Pappas, and was known previouslyin some PEL cases.

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mark kisin harvard - clay mathematics institute · harvard 1. main theorem let (g,x) be a shimura...

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