mark kisin harvard · 2015. 7. 29. · mark kisin harvard 1. the beginning. 2. the beginning. let...

Integral models of Shimura varieties I

Mark Kisin

Harvard

1

The beginning.

2

The beginning.

Let X be the GL2(R) conjugacy class X

h0 : C× → GL2(R); z = a + ib 7→(a −bb a

)consists of the upper and lower half planes (take the orbit of the point i).

2

The beginning.


h0 : C× → GL2(R); z = a + ib 7→(a −bb a


Let Af be the finite adeles over Q. For K ⊂ GL2(Af) compact open,

XK = GL2(Q)\X × GL2(Af)/K

is a modular curve.

2

The beginning.


h0 : C× → GL2(R); z = a + ib 7→(a −bb a




is a modular curve.

For h ∈ X, becomes R2 a one dimensional complex vector space.

2

The beginning.


h0 : C× → GL2(R); z = a + ib 7→(a −bb a




is a modular curve.


Eh = R2/Z2 is an elliptic curve.

2

The beginning.


h0 : C× → GL2(R); z = a + ib 7→(a −bb a




is a modular curve.



For any elliptic curve E over C let

T (E) = lim← n

E[n]

and V (E) = T (E)⊗Q.

2

The beginning.


h0 : C× → GL2(R); z = a + ib 7→(a −bb a




is a modular curve.



For any elliptic curve E over C let

T (E) = lim← n

E[n]

and V (E) = T (E)⊗Q.To (h, g) ∈ X × GL2(Af) we attach Eh considered up to isogeny and

εh,g : V (Eh)∼−→ A2

fg→ A2

f mod K.

The pair (Eh, εh,g) depends only on the image of (h, g) in XK.

2

Moduli space of polarized abelian varieties.

3


Let V be Q-vector space equipped with a perfect alternating pairing ψ.

3



Take G = GSp(V, ψ) and let X = S± be the Siegel double space:

S± = {h : C× → GSp(VR, ψ)} such that

3





1. VC = V −1,0 ⊕ V 0,−1 z · v = z−pz−q v ∈ V p,q

3





1. VC = V −1,0 ⊕ V 0,−1 z · v = z−pz−q v ∈ V p,q

2. The symmetric pairing

VR × VR → R; (x, y) 7→ ψ(x, h(i)y)

is definite.

3





1. VC = V −1,0 ⊕ V 0,−1 z · v = z−pz−q v ∈ V p,q

2. The symmetric pairing

VR × VR → R; (x, y) 7→ ψ(x, h(i)y)

is definite.

If VZ ⊂ V is a Z-lattice, and h ∈ S±, then V −1,0/VZ is a (polarized)abelian variety, which leads to an interpretation of

ShK(GSp, S±) = GSp(Q)\S± × GSp(Af)/K

as a moduli space for polarized abelian varieties with level structure.

3

Review of Shimura varieties:

Let G be a connected reductive group over Q and X a conjugacy class ofmaps of algebraic groups over R

h : S = ResC/RGm → GR.

which gives on real points

h : C× → G(R).

We assume that (G,X) is a Shimura datum , which means

4





h : C× → G(R).


1. The action of C× on LieGC by conjugation is via the characters

z 7→ z/z, 1, z/z.

That is, LieGC is a Hodge structure of weights (−1, 1), (0, 0), (1.− 1).

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h : C× → G(R).



z 7→ z/z, 1, z/z.


2. This implies h(−1) is central. So h(i) acts by an involution.

4





h : C× → G(R).



z 7→ z/z, 1, z/z.



We require it be a Cartan involution: Twisting the real structure on Gby h(i) gives the compact form of GR.

4





h : C× → G(R).



z 7→ z/z, 1, z/z.




3. h is non-trivial in each factor of GadQ .

4





h : C× → G(R).



z 7→ z/z, 1, z/z.




3. h is non-trivial in each factor of GadQ .

For us the consequences will be more important than the axioms.

4

h : C× → G(R).

We assume that (G,X) is a Shimura datum .

5

h : C× → G(R).


Let K ⊂ G(Af) a compact open subgroup.

5

h : C× → G(R).



A theorem of Baily-Borel asserts that (for K small enough)

ShK(G,X) = G(Q)\X ×G(Af)/K

has a natural structure of an algebraic variety over C.

5

h : C× → G(R).





has a natural structure of an algebraic variety over C.In fact ShK(G,X) has a model over a number field E = E(G,X) - thereflex field. In particular it does not depend on K (Shimura, Deligne, ...).

5

h : C× → G(R).






The conjugacy class of

µh : C× z 7→z×1↪→ C× × C× = S(C)→ GC

does not depend on h.

5

h : C× → G(R).






The conjugacy class of

µh : C× z 7→z×1↪→ C× × C× = S(C)→ GC

does not depend on h.

E(G,X) is the field of definition of this conjugacy class.

We will again denote by ShK(G,X) this algebraic variety over E(G,X).

5

Abelian varieties with extra structure:

6


Heuristically Shimura varieties can be regarded as moduli spaces of “abelianmotives”. In particular, they carry variations of Hodge structure:

6



For any Shimura datum (G,X), if G → GL(V ), V a Q vector space, weget a VHS

V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)

6





If s = (h, g) ∈ ShK(G,X) then

h : C× → G(R) y VR

gives a bigrading

Vs ⊗ C = ⊕V p,q; h(z)v = z−pz−qv, v ∈ V p,q

6





If s = (h, g) ∈ ShK(G,X) then

h : C× → G(R) y VR

gives a bigrading

Vs ⊗ C = ⊕V p,q; h(z)v = z−pz−qv, v ∈ V p,q

The condition that C× acts on LieGC via

z 7→ z/z, 1, z/z.

implies Griffiths transversality.

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A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.

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Such a morphism induces

ShK(G,X)→ ShK′(G′, X ′)

for suitable K, K′.

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If (G,X) ↪→ (GSp(V ), S±), as before, then ShK(G,X) ↪→ ShK′(GSp, S±)and the LHS carries a family of abelian varieties, which are equipped witha collection of Hodge cycles:

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Let V ⊗ = ⊕n,mV ⊗n ⊗ V ∗⊗m. Then G ⊂ GSp(V, ψ) is the stabilizer of acollection of tensors {sα} ⊂ V ⊗.

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The sα are fixed by G hence by h(C×) for h ∈ X.

sα ∈ V ⊗ ∩ (V ⊗C )0,0

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The sα are fixed by G hence by h(C×) for h ∈ X.

sα ∈ V ⊗ ∩ (V ⊗C )0,0

So the family of abelian varieties on ShK(G,X) carries a collection of Hodgecycles coming from the sα.

7








The sα are fixed by G hence by h(C×) for h ∈ X.sα ∈ V ⊗ ∩ (V ⊗C )0,0


Such (G,X) are called of Hodge type.

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The sα are fixed by G hence by h(C×) for h ∈ X.sα ∈ V ⊗ ∩ (V ⊗C )0,0


Such (G,X) are called of Hodge type.

If these extra structures can be taken to be endomorphisms of the abelianvariety, then (G,X) is called of PEL type.

8

Example: Let W be a Q-vector space with a quadratic form such that

SO(W )R = SO(n, 2)

9


SO(W )R = SO(n, 2)

There is a Shimura datum (SO(W ), X). If h ∈ X

Cent(h) = SO(n)× SO(2) ⊂ SO(W )R.

9


SO(W )R = SO(n, 2)



One has an extension

1→ Gm → CSpin(W )→ SO(W )→ 1

9


SO(W )R = SO(n, 2)




1→ Gm → CSpin(W )→ SO(W )→ 1

and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)

with dimV = 2n+1.

9


SO(W )R = SO(n, 2)




1→ Gm → CSpin(W )→ SO(W )→ 1


with dimV = 2n+1.

When n = 19, the variety ShK(SO(W ), X) carries a family of K3-surfaces.For general n the motivic meaning is unclear.

9


SO(W )R = SO(n, 2)




1→ Gm → CSpin(W )→ SO(W )→ 1


with dimV = 2n+1.


(SO(W ), X) is an example of a Shimura datum of abelian type - these are“quotients” of ones of Hodge type;

9


SO(W )R = SO(n, 2)

There is a Shimura datum (SO(W ), X). If h ∈ XCent(h) = SO(n)× SO(2) ⊂ SO(W )R.


1→ Gm → CSpin(W )→ SO(W )→ 1


with dimV = 2n+1.



More precisely (G,X) is called of abelian type, if there exists a datum(G′, X ′) of Hodge type and a morphism G′der → Gder inducing

(G′ad, X ′ad)∼−→ (Gad, Xad).

9


SO(W )R = SO(n, 2)

There is a Shimura datum (SO(W ), X). If h ∈ XCent(h) = SO(n)× SO(2) ⊂ SO(W )R.


1→ Gm → CSpin(W )→ SO(W )→ 1


with dimV = 2n+1.


More precisely (G,X) is called of abelian type, if there exists a datum(G′, X ′) of Hodge type and a morphism G′der → Gder inducing

(G′ad, X ′ad)∼−→ (Gad, Xad).

They include almost all (G,X) with G a classical group.

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Arithmetic properties: We’re interested in ShK(G,X)(Fp)

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Arithmetic properties: We’re interested in ShK(G,X)(Fp)Some applications/motivation

1. Compute the Hasse-Weil zeta function of ShK(G,X) in terms of auto-morphic L-functions (Langlands).

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2. Arakelov intersection theory a la Gross-Zagier, Kudla: This relates theintersection theory of arithmetic cycles on ShK(G,X) to special valuesof derivatives of L-functions.

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3. Honda-Tate theory; Every abelian variety over Fp is isogenous to thereduction of a CM abelian variety. Can we control ShK(G,X)(Fp) interms of CM points ? (This turns out to be an input into 1.)

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4. Testing ideas about motives over Fp.

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4. Testing ideas about motives over Fp.

To do this we need integral models

11

Integral models: (Talk II)

Suppose now G has a reductive model GZp over Zp.

12



Let K = KpKp, where Kp = GZp(Zp) (Kp hyperspecial) and Kp ⊂ G(Ap

f) is

compact open Apf - finite adeles with trivial p-component.

12




f) is


Conjecture. (Langlands-Milne) If Kp is hyperspecial, and λ|p, is aprime of E = E(G,X). Then there is a smooth OEλ-scheme S K(G,X)extending ShK(G,X)

12




f) is



such that the G(Apf)-action on

ShKp(G,X) = lim← Kp

ShKpKp(G,X)

extends toS Kp(G,X) = lim

← KpS KpKp(G,X),

12




f) is





ShKpKp(G,X)


← KpS KpKp(G,X),

and S Kp(G,X) satisfies the extension property.

12




f) is





ShKpKp(G,X)


← KpS KpKp(G,X),


The extension property: If R is regular, formally smooth over OEλ then

S Kp(G,X)(R)∼−→ S Kp(G,X)(R[1/p]).

12

Conjecture. (Langlands-Milne) If Kp is hyperspecial, and λ|p, is aprime of E = E(G,X). Then there is a smooth OEλ-scheme S K(G,X)extending ShK(G,X) such that the G(Ap

f)-action on


ShKpKp(G,X)


← KpS KpKp(G,X),



S Kp(G,X)(R)∼−→ S Kp(G,X)(R[1/p]).

13

Conjecture. (Langlands-Milne) If Kp is hyperspecial, and λ|p, is aprime of E = E(G,X). Then there is a smooth OEλ-scheme S K(G,X)extending ShK(G,X) such that the G(Ap

f)-action on


ShKpKp(G,X)


← KpS KpKp(G,X),



S Kp(G,X)(R)∼−→ S Kp(G,X)(R[1/p]).

Hyperspecial Kp subgroups exist if and only if G is quasi-split at p andsplit over an unramified extension. This implies Kp is maximal compact.Usually one can only expect smooth models in this case.

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Example: Take (G,X) = (GSp, S±), defined by (V, ψ) as above.

14


Let VZ ⊂ V be a Z-lattice, and Kp the stabilizer of

VZp = VZ ⊗Z Zp ⊂ V ⊗Qp.

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Kp is hyperspecial if and only if a scalar multiple of ψ induces a perfect,Zp-valued pairing on VZp. Then Kp = GSp(VZp, ψ)(Zp).

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The choice of VZ makes ShK(GSp, S±) as a moduli space for polarizedabelian varieties, which leads to a model S K(GSp, S±) over O(λ).

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The choice of VZ makes ShK(GSp, S±) as a moduli space for polarizedabelian varieties, which leads to a model S K(GSp, S±) over O(λ).

The S K(GSp, S±) are smooth over O(λ) if and only if the degree of thepolarization in the moduli problem is prime to p. This corresponds to thecondition that ψ induces a perfect pairing on VZp.

14

The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:

15


If R is an OEλ-algebra, and A an abelian scheme over R, set

V p(A) = (lim← (n,p)=1

A[n])⊗Q

15



V p(A) = (lim← (n,p)=1

A[n])⊗Q

This is a (pro)-etale group scheme.

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V p(A) = (lim← (n,p)=1

A[n])⊗Q


Attached to x ∈ S Kp(GSp, S±)(R) one has an abelian scheme Ax/R andan isomorphism

εx : V p(Ax)∼−→ V ⊗Q Ap

f .

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V p(A) = (lim← (n,p)=1

A[n])⊗Q




f .

Let R be a mixed characteristic dvr, and x ∈ S Kp(GSp, S±)(R[1/p]).

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V p(A) = (lim← (n,p)=1

A[n])⊗Q




f .


Let K be an algebraic closure of K = R[1/p]. The existence of εx implies

Gal(K/K) acts trivially on V p(Ax), and in particular, so does the inertiasubgroup.

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V p(A) = (lim← (n,p)=1

A[n])⊗Q




f .




This implies Ax has good reduction (Neron-Ogg-Shafarevich), so x extendsto an R-point.

15



V p(A) = (lim← (n,p)=1

A[n])⊗Q




f .




This implies Ax has good reduction (Neron-Ogg-Shafarevich), so x extendsto an R-point.

One can show the same works for any R regular, formally smooth overOEλ.

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Theorem. If p > 2, Kp hyperspecial and (G,X) is of abelian type, thenShKp(G,X) admits a smooth integral model

S Kp(G,X) = lim← Kp

S KpKp(G,X)

as in the conjecture.

16



S KpKp(G,X)


Many PEL cases due to Kottwitz.

16



S KpKp(G,X)



In the case of Hodge type, S Kp(G,X) is given by taking the normalizationof the closure of

ShKp(G,X) ↪→ ShK′p(GSp, S±) ↪→ S K′p(GSp, S±)

into a suitable moduli space of polarized abelian varieties.

16



S KpKp(G,X)






The idea for such a construction goes back to Milne, Faltings, Vasiu ... Thedifficulty is to show that the resulting scheme is smooth.

16



S KpKp(G,X)






The idea for such a construction goes back to Milne, Faltings, Vasiu ... Thedifficulty is to show that the resulting scheme is smooth.

This uses deformation theory of p-divisible groups, which can be viewed asp-adic analogues of Hodge structures of weight 1.

16



S KpKp(G,X)


17



S KpKp(G,X)


Theorem. (Madapusi): Under the same assumptions, S Kp(G,X) hasgood toroidal and minimal compactifications.

17



S KpKp(G,X)



In particular, ShKp(G,X)/E proper =⇒ S Kp(G,X)/OEλ proper.

17



S KpKp(G,X)




When (G,X) is of PEL type this is due to Kai-Wen Lan

17



S KpKp(G,X)




When (G,X) is of PEL type this is due to Kai-Wen Lan

Theorem. (MK-Pappas) Good (not smooth in general) models existwhen Kp is parahoric, GQp splits over a tamely ramified extension ofQp, and p > 2.

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Mod p points (Talk III): Suppose (G,X) is of Hodge type. We can alwaysmake a model of ShK(G,X) by taking the closure in a moduli space forabelian varieties, as before.

18


This gives rise to a notion of when two points in S Kp(G,X)(Fp) are isoge-nous.

18



A point of (h, g) ∈ ShKp(G,X)(C) is called special if h : C× → G(R)factors through T (R), where T ⊂ G is a subtorus, defined over Q. Thesecorrespond to CM abelian varieties.

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Theorem. (MK, Madapusi, Shin) Suppose GQp is quasi-split. Thenany x ∈ S K(G,X)(Fp) is isogenous to the reduction of a special point.

(some PEL cases: Kottwitz, Zink).

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Theorem. (MK, Madapusi, Shin) Suppose GQp is quasi-split. Thenany x ∈ S K(G,X)(Fp) is isogenous to the reduction of a special point.

(some PEL cases: Kottwitz, Zink).

One has the following result, conjectured by Langlands-Rapoport in greatergenerality.

Theorem. Suppose p > 2, Kp is hyperspecial, and (G,X) of abeliantype. There exists a bijection

S Kp(G,X)(Fp)∼−→

∐ϕ

Iϕ(Q)\Xp(ϕ)×Xp(ϕ)

compatible with the action G(Apf) and Frobenius,

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∐ϕ



The expression on the right is purely group theoretic, in the sense that itdoesn’t involve any algebraic geometry.

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∐ϕ




Xp(ϕ)←→ p-power isogenies

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∐ϕ





Xp(ϕ)∼−→ G(Ap

f)←→ prime to p-isogenies

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∐ϕ





Xp(ϕ)∼−→ G(Ap

f)←→ prime to p-isogenies

Iϕ(Q)←→ automorphisms of the AV up to isogeny +extra structure

and Iϕ is a compact (mod center) form of the centralizer Gγ0.

where γ0 ∈ G(Q) corresponds to the conjugacy class of Frobenius.

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mark kisin harvard · 2015. 7. 29. · mark kisin harvard 1. the beginning. 2. the beginning. let...

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