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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.
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).
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.
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).
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.
For h ∈ X, becomes R2 a one dimensional complex vector space.
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).
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.
For h ∈ X, becomes R2 a one dimensional complex vector space.
Eh = R2/Z2 is an elliptic curve.
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).
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.
For h ∈ X, becomes R2 a one dimensional complex vector space.
Eh = R2/Z2 is an elliptic curve.
For any elliptic curve E over C let
T (E) = lim← n
E[n]
and V (E) = T (E)⊗Q.
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).
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.
For h ∈ X, becomes R2 a one dimensional complex vector space.
Eh = R2/Z2 is an elliptic 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
Moduli space of polarized abelian varieties.
Let V be Q-vector space equipped with a perfect alternating pairing ψ.
3
Moduli space of polarized abelian varieties.
Let V be Q-vector space equipped with a perfect alternating pairing ψ.
Take G = GSp(V, ψ) and let X = S± be the Siegel double space:
S± = {h : C× → GSp(VR, ψ)} such that
3
Moduli space of polarized abelian varieties.
Let V be Q-vector space equipped with a perfect alternating pairing ψ.
Take G = GSp(V, ψ) and let X = S± be the Siegel double space:
S± = {h : C× → GSp(VR, ψ)} such that
1. VC = V −1,0 ⊕ V 0,−1 z · v = z−pz−q v ∈ V p,q
3
Moduli space of polarized abelian varieties.
Let V be Q-vector space equipped with a perfect alternating pairing ψ.
Take G = GSp(V, ψ) and let X = S± be the Siegel double space:
S± = {h : C× → GSp(VR, ψ)} such that
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
Moduli space of polarized abelian varieties.
Let V be Q-vector space equipped with a perfect alternating pairing ψ.
Take G = GSp(V, ψ) and let X = S± be the Siegel double space:
S± = {h : C× → GSp(VR, ψ)} such that
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
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
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).
4
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
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).
2. This implies h(−1) is central. So h(i) acts by an involution.
4
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
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).
2. This implies h(−1) is central. So h(i) acts by an involution.
We require it be a Cartan involution: Twisting the real structure on Gby h(i) gives the compact form of GR.
4
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
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).
2. This implies h(−1) is central. So h(i) acts by an involution.
We require it be a Cartan involution: Twisting the real structure on Gby h(i) gives the compact form of GR.
3. h is non-trivial in each factor of GadQ .
4
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
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).
2. This implies h(−1) is central. So h(i) acts by an involution.
We require it be a Cartan involution: Twisting the real structure on Gby h(i) gives the compact form of GR.
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).
We assume that (G,X) is a Shimura datum .
Let K ⊂ G(Af) a compact open subgroup.
5
h : C× → G(R).
We assume that (G,X) is a Shimura datum .
Let K ⊂ G(Af) a compact open subgroup.
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).
We assume that (G,X) is a Shimura datum .
Let K ⊂ G(Af) a compact open subgroup.
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.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).
We assume that (G,X) is a Shimura datum .
Let K ⊂ G(Af) a compact open subgroup.
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.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, ...).
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).
We assume that (G,X) is a Shimura datum .
Let K ⊂ G(Af) a compact open subgroup.
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.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, ...).
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
Abelian varieties with extra structure:
Heuristically Shimura varieties can be regarded as moduli spaces of “abelianmotives”. In particular, they carry variations of Hodge structure:
6
Abelian varieties with extra structure:
Heuristically Shimura varieties can be regarded as moduli spaces of “abelianmotives”. In particular, they carry variations of Hodge structure:
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
Abelian varieties with extra structure:
Heuristically Shimura varieties can be regarded as moduli spaces of “abelianmotives”. In particular, they carry variations of Hodge structure:
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)
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
Abelian varieties with extra structure:
Heuristically Shimura varieties can be regarded as moduli spaces of “abelianmotives”. In particular, they carry variations of Hodge structure:
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)
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.
6
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
Let V ⊗ = ⊕n,mV ⊗n ⊗ V ∗⊗m. Then G ⊂ GSp(V, ψ) is the stabilizer of acollection of tensors {sα} ⊂ V ⊗.
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
Let V ⊗ = ⊕n,mV ⊗n ⊗ V ∗⊗m. Then G ⊂ GSp(V, ψ) is the stabilizer of acollection of tensors {sα} ⊂ V ⊗.
The sα are fixed by G hence by h(C×) for h ∈ X.
sα ∈ V ⊗ ∩ (V ⊗C )0,0
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
Let V ⊗ = ⊕n,mV ⊗n ⊗ V ∗⊗m. Then G ⊂ GSp(V, ψ) is the stabilizer of acollection of tensors {sα} ⊂ V ⊗.
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
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
Let V ⊗ = ⊕n,mV ⊗n ⊗ V ∗⊗m. Then G ⊂ GSp(V, ψ) is the stabilizer of acollection of tensors {sα} ⊂ V ⊗.
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.
Such (G,X) are called of Hodge type.
7
V = G(Q)\V ×X ×G(Af)/K→ ShK(G,X)
A morphism of Shimura data (G,X)→ (G′, X ′) is a map G→ G′ whichinduces X → X ′.
Such a morphism induces
ShK(G,X)→ ShK′(G′, X ′)
for suitable K, K′.
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:
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.
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
Example: Let W be a Q-vector space with a quadratic form such that
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
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
9
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)
with dimV = 2n+1.
9
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)
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
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)
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.
(SO(W ), X) is an example of a Shimura datum of abelian type - these are“quotients” of ones of Hodge type;
9
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)
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.
(SO(W ), X) is an example of a Shimura datum of abelian type - these are“quotients” of ones of Hodge type;
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
Example: Let W be a Q-vector space with a quadratic form such that
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.
One has an extension
1→ Gm → CSpin(W )→ SO(W )→ 1
and(SO(W ), X)← (CSpin(W ), X ′) ↪→ (GSp(V ), S±)
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;
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.
10
Arithmetic properties: We’re interested in ShK(G,X)(Fp)
11
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).
11
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).
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.
11
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).
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.
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.)
11
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).
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.
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.)
4. Testing ideas about motives over Fp.
11
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).
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.
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.)
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
Integral models: (Talk II)
Suppose now G has a reductive model GZp over Zp.
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
Integral models: (Talk II)
Suppose now G has a reductive model GZp over Zp.
Let K = KpKp, where Kp = GZp(Zp) (Kp hyperspecial) and Kp ⊂ G(Ap
f) is
compact open Apf - finite adeles with trivial p-component.
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
Integral models: (Talk II)
Suppose now G has a reductive model GZp over Zp.
Let K = KpKp, where Kp = GZp(Zp) (Kp hyperspecial) and Kp ⊂ G(Ap
f) is
compact open Apf - finite adeles with trivial p-component.
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(Apf)-action on
ShKp(G,X) = lim← Kp
ShKpKp(G,X)
extends toS Kp(G,X) = lim
← KpS KpKp(G,X),
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Integral models: (Talk II)
Suppose now G has a reductive model GZp over Zp.
Let K = KpKp, where Kp = GZp(Zp) (Kp hyperspecial) and Kp ⊂ G(Ap
f) is
compact open Apf - finite adeles with trivial p-component.
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(Apf)-action on
ShKp(G,X) = lim← Kp
ShKpKp(G,X)
extends toS Kp(G,X) = lim
← KpS KpKp(G,X),
and S Kp(G,X) satisfies the extension property.
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Integral models: (Talk II)
Suppose now G has a reductive model GZp over Zp.
Let K = KpKp, where Kp = GZp(Zp) (Kp hyperspecial) and Kp ⊂ G(Ap
f) is
compact open Apf - finite adeles with trivial p-component.
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(Apf)-action on
ShKp(G,X) = lim← Kp
ShKpKp(G,X)
extends toS Kp(G,X) = lim
← KpS KpKp(G,X),
and S Kp(G,X) satisfies the extension property.
The extension property: If R is regular, formally smooth over OEλ then
S Kp(G,X)(R)∼−→ S Kp(G,X)(R[1/p]).
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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
ShKp(G,X) = lim← Kp
ShKpKp(G,X)
extends toS Kp(G,X) = lim
← KpS KpKp(G,X),
and S Kp(G,X) satisfies the extension property.
The extension property: If R is regular, formally smooth over OEλ then
S Kp(G,X)(R)∼−→ S Kp(G,X)(R[1/p]).
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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
ShKp(G,X) = lim← Kp
ShKpKp(G,X)
extends toS Kp(G,X) = lim
← KpS KpKp(G,X),
and S Kp(G,X) satisfies the extension property.
The extension property: If R is regular, formally smooth over OEλ then
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
Example: Take (G,X) = (GSp, S±), defined by (V, ψ) as above.
Let VZ ⊂ V be a Z-lattice, and Kp the stabilizer of
VZp = VZ ⊗Z Zp ⊂ V ⊗Qp.
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Example: Take (G,X) = (GSp, S±), defined by (V, ψ) as above.
Let VZ ⊂ V be a Z-lattice, and Kp the stabilizer of
VZp = VZ ⊗Z Zp ⊂ V ⊗Qp.
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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Example: Take (G,X) = (GSp, S±), defined by (V, ψ) as above.
Let VZ ⊂ V be a Z-lattice, and Kp the stabilizer of
VZp = VZ ⊗Z Zp ⊂ V ⊗Qp.
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).
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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Example: Take (G,X) = (GSp, S±), defined by (V, ψ) as above.
Let VZ ⊂ V be a Z-lattice, and Kp the stabilizer of
VZp = VZ ⊗Z Zp ⊂ V ⊗Qp.
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).
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.
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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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 .
Let R be a mixed characteristic dvr, and x ∈ S Kp(GSp, S±)(R[1/p]).
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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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 .
Let R be a mixed characteristic dvr, and x ∈ S Kp(GSp, S±)(R[1/p]).
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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The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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 .
Let R be a mixed characteristic dvr, and x ∈ S Kp(GSp, S±)(R[1/p]).
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.
This implies Ax has good reduction (Neron-Ogg-Shafarevich), so x extendsto an R-point.
15
The extension property for S Kp(GSp, S±), (Kp hyperspecial) is motivatedas follows:
If R is an OEλ-algebra, and A an abelian scheme over R, set
V p(A) = (lim← (n,p)=1
A[n])⊗Q
This is a (pro)-etale group scheme.
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 .
Let R be a mixed characteristic dvr, and x ∈ S Kp(GSp, S±)(R[1/p]).
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.
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.
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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.
Many PEL cases due to Kottwitz.
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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.
Many PEL cases due to Kottwitz.
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.
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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.
Many PEL cases due to Kottwitz.
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.
The idea for such a construction goes back to Milne, Faltings, Vasiu ... Thedifficulty is to show that the resulting scheme is smooth.
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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.
Many PEL cases due to Kottwitz.
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.
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.
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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.
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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.
Theorem. (Madapusi): Under the same assumptions, S Kp(G,X) hasgood toroidal and minimal compactifications.
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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.
Theorem. (Madapusi): Under the same assumptions, S Kp(G,X) hasgood toroidal and minimal compactifications.
In particular, ShKp(G,X)/E proper =⇒ S Kp(G,X)/OEλ proper.
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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.
Theorem. (Madapusi): Under the same assumptions, S Kp(G,X) hasgood toroidal and minimal compactifications.
In particular, ShKp(G,X)/E proper =⇒ S Kp(G,X)/OEλ proper.
When (G,X) is of PEL type this is due to Kai-Wen Lan
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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.
Theorem. (Madapusi): Under the same assumptions, S Kp(G,X) hasgood toroidal and minimal compactifications.
In particular, ShKp(G,X)/E proper =⇒ S Kp(G,X)/OEλ proper.
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
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.
This gives rise to a notion of when two points in S Kp(G,X)(Fp) are isoge-nous.
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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.
This gives rise to a notion of when two points in S Kp(G,X)(Fp) are isoge-nous.
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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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.
This gives rise to a notion of when two points in S Kp(G,X)(Fp) are isoge-nous.
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.
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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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.
This gives rise to a notion of when two points in S Kp(G,X)(Fp) are isoge-nous.
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.
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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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,
The expression on the right is purely group theoretic, in the sense that itdoesn’t involve any algebraic geometry.
19
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,
The expression on the right is purely group theoretic, in the sense that itdoesn’t involve any algebraic geometry.
Xp(ϕ)←→ p-power isogenies
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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,
The expression on the right is purely group theoretic, in the sense that itdoesn’t involve any algebraic geometry.
Xp(ϕ)←→ p-power isogenies
Xp(ϕ)∼−→ G(Ap
f)←→ prime to p-isogenies
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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,
The expression on the right is purely group theoretic, in the sense that itdoesn’t involve any algebraic geometry.
Xp(ϕ)←→ p-power isogenies
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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