SHIMURA VARIETIES
BEN P. GREEN
Disclaimer. These are rough notes from a talk I gave on Shimura varieties. None of the
work is my own and their may be some mistakes; use at your own risk!
1. The Definition
Let G/Q be a reductive algebraic group and let S = ResC/R Gm . Recall that the adjoint
group Gad of G is the linear group which is the image of G under the adjoint representation.
Note that S(R) = C× and S(C) = C× × C× . A Shimura datum is a pair (G, X) where
X is a G(R)-conjugacy class of homomorphisms
h : S → GR
satisfying the following conditions1
(1) For all h ∈ X, gC decomposes as a direct sum of three components where the action
of S given by h on each component is by the characters z/z̄, 1, z̄/z respectively.
(2) Ad(h(i)) induces a Cartan involution on Gad
R .
ad
(3) GR does not admit a factor H/Q s.t. the projection of h on H is trivial.
These conditions ensure that X is a finite union of bounded symmetric domains. We define
b the double coset space
for any sufficiently small open compact subgroup K of G(Q)
b
ShK (G, X) = G(Q)\X × G(Q)/K
The varieties ShK (G, X) are complex algebraic varieties and they form an inverse system
over all sufficiently small compact open subgroups K. We call the inverse limit Sh(G, X) a
b We have the following useful
Shimura Variety, it admits a natural right action of G(Q).
fact.
Fact 1. Let X + and Gad (R)+ be connected components, let G(R)+ be the preimage of
Gad (R)+ under the adjoint map, and let G(Q)+ = G(R)+ ∩ G(Q). Then
b
(1) The set G(Q)+ \G(Q)/K
is finite for any open compact subgroup K.
Date: 21/5/2015.
1Note the exact conditions will not really be of importance to us as we are interested only in a specific
datum.
1
2
BEN P. GREEN
b
(2) Let g1 , . . . , gk be representatives for G(Q)+ \G(Q)/K,
and let Γi = G(Q)+ ∩gi Kgi−1 .
Then
k
G
ShK (G, X) =
Γi \X +
i=1
As we don’t want to delve too much into the motivation behind the definition it will be
illustrative to give an example. Let G = GL2 and let
a b
h0 (a + ib) =
−b a
Then (G, X) is a Shimura datum where X is the GL2 (R)-conjugacy class of h0 . Note that
there is a bijection
X → C\R
gh0 g −1 7→ g · i
We therefore have two connected components for X, the upper and lower half planes. This
indicates why modular curves are examples
of Shimura varieties. Note G(Q)+ = GL+
2 (Q)
Q
+
and we can take X = H. If K = p Kp with Kp ⊆ GL2 (Zp ) and Kp = GL2 (Zp ) for all
but finitely many p then
b = GL+ (Q) · K
GL2 (Q)
2
Consequently fact 1 says that
ShK (G, X) = Γ\H
where
Γ = K ∩ GL+
2 (Q) ⊆ SL2 (Z)
For example if Kp = SL2 (Zp ) ∀p then Γ = SL2 (Z) and ShK (G, X) = Y (1). Other level
structures can be defined similarly.
To (briefly) generalise the above, let B be a quaternion algebra over a totally real field F0 .
If G0 is the group whose F0 points are given by B × let G = ResF0 /Q (G0 ), so G(Q) = B × .
One has that
G(R) = |H × ·{z
· · × H} × GL2 (R) × · · · × GL2 (R)
|
{z
}
h
s
where h + s = [F0 : Q]. We define
a b
a b
h0 (a + ib) = (1, . . . , 1,
,...,
)
−b a
| {z } −b a
|
{z
}
h
s
If we define X to be the G(R)-conjugacy class of h0 , (G, X) is a Shimura datum provided
that s > 0. In particular we could take F0 = Q and B to be an indefinite quaternion
algebra.
It will be of later importance to us to define the reflex field E of a Shimura variety; Shimura
SHIMURA VARIETIES
3
varieties always have a canonical model defined over their reflex field. Given an h ∈ X we
define a cocharacter over C by
c
h
− S ⊗R C −
→ GC
µh : Gm → Gm × Gm →
Here the first map is just z 7→ (z, 1), and the map c is the isomorphism whose inverse is
induced by the map
R ⊗R C → R × R, r ⊗ z →
7 (rz, rz)
for any C-algebra R. The conjugacy class C of µh is independent of h. Now let T be a
maximal torus of G and let
W = NG (T )/CG (T )
be its absolute Weyl group. Then the maps
{W -conjugacy classes in Hom(Gm , TQ )} → {G(Q)-conjugacy classes in Hom(Gm , GQ )}
→ {G(C)-conjugacy classes in Hom(Gm , GC )}
are all isomorphisms so we can regard C as an element of the first set. We define the reflex
field E of (G, X) as the field of definition of C, i.e. the fixed field inside Q of the subgroup
of Gal(Q/Q) fixing C. Note if F splits T then E ⊆ F , so E is a number field. In particular
if we take the datum we had previously for GL2 we see that E = Q.
2. A Particular Shimura Variety
Having given a definition of a Shimura variety we are now ready to give the main
definition of section 8 of Scholze. We fix the following;
•
•
•
•
F0 - a totally real field of even degree over Q
τ - an infinite place of F0
x0 - a finite place of F0
K - an imaginary quadratic field s.t. the rational prime below x0 split in K
We get a CM field F = KF0 and two places x, xc of F over F0 (the two places above x0 .
Fix also an embedding K ,→ C.
Fact 2. Given the above data and n ∈ N there exists a central division algebra D/F
of dimension n2 , an involution ∗ on D of the second kind (i.e. not F -linear), and a
homomorphism
h0 : C → D R
s.t. x 7→ h0 (i)−1 x∗ h0 (i) is a positive involution2 and such that the following properties
hold;
(1) D splits at all places of F different from x, xc .
0
2This means if the involution is x 7→ x0 that Tr
DR /R (x · x) > 0 ∀x ∈ DR .
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BEN P. GREEN
(2) If G0 /F0 is the algebraic group defined by
G0 (R) = {g ∈ (D ⊗F0 R)× : gg ∗ = 1}
then G0 is quasi-split at all finite places of F0 that do not split in F , is a unitary
group of signature (1, n − 1) at τ , and is a unitary group of signature (0, n) at all
other infinite places.
Note that any infinite place τ 0 of F0 induces an isomorphism D ⊗F0 R ∼
= Mn (C), and
under this identification
Y
h0 (z) = (diag(z, . . . , z, z̄, . . . , z̄)) 0 ∈ DR ∼
Mn (C)
=
τ
τ0
We say that G0 has signature (p, q) at τ 0 if it has p z’s and q z̄ 0 s. One then sees that G0
is isomorphic to a unitary group U (p, q) at τ 0 .
We can define a reductive group G/Q by
G(R) = {g ∈ (D ⊗Q R)× : gg ∗ ∈ R× }
Restricting h0 to C× and considering it as a morphism of algebraic groups over R gives a
map
h : S → GR
If X is the G(R)-conjugacy class of h, (G, X) is a Shimura datum. We denote the Shimura
variety of level K by ShK . We want to compute the reflex field of the ShK . Over C we
have the following decomposition


Y
GC = 
GLn  × Gm
F0 ,→C
Q
where the projections to GLn are given by the projection of D ⊗Q C = F ,→C Mn (C) to
the factor corresponding to the embedding F0 ,→ C and the fixed embedding K ,→ C. The
projection to Gm is the map g 7→ gg ∗ . Recall we have the map
Gm → Gm × Gm ∼
= S ⊗R C
given before. Precomposing hC with this morphism gives
µ : Gm → GC
Note that h(z) = hC (z, z̄). We see therefore that our assumptions imply it is given by
z 7→ diag(z, 1, . . . , 1) on the τ factor, by z 7→ 1 on the GLn factor, and z 7→ z on the
Gm -factor. Consequently the reflect field E is canonically isomorphic to F ; the infinite
place τ of F0 and the embedding K ,→ C give a canonical embedding of F into C which
identifies F with E.