Méthodes explicites pour les groupes arithmétiques

Q
Z Zn
SL2(Z)
G⊂GLnQ
G(Z) = G(Q)∩GLn(Z)G⊂GLnΓ⊂G(Q)
G(Z) Γ ∩G(Z) Γ
G(Z)
K⊂G(R)
X=G(R)/K G(R)
Γ⊂G(Q)X C ⊂X
{γ∈Γ|γC ∩C6=∅}XG
F F
i, j
i2=a, j2=b ij =−ji a, b ∈F×
a,b
Fw=x+yi +zj +tij
B
w=x−yi −zj −tij
trd(w) = w+w= 2x
nrd(w) = ww =ww =x2−ay2−bz2+abt2

w X2−trd(w)X+ nrd(w)
B×B×
1
M2(F)
1,1
Fi=−1 0
0−1j=0−1 1 0 
R H =−1,−1
R
M2(R)H R
B F
Q Q AG(A) = (B⊗QA)×
1
G(Q) = B×
1
G(Q)
GLnF
ZF
B F
ZFO ⊂ B F O=B
O×
1={w∈ O | nrd(w)=1}⊂O×G(Q)
G(Q)O×
1
X
G(R)=(B⊗QR)×
1
F⊗QR=Rr×Cc
r+ 2c F
B⊗QR∼
=M2(R)d×Hr−d× M2(C)c
σ F B ⊗F,σ R
2×2
G(R)∼
=SL2(R)d×H×
1r−d×SL2(C)c
H H×
1
SO2(R) SL2(R) SU2(C)
SL2(C)G(R)
K= SO2(R)d×H×
1r−d×SU2(C)c
X=G(R)/K ∼
=Hd
2× Hc 3
SL2(R)/SO2(R)∼
=H2SL2(C)/SU2(C)∼ =H3H2H3
2 3
−1 2 3
X=H2X=H3
Γ\X
3
Γ
H∗(Γ,Z) = H∗(Γ\X, Z)

δ∈G(Q)Tδ
H∗(Γ, M)Tδ −−−−→ H∗(Γ, M)
res
yx
cores
H∗(Γ ∩δΓδ−1, M)−−−−→
˜
δ
H∗(δ−1Γδ∩Γ, M)
res cores ˜ δ
δ
G(Q)
G(Q)
G(Q)
GL2
GL2
X
Γ\X
XΓ
XΓ(C)∼
=Γ\X
G(Q)
SL2(Z)
X
X
SL2(ZF)F

GL2
GL2
O
O×
1
X
X
X

SL(2,C)