Sous-groupes finis de GLn(Q)

GLn(Q)
GLn(Q)
GL2(Q)
Q

GLn(Q)
GL2(Q)
p

[x]
x v`(n) n A v`(A)
 A G 6H H
G H ∧
n 
M(n, ) = r+v`(r!) =
∞
X
i=0 n
i(−1)
r= [ n
`−1]M(n) = Q`M(n,`)
GGLn(Q)|G|M(n)
GLn(Q)
M(n,`)
C R
Q
G  
G v`(G)
G6GLn(Q)
GLn(Fp) On(Fp)p
GLn(Q)M(n,`)
GLn(Q)
GLn(Fp)
Qn
RZ Qn
RQ Qn
R n
Qn

Z
=⇒ F ZR
R=Pe∈F ZeQn= VectQPe∈F Ze=Pe∈F Qe
FQnQB= (ei)i∈[[1,n]]
xFx=Pqieiqi
d qi
n
M
i=1
Zei⊂R⊂1
d
n
M
i=1
Zei
R
n
=⇒ B ZR
Q
BQ Qn
Z
Q
GGLn(Q)
GGLn(Z)
GQn
R=Pg∈GgZn
GZR G
qQn
G6O(q) O(q)
GLn(Q)q
q(x) = Pn
i=1 x2
ix∈Qn
QnPg∈Gq◦g
Fp
m>3
ϕ: GLn(Z)−→ GLn(Z/mZ)
Ker ϕ
g∈Ker ϕ g 6= 1 q
gq= 1 g
q

h∈ Mn(Z)
d > 0g= 1 + mdh gq= 1
q
X
k=1 q
k(md)khk= 0
m qh
qh +q
2mdh2+· · ·
|{z} m
= 0
h m q
q=m q m >3q
2
h+q
2dh2+· · ·
|{z} q
= 0
q h
GGLn(Z)G∩Ker ϕ={1}
m ϕG
 > 2p(Z/2Z)
×
v`(GLn(Fpr)) = (1 + v`(r)) n
τ+v`n τ!
τ=`−1
r∧(`−1)
p
r= 1 M(n, )
i > 0
v`(pi−1) = (1 + v`(i)−1i
0
p 2 pi≡1 (mod )
p`i ≡1 (mod 2)ϕ(2) = (−1) |i  −1|i
pi≡1 (mod )−1|i
i m(−1)km 
k s =pm(`−1)
k= 0 |pi−12-pi−1(−1) -m(−1) k > 0
s`k−1−1 = uk-u
s`k=1 + uk`= 1 + uk+1 +
`
X
j=2 
jujjk
| {z }
`k+2 ` > 2