Groupe des nombres complexes de module 1. Sous - IMJ-PRG

U
(U,×)C∗
zexp z=P∞
n=0 zn
n!
(C,+) (C∗,×)
t∈Rcos sin
eit
π
tcos t= 0 eiπ + 1 = 0
(C,+) (C∗,×)
(R,+) (U,×)
2πZUR/2πZ
z
z=reiθ r∈R∗
+θ∈R/2πZr z θ
∀t∈R, n ∈N,(cos t+isin t)n=
cos(nt) + isin(nt)
∀t∈R,cos t=eit+e−it
2,sin t=eit−e−it 2i
sinntcosnt
PN
n=0 sin(nx)

U
U U Un={z∈ U |
zn= 1}n∈N
U
p p Op=Sα∈NUpα
OpUpαOp
USn∈NUn
Q/Z
n
Un
Unn
Z/nZ C Un
n
Unµ∗
nn
µ∗
3={j, j2}, µ∗ 4={i, −i}
Sl(E)Un
µ∗
n={e2ikπ/n |1≤k≤n, k ∧n= 1} Un=td|nµ∗
d
ϕ(n)µ∗
nξ∈µ∗ nµ∗ n
ξk1≤k≤n k ∧n= 1
n=Pd|nϕ(d)
nΦn=Qξ∈µ∗
n(X−ξ)
Φ1=X−1 Φ2=X+ 1 Φ3=X2+X+ 1
Φnϕ(n)
Xn−1 = Qd|nΦd
Φn∈Z[X]
ΦnQ

ξ∈µ∗
nQΦn
[Q(ξ) : Q] = ϕ(n)
Fq
E
E
(u, v)R(u0, v0)r r(u) = u0r(v) = v0
(u, v)
u v A
ϕ:A → SO(E),(u, v)7→ r r
u v ϕ
A(u, v)+(u0, v0) = ϕ−1(ϕ(u, v)◦ϕ(u0, v0))
ϕ
(u, v)+(v, w)=(u, w)
E
E
E
(u, v)θ∈R/2πZ
ϕ(u, v)cos θ−sin θ
sin θcos θθ
(u, v)
SO(E)≈R/2πZSO2(R)≈ U

G
G G C∗
G
Cb G
G=SnG
G=Z/nZk7→ e2ikπ/n G
χ:G→CG/D(G)→
CG
|G|=nb G
Un
\
Z/nZ→ Un, χ 7→ χ1
H
\
G×H≡ˆ G׈ H
H G H
G
G
a1,...aka1|. . . |aka1>1G≈Z/a1Z×
· · · × Z/akZ
p
n p n
p
n
p=
0a≡0 [p]
1a p
−1a p
Fp(p+ 1)/2
∀a∈Z,a
p≡a(p−1)/2[p]
−1
N= (−1)(p−1)/22 N= (−1)(p2−1)/8

p, q
p
qq p= (−1)p−1
2
q−1
2
p p =x2−6y2
x, y ∈Zp y p x p2
p6≡(xy−1)2[p]6
p= 1 (−1)(p2−1)/8(−1)(p−1)/2p 3=
1p