Courbes elliptiques et courbes modulaires non commutatives

R

P20
y2=x3+x2−x
y2=x3+x2−tx t ∈R
t= 0 0
r: Λ →V
VC1 Λ Z2
Λ0r0
//
φ
V0
ψ
Λr//V
φ ψ C
R
r:Z2→Cr(1,0) = 1
r(0,1) = τ∈H={z∈C|Im(z)>0}
[Z+τZ]
Λtau0→Λtau
Z2
g=a c
b d ∈M2(Z)

τ τ0
τ0=c+dτ
a+bτ .
g∈GL2(Z)
Y=H/GL2(Z) = H/SL2(Z)
SL2(Z)
Λ=[Z+τZ]T=C/Λ
P2
C
P(z) = 1
z2+X
ω∈Λ,ω6=0
(1
(z−ω)2−1 ω2)
CΛ
P0(z)2= 4P3(z)−g2(Λ)P(z)−g3(Λ)
g2g3Λ
C/Λ→P2
z6= 0 7→ (P(z) : P0(z) : 0)
07→ (0 : 1 : 0) = ∞
y2= 4x3−g2x−g3,
C/ΛE/CE3 ∞ =
0∈C/Λ
EC
P2
C
E:R→EC
r:Z2→C7→ C/r(Z2)⊂P2.
P:EC→R
E7→ {ΛE→Lie(E)}
Lie(E) = T0(E) ΛE=Ker(exp : Lie(E)→E(C))
EC∼
=R.
C
Y=H/SL2(Z).

E
Y
H/SL2(Z)∼
=C
y2=x3+x2−x0
H/SL2(Z) = C→P1
C
τ→ ∞
C[Z+τZ]
βH
r:Z2→C
R⊂C
τ β
C/[Z+τZ]
C/r(Z2)
r(Z2)
C∗AC
σ σ :C→C
(xy)∗=y∗x∗kx∗xk=kxk2x, y ∈A.
ρ:A→C
ρ(1) = 1
X C(X)
X C∗
kfk=supx∈Xkf(x)k