Groupes et Géométrie

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(E, V )
ERVΘE:E×E→V: (x, y)7→ xy
ΘE
x:{x} × E→V x ∈E
xy +yz =xz x, y, z ∈E.
(F, W ) (E, V )F E W
VΘF= ΘE|W×W(F1, W1) (F2, W2) (E, V )
W1W2V
Rn
Rnx+V V
Rnx∈RnR R2R3
Rn
A, B, C E
(x, y, z)∈R3x+y+z= 1 X E
XX =x XA +y XB +z XC X =xA +yB +zC (x, y, z)
X(A, B, C)E
D A B C y
z
xA +yB +zC D
y
z= 1 1 3A+1 3B+1 3C E
xA+yB +zC E z ≥0
A, B, C
fa,b,c,d :P1(R)→P1(R)P1(R) = R∪ {∞}
x7→ ax +b
cx +da b
c d∈Gl2(R)
∞
[x1, x2;x3, x4] = (x1−x3)(x2−x4)
(x2−x3)(x1−x4)x1, x2, x3∈R.
fa,b,c,d =fa0,b0,c0,d0(a0, b0, c0, d0) = λ(a, b, c, d)λ∈R∗
[x1, x2;x3, x4] = f(x4)f(x1, x2, x3) (∞,0,1)
[x1, x2;x3, x4]=[g(x1), g(x2); g(x3), g(x4)] g
Di, i = 1,2,3,4
O Dx, Dyxi=Di∩Dxyi=Di∩Dy
O xiyii= 1,2,3,4
[x1, x2;x3, x4]=[y1, y2;y3, y4].
z16=z4arg([z1, z2;z3, z4]) ≡α−βmod 2π α z3
z1−z3z2−z3β z4z1−z4z2−z4
0π2π
[z1, z2;z3, z4]z1, z2, z3, z4
z1, z2, z3, z4
P Gl2(Fp)P1(Fp)
P Gl2(Fp)→Sp+1.
p= 2 p= 3