Jeudi 27 novembre 2014

p
BQB
4Qa b ∈Q
B=Qhi, ji, i2=a, j2=b, ij =−ji .
B B ⊗QR∼
=M2(R)
B M2(Q)
B D > 1D=Qpp p
B⊗QQp∼
=M2(Qp)
OBO4B
ZO
B⊗QR∼
=M2(R)
ψ:B →M2(R).
M2(R)C2
a b
c d  w1 w2=aw1+bw2 cw2+dw2
u=w1
w2∈C2w1/w2/∈RuO
Λu:= ψ(O)u ,

C2u
Au:= C2/Λu.
µ∈B µ2=−1/D
E: Λu×Λu−→ Z,
(ψ(α)u, ψ(β)u)7−→ trd(µα ¯ β)
α=x+iy +jz +ijw ¯α:= x−iy −jz −ijw trd(α)=2x ER
REC2
E(√−1·,√−1·) = E(·,·)
ERΛu
ER
⇐⇒ E(√−1·,·)>0Au
ψ µ (Au, Eu, ψ) (Av, Ev, ψ)
u=w1
w27→ w1/w2 1u=τ 1
O(1) O(Aτ, Eτ, ψ) (Aτ0, Eτ0, ψ)
ψ()τ=τ0∈ O(1) a b
c d ·τ=aτ +b
cτ +d
X=ψ(O(1))\H
Aτ
A
AEnd0(A) = End(A)⊗Q
µ−1∈ O

End(A)⊗Q
Q
K
B Q
A
M2(K)K
Q×Q
M2(Q)
B
M2(Q)
ΛτO
B B
ΛτB
N
B
F B
F
σ:F →RB⊗F,σ R
HM2(R)
End0(A)
B A
E1×E2
K Q End0(A)
M2(K) End0(A)B
A

A
End0(A)K
A E1×E2
K Q
A
A E1×E2
Q
X ψ :B →M2(R)µ B
Eτ
XA2,1
ψ˜
E X A2,1M2
CC
((Jac(C)),Θ) E˜ EM2
A2,1P3
CM2
A2,1M2
?_
(Jac(C),Θ)
oo
X[4:1] [2:1]
//˜
E
?
OO
E
?_ oo
?
OO
X
M2(K)
A2,1
X
X
X(Aτ, Eτ, ψ)
Eτψ X

H